EPR and Steering in Quantum Mechanics (QM)...

Version: 16 Februari, 2017.
Status: Almost done...

Sometimes I like to write some simple notes. I need to pick this one up, too, again,
since I created it a while ago, but never finished it (That's typically me).
But the subject is simply much too interesting, so I will give it a go again...
I will rewrite this one...
So, all the text below will be completely revisited soon.

Some ideas on Quantum Entanglement and non-locality were re-discovered the last 15 or 20 years (or so),
partly based on idea's of Schrodinger on "EPR steering", which were expressed in 1935, but with some
new flavours too.

There indeed exists a subtle difference between, what we describe as "entanglement", "Bell non-locality",
and "Steering".

I like to say something on those subjects, since it's absolutely facinating stuff.
So, in case you rather unfamiliar with such subjects, this note might be of interest.

The first five chapters will describe some well-know effects of "entanglement", that traditionally led
to the socalled EPR paradox. So, these first five chapters are a bit "old-skool", I think.

While after many recent efforts were, and are, spend on finding the "essence" of steering, entanglement,
and nonlocality, it now seems that the views that were deveoped in the years before the '90's,
probably needed quite some "revision", especially due to all the research since the 2000's.

However, the first five chapters will present the (pre 90's) "old-skool" idea's first,
since this way probably still remains the best practice to present such material.

Chapter 1 is about the famous EPR paradox, and that phenomenon is a important theme for
this entire text. In this starting chapter, entanglement, and possibly even "steering",
will be presented in a "strange case".

Then, chapter 2 will spend a few words on Quantum Teleportation (QT), which is a real effect.
Here too, entanglement and possibly "non-locality" are demonstrated.

Chapters 3 and 4 might be viewed as a nano introduction to QM, showing the most basic concepts
and notations used in QM. This might help if you like to read professional articles on the various subjects.

Chapter 5 will be a short intro to the "Bell inequalities", which in fact are stochastic criteria
to decide if measurement results are due to non-local effects of QM, are are due to local mechanisms,
like the "Hidden variables", which in fact represent a local realistic view on reality.

In chapter 6, I will try to decribe a few specifics of "EPR steering", but it will be of a very "lightweight"
nature, and can only be of interest if you are really unfamiliar with the subject.

Inevitably, a "problem" pertinent to any interpretation of QM must be addressed: In chapter 7, I will try to
say something useful on the "measurement" problem, and the role of the "observer".

In chapter 8, I like to touch upon some quite radical ideas on the interpretation of QM, namely
some new "parallel" theories like "MIW" (and ofcourse also the older Everett's MWI theory).

Chapter 10, is a very simple intro on some newer ideas, that SpaceTime itself can be "build" from entanglement,
as well as that entanglement might play a large role in new gravitational models.
Here, one central theme is the ER=EPR paradigm. I personally think that this is a promising path in physics.
Whatever you might think of it...., it certainly is quite spectecular.

Due to my own limitations, the whole text is quite elementary, and it almost exclusively uses
(as we say in Dutch) a sort of "Jip en Janneke" languange. Sorry for that !

So, you are warned! But if you want to try it anyway..., then let's see what this is all about.

Contents at a glance:

Chapter 1. Introduction to the EPR paradox.
Chapter 2. A few words on the original Quantum Teleportation article.
Chapter 3. A nano intro to QM: some important concepts and notations in QM.
Chapter 4. A nano intro to QM: pure states and mixed states.
Chapter 5. The inequalities of Bell, or Bell's theorem.
Chapter 6. A few words on Steering, Entanglement, and Bell non-locality.
Chapter 7. A few words on the measurement problem.
Chapter 8. Some "Many Worlds" interpretations in QM.
Chapter 9. A few words on entropy.
Chapter 10. Newer ER=EPR models and idea's.

Short Appendices: appliances

A1: Quantum Radar.

Chapter 1. Introduction to the EPR paradox.

1.1 Introduction and Some background information.

We know that it is not allowed for information, to "travel" faster than "c".
However, there exists certain situations in Quantum Mechanics (QM), where it appears that this rule is broken.
I immediately haste to say, that virtually all physicists believe that the rule still holds,
but that something else is at work.

What that something else precisely is, is not fully clear yet, although some well-funded idea's, do exist.

QM uses several flavours to mathematically describe entities (e.g. a particle), properties (e.g. position, spin),
and events. One such flavour which is often used is the Dirac (vector) notation.

In chapter 3, for folks who are not too aquinted with QM, I will introduce some highlights of QM,
and the "wavefunction", and something on the Dirac notation.
For now, however, even if you would have only a basic notion of "vectors", then we will come a long way in this chapter.

Suppose we have a certain observable (propery) of a particle. Suppose that this observable indeed can be measured
by some measuring device.
For the purpose of this sort of text, the "spin" of a particle is often used as the characteristic observable.
This spin, resembles an angular magnetic momentum, and can be either be "up" (or often written as "+" or "1"),
or "down" (or written as "-" or "0"), when measured along a certain direction (like for example the z-axis in R³.

Most physicists agree on the fact that the framework QM is intrinsically "probabilistic".

It means that if the spin of a particle is unmeasured, the spin is a linear combination of both
"up" and "down" at the same time.
Actually, it resembles a "vector" in 2 dimensional space (actually a 3D Bloch sphere), so in general,
such a state can be written as:

|φ> = a|1> + b|0> (equation 1)

(Note: arguments can be found to speak of a 3D Bloch sphere, but we don't mind about this, at this moment.)

where |1> and |0> represents the basis vectors of such superposition.

This is indeed remarkable by itself! But this is how it fits in the framework of QM. There is a certain probability of
of finding |φ>=|1> and a certain probability of finding |φ>=|0> when a measurement is done.

For those "probabilities", it must hold that |a|² + |b|² =1, since the total of the probabilities
must add up to one.

By the way, the system described in equation 1, is often called a "qubit", as the Quantum Mechanical "bit" in Quantum Computing.

Similarly, a "qutrit" can be written as a linear combination of |0> and |1> and |2>, which are three orthogonal
basis states. It all really looks like it is in vector calculus. The qutrit might thus be represented by:

|φ> = a|0> + b|1> + c|2> (equation 2)

However, in most discussions, the qubit as in equation 1, plays a central role.

Now, suppose we have two non-interacting systems of two qubits|φ₁> and |φ₂> (close together).
Then their combined state, or "product state" (that is: when they are not entangled), might be expressed by:

|Ψ> = |φ₁> ⊗ |φ₂> = a₀₀ |00> + a₀₁ |01> + a₁₀ |10> + a₁₁ |11> (equation 3)

Such a state is also called "seperable", because the combined state is a "product" of the individual states.
If you have such a product state, it's possible "to factor out" (or seperate) each individual system from the combined equation.

Note:
When we say of equation 1, |φ> = a|1> + b|0>, that it means that it is a linear combination of both
"up" and "down" at the same time, thus simultaneously, then this statement is a common interpretation in QM.
Most folks do not question this interpretation, however, some folks still do.

In some cases, such an equation as equation 3, does not work for a combined system of particles. In such case,
the particles are fully "intertwined", and in such a way, that a measurement on one, affects the state of the second one.
The latter statement is extremely remarkable, and is what people nowadays would call "steering", as a special subset
of the more general term 'entanglement'.

Suppose we start out with a quantum system with spin 0. Now suppose further that it decays in two particles.
Since the total spin was "0", it must be true that the sum of the spins of the new particles is zero too.
But, we cannot say that one must have spin "up", and the other must have spin "down". However, we can say that
their combination carries zero spin.
It may appear strange, but a good way to denote the former statement, is by the following equation:

|Ψ> = 1/√2 . ( |01> + |10> ) (equation 4)

Note that this is a superpostion of two states, namely |01> and |10>.
In QM talk, we say that we have a probability of "½" to measure |01> for both particles, and likewise,
a probability of "½" to measure |10> for both particles. That is, after measuerement.

Note that an expression as |10> actually seems to say that one particle is found to be "up",
while the other is found to be "down".
But the superposition means, that both particles can be in any state, at the same time.
Before measurement, we simply do not know. We only know that |Ψ> is |Ψ>.

1.2. Describing "the apparent strange case" of the EPR paradox.

A more historical perspective is presented in section 1.3. But I like to "jumpstart" directly
to what is known as the "EPR paradox".

The following "case" uses two persons, namely Alice and Bob, each at seperate remote locations.
Each have one member particle, of an entangled system (like equation 4), in their labs.

The description of the "case" below, is not without "critism":

-One important question is, if one would consider the case using a "pure states" interpretation,
or a "mixed state" interpretation. To better understand that, we should already know have studied
chapter 4. For now, however, we take for granted that "this case" deals with one pure state,
as decribed by equation 4.
However, in a real experimental setup, we would have many experiments, and so introducing
an "ensemble" anyway. This also means that to certain degree, "statistics" and "correlations"
comes into play too.

-Furthermore, as the spacetime seperation increases, some may also express doubts as to how
real (or effective) a decription as equation 3 remains to be "true".
However, this argument is rather weak. Entanglement is really quite established,
and confirmed, even over large distances like in Quantum Teleportation experiments.
Only dissipation of the entangled state, due to interactions with the environment (decoherence),
might weaken or destroy the entanglement (possibly even very abruptly).

-Also, one may argue that both Alice or Bob will simply measure "up" or "down"
for their member particle, and "no strings attached".
A "down to Earth" vision states that, nothing what Bob does, or what Alice does (or measures),
will change a thing on their private measurements. One might suspect, that only if Alice informs Bob,
or the other way around, one might find correlations.
Such viewpoints probably complicates on how to interpret the results.

However, the "steering" of Alice's findings, on Bob's member particle, is believed to be true, since
experimental results support this view.

Whatever is true. or what we need to be careful of, I like to present the "case" in it's original form.

However, it will be a simplification of the original idea, and later experimental setups.
Fortunately, it is quite accepted to present "the case" this way.

The Strange Case:

Look again at equation 4. Both states, |01> and |10>, seem to have an equal probability to be "true" or "found",
after an measurement has been performed.

If any measurement is performed, the "state" reduces to |01> or |10>, independend of any distance.
This is the heart of the apparant "problem".

Note: The whole system seems to be "pure", in a "superposition", while each term seems to be "mixed".
The differences will be touched upon in chapter 4 (on pure- and mixed states).

Experiment:

Suppose we have an etangled system again, which can be described by equation 4.

Before we do any measurement, suppose we have a way to seperate the two particles. Let's say that the distance
seperating the particles, gets really large.

Alice is in location 1, where particle 1 is moving to, and Bob is in location 2, where particle 2 just arrived.
Now, Alice performs a measurement to find the spin of particle 1.
The amazing thing is, that if she measures "up" along a certain axis, then Bob must find "down" at the same axis.
Do not think too lightly on this. We started out by saying that (in QM language), both states, |10> and |01>,
have an equal probability to be "true".
The total state, is always a superposition of |01> and |10> where each have an equal probability.

How does particle 2 knows, that particle 1 was found by Alice, to be in the "1" state, so that particle 2
now knows that it must be "0"?

You might say that particle 1 "quickly" informs particle 2 of the state of affairs.
But this gets very weird if the distance between both particles is so large, that only a signal (of some sort)
faster than the speed of light, is involved. That is quite absurd ofcourse.

Notes:

(1). Equation 4 is not a socalled mixed state. It's a superposition, and a pure state.
The probabilities calculated with mixed state, go a little "different" compared to true pure states.
See chapters 3 and 4 for a comparison between mixed- and pure states.

(2). The EPR paradox was first concieved by Einstein, Podolski and Rosen (1935).
More information can be found in many sections below.

The apparent paradox is that a measurement on either of the particles seems to "collapse" the state
|Ψ> = 1/√2 . ( |01> + |10> ), thus of the entire entangled system, into |01> or |10>.
But the superposition (equation 4) was in effect, all the time. Why that "collapse", which always determines the
state of the second particle?

In effect, if you observe one particle along some measurement axis, then the other one is always found
to be the opposite.

This seems to happen instantaneously, for which we have no classical explanation.

The effect has been experimentally confirmed, by Stuart Freedman (et al) in the early 70's, and quite famous are
are the Aspect's experiments of the early '80's.
However, since the experiments were statistical of character, they were not fully "loophole" free.
Later more on this. By the way, the first loophole free experiments were done in 2015 (Delft), almost
conclusively confirming the "strange" effect as described above.

In this setting, it really looks as if Alice "steers" what Bob can find.

- One (temporary) explanation with a certain consensus among physicists:

It would not be good to keep the "apparant paradox", fully open, at this point.

It's true that much details must be worked out further, since all descriptions above,
are presented in a very simple and incomplete manner.

If the distance between Alice and Bob is sufficiently large, then if you would assume that the first measured
particle, informs the second particle on which state it must "take", then such signal must go faster than
the speed of light. This is quite is unacceptable, for most physicists.

In the thirties of the former century, several models were proposed (Einstein: see chapter 2), of which
the (Local) Hidden variables theory was the most prominent one.
Essentially it is this: At the moment the entangled pair (as in section 1.2) is created, a "hidden contract"
exists which fully specify their behaviours in what only seems to be "non-local" events.
It's only due to our lack of knowledge of those hidden variables, which make us think of "a spooky action at a distance".
In a way, this hypothesis is a return to local realism.

- A Modern understanding (without full consensus among physicists):

A modern understanding lies in the "superposition" of the entangled state as expressed by equation 4.
Alice may measure her qubit, and she finds either "up" or "down", each with a 50% probability.

She knows nothing about Bob's measurement, if he indeed did something on that at all.

One modern interpretation then says, that Alice does not know for certainty what Bob's finding is,
or will be, unless Bob's does his measurement at his member particle (along the same direction).
Now, the magic actually sits in the words "unless Bob's does his measurement", which also implies that
Alice and Bob (at a later time) compare their results.
This magic thus sits in the entanglement, or non-locality, where both terms are rather similar,
when considering pure states.

Now, researchers are still faced with the astonishing inner workings of entanglement, non-locality,
and steering, of which I hope that this simple note can shed some light on.

Another, but related, puzzle:

I also have to mention, that many researchers say, that to know that both particles truly form
a "pure state" (like equation 4), you need access to both particles.
If you can only observe one particle, you will effectively "see" a "traced out" entity, which,
in case of a system like equation 4, will reduce to a "mixed state".

If one tries to "seperate" the equation for one particle, from the entangled pair, some weird
results comes to surface.

Using a well know mathematical operation, that is, "tracing out" one particle from equation 4,
returns the "reduced density matrix" (see chapter 3 and 4):

ρ = ½ ( |0 > < 0| + |1 > < 1| ) (equation 5)

Although we haven't covered how to do a partial trace yet, equation 5 above is a mixed state,
meaning a statistical ensemble (see chaper 4). Say that we did the partial trace for Bob's particle,
then the density matrix simply tells us there is 50% chance for finding |1 >, and 50% chance for finding |0 >.

This fact is pretty amazing, since the "probability calculation" seems a local effect.

These sorts of statements can be confusing, but that's also due to the fact that we still miss
some important theory at this point. In later chapters, I will set things straight.

-Other views:

I also must make clear at this point, that some physicists do not see "non-locality" as the mechanism
at work here. Some still stick to the socalled "Hidden variables" (see 1.3 and Chapter 6),
and some even consider quite "exotic" theories like "parallel World" interpretations.

First, we need some more information on the historical setting, and the "spooky action at a distance",
as it was percieved in the '30's, and even up to the '90's of the former century.

So, there probably is "a strange effect", and there is a consensus among many physicists, that entanglement,
and non-locality (and steering), form the basis of the "effect" as described above, and which is rather
quite un-like as the world we know from classical physics.

And, as a sort of summery of critism: there is always a "but"...

It must be stressed, that some physicists say that EPR may appear non-local, but since "faster than light" (FTL),
is not possible, we don't really have "non-locality", but there is something else going on....
For example, "hidden variables", or something else we are not aware of yet.
Thus take notice that there are physicists who are quite weary of "non-locality".
It seems that those physicists, measure non-locality against the possibility of FTL.
Since suggesting FTL is quite "sinful" in physics, non-locality must be false.

However, please take notice of the fact that most physicists view "non-locality"
as the best explanation.

Still no definite answer here! So let's proceed with some more historical background, and then focus on more
modern insights.

1.3 EPR and possible alternatives.

Quite a few famous scientists contributed to QM, roughly in the period 1890-1940. Ofcourse, also in later decades,
countless refinements and discoveries took place.
However, the original basic fundaments of QM were laid in the forementioned period.

Eistein contributed massively as well. My impression is, that his original positivism towards the theory, slowly
diminished to a certain extend, and mainly in the field of the interpretation of QM, and more importantly, to the question
as to which extend the theory of QM truly represents "reality".

Together with a few colleques, in 1935, he published his famous "EPR article":

"Can A Quantum-Mechanical Description of Physical Reality Be Considered Complete? (1935).

There are many places where you can find this classical article, for example:

EPR paradox (1935) (jeti.uni-freiburg.de)

Even in this title you can already see some important themes which occupied Einstein: "Physical Reality" and "Complete".
There are several circumstances and theoretical QM descriptions, which troubled Einstein.
Here, I like to describe (in a few words), the following four themes:

(1):
As an example of Einstein's doubts, may serve the determination of "position" and "momentum", which are
quite mundane properties in the Classical world.

However, with socalled Quantum Mechanical "non-commuting observables", it is not possible to measure (or observe)
them simultaneously with unlimited precision. This is especially fairly quickly to deduce,
using a wave-function notation of a particle.
It is also expressed by one of Heisenberg's uncertainty principles: Δ p Δ v > ½ ℏ
This relation actually says, that if you are able to measure the velocity (v) very precisely,
then the momentum (p) will be (automatically) very imprecise. And vice versa.

These sort of results of QM, made Einstein (quite rightfully) to question, as to how much "reality" we can
attribute to these results of QM.

(2:)
Then we have the problem of "local realism" too. In a classical view, "local realism" is only natural.
For example, if two billiard balls collide, then that's an "action" which causes momentum to be exchanged
between those particles. As another example: a charged particle in an electric Field, "notices" the local effect
of that field, and it may influence it's velocity.

As we have seen in section 1.2, the measurement of one particle of an entangled pair, seem to directly (instantaneously)
have an effect on the measurement the other particle, even if the distance is so large that the speed of light
cannot convey information of the first particle to the second one, in time. This is an example of "non-locality".

Many others had strong reservations to non-locality" too.

Quite a few "conservative" (in some respects) hypothesises emerged, most notably the "Hidden variables" theory.
In a nutshell it means this: At the moment the entangled pair (as in section 1.2) is created, a "hidden contract"
exists which fully specify their behaviours in what only seems to be "non-local" events.
It's only due to our lack of knowledge of those hidden variables, which make us think of "a spooky action at a distance".
In a way, this hypothesis is a return to local realism.
I must say that some alternatives to the "Hidden variables" existed too.

In 1964, the physicist John Stewart Bell proposed his "Bell inequality", which is a mathematical derivation
which, in principle, would make it possible if a "local realistic theory" could produce the same results as QM.
The Bell theorem was revised at a later moment, making it even a stronger argument for a conclusive test.

Although Bell's theorem is not controversial among physicists, still a few have reservations.

The revised "Bell inequality" have indeed be put to the test in various experiments, in favour of QM.
These tests seem to invalidate Local theories, like the Local Hidden variables, and promote the
non-local features of QM.

(3):
The EPR authors also have some serious "doubts" on how to handle an entangled system, such a described above in 1.2.
For example, if Alice would like to change the set of basic vectors, how would it affect Bob's system?
In fact, especially these doubts form the basis of Einstein's sceptism on the representation of QM on reality.

(4):
This theme is again about an entangled system. This time, the EPR authors considered entanglement
mainly with respect to position and momentum. According to QM, both observables cannot be sharply observed
simultaneously. The authors then provide arguments as to why QM fails to give a complete description of reality.
Given the fact that QM was fairly new at that time, it seems to be a quite understandable viewpoint,
although various physicists strongly disagreed with those arguments.

As of the '90's, it seems to me that more and more people started to doubt the argumentation of the EPR authors,
partly due to newer insights or theoretical developments. But, as already mentioned, also in the period of the '30's,
some physicists fundamentally disagreed with Einstein's views (like for example Bohr).

Before we go to EPR steering and some other great proposals, let's take a look at a nice example which
has hit the spotlights the last few decades, namely "Quantum Teleportation".

I really do not have a particular reason for this example. But it exhibits strong characteritics of "non-locality",
and something which many folks call "the EPR channel". And amazingly, we will see that we need to use classical bits
and a classical channel too!

Chapter 2: a few words on the first article on Quantum Teleportation (QT) (1993).

The following classical article, published in 1993:

Teleporting an Unknown Quantum State via Dual Classical and Einstein–Podolsky–Rosen Channels (1993)

by Charles H. Bennett, Gilles Brassard, Claude Crépeau, Richard Jozsa, Asher Peres, and William K. Wootters.

really started to set the QT train in motion.

You can find that article here (researcher.watson.ibm.com).

Quantum Teleportation is not about the "teleportation" of matter, like for example a particle.
It's about teleporting the information which we can associate with that particle, like the state of it's spin.
For example, the state of the system described by equation 1 above.

A collarly of "Quantum Information Theory" says, that "unknown" Quantum Information cannot be cloned.
This means that if you would succeed in teleporting Quantum Information to another location,
the original information is lost. This is also often referred to as the "no-cloning theorem".

It might seem rather bizar, since in the classical world, many examples exists where you can simply copy
unknown information to another location (e.g. copying the content of a computer register, to another computer).

In QM, it's actually not so bizar, because if you look at equation 1 again, you see an example of an unknow state.
It's also often called a "qubit" as the QM representative of a classical "bit".

Unmeasured, it is a superposition of the basis states |0> and |1>, using coefficients "a" and "b".
Indeed, unmeasured, we do not know this state. If you would like to "copy" it, you must interact with it,
meaning that in fact you are observing it (or measuring it), which means that it flips into
one of it's basis states. So, it would fail. Hence, the "no-cloning theorem" of unknown information.

Note that if you would try to (stronly) "interact" with a qubit, it collapses (or flips) from the superpostion
into one of the basis states.

Instead of the small talk above, you can also formally work with an Operator on the qubit, which tries to copy it,
and then it gets proven that it can't be done.

One of the latest "records" in achieved distances, over which Quantum Teleportation succeeded, is about 150 km.

What is it, and how does an experimental might look like?

Again, we have Alice and Bob. Alice is in Lab1, and Bob is in Lab2, which is about 100km away from Alice.
Suppose Alice is able to create an "entangled 2 particle system", with respect to the spin.
So, the state might be written as |Ψ> = 1/√2 ( |01> + |10> ), just like equation 4 above.

It's very important to realize, that we need this equation (equation 4) to describe both particles,
just as if "they are melted into one entity".

As a side remark, I like to mention that actually four of such (Bell) states would be possible, namely:

|Ψ¹> = 1/√2 ( |00> + |11> )
|Ψ²> = 1/√2 ( |00> - |11> )
|Ψ³> = 1/√2 ( |01> + |10> )
|Ψ⁴> = 1/√2 ( |01> - |10> )

In the experiment below, we can use any of those, to describe an entangled pair in our experiment.

Now, let's return to the experimental setup of Alice and Bob.
Let's call the particle which Alice claims, "particle 2", and which Bob claims "particle 3".
Why not 1 and 2? Well, in a minute, a third particle will be introduced. I like to call that "particle 1".
This new particle (particle 1), is the particle which "state" will be teleported to Bob's location.

At this moment, only the entangled particles 2 and 3, are both at Alice's location.

Next, we move particle 3 to Bob's location. The particles 2 and 3, remain entangled, so they stay
strongly correlated.

After a short while, particle 3 arrived at Bob's Lab.

Next, a new particle (particle 1), a qubit, is introduced at Alice's location.
In the picture below, you see the actions above, be represented by the subfigures 1, 2, and 3.

The particles 2 and 3, are ofcourse still entangled. This situation, or non-local property, is often also expressed
(or labeled) as an "EPR channel" between the particles.
This is presumably not to be understood as a "real channel" between the particles, like in the sense of a
channel in the classical world.

In chapter 2, we try to see what physicists are suggesting today, of which physical principles may
be the source for the "EPR channel/non-locality" phenomenon.

Let's return to the experimental setup again. Suppose we have the following:

-The entangled particles, Particles 2 and 3, are collectively described by:

|Ψ_2,3> = 1/√2 ( |01> - |10> )

-The newly introduced particle, Particle 1 (a qubit) is decribed like we already saw in equation 1, thus by:

|φ₁> = a|1> + b|0>

Also note the subscripts, which may help in distinguishing the particles.

At a certain moment, when particles 1 and 2 are "really close", (as in subfigure 4 of the figure above),
we have a 3 particle system, which have to be described using a product state, as in:

| θ₁₂₃> = |φ₁> ⊗ |Ψ_2,3 > (equation 6)

Such a product state, does not imply a "strong" measurement or interaction, so the entanglement still holds.
Remember, we "are" still in the situation as depicted in subfigure 4 of the figure above.
We now try to rewrite our product state in a more convienient way. If the product is expanded,
and some some re-arrangements are done, we get an interresting endresult.
It's quite a bit math, and does not add value to our understanding, I think, so I will represent this endresult
in a sort of "pseudo Ket" equation:

| θ₁₂₃> = |φ₁> ⊗ |Ψ_2,3 >
=
½ x [ Φ₁₂ (-a |1₃ > - b |0₃ >) + Φ₁₂ (-a |1₃ > + b |0₃ >) + Φ₁₂ ( a |0₃ > + b |1₃ >) + Φ₁₂ ( a |0₃ > - b |1₃ >) ] (equation 7)
=
½ x Φ₁₂ [ (-a |1₃ > - b |0₃ >) + (-a |1₃ > + b |0₃ >) + ( a |0₃ > + b |1₃ >) + ( a |0₃ > - b |1₃ >) ] (equation 8)

Note the factor "Φ₁₂".

We have managed to "factor out" the state of particles 1 and 2 into the "Φ₁₂" term. At the same time,
the state of particle 3 "looks like" a superpostion of four qubit states.Indeed. Actually, it is a superposition.

Now, Alice performs a masurement on particle 1 and particle 2. For example, she uses a laser, or EM radiation
to alter the state of "Φ₁₂".

This will result in the fact that "Φ₁₂" will collapse (or "flip") into another state.

It will immediately have an effect on Particle 3, and Particle 3 will collapse (or be projected, or flip) into one
of the four qubit states as we have seen in equations 7 and 8 above.

Ofcourse, the Entanglement is gone, and so is the EPR channel.

Now note this: While Alice made her measurement, a quantum gate recorded the resulting "classical" bits
that resulted from that measurement on Particles 1 & 2.
Before that measurement, nothing was changed at all. Particle 1 still had it's original ket equation |φ₁> = a|1> + b|0>
We only smartly rearranged equation 6 into equation 7 or 8, that's all.

Now, it's possible that you are not aware of of the fact that "quantum gates" do exists, which functions as experimental devices,
by which we can "read out" the classical bits that resulted from the measurement of Alice.
This is depicted in subfigures 5 and 6 in the figure above.

These bits can be transferred in a classical way, using a laser, or any sort of other classical signalling,
to Bob's Lab, where he uses a similar gate to "reconstruct" the state of Particle 3, exactly as the state of
particle 1 was directly before Alice's measurement.

It's an amazing experiment. But it has become a reality in various real experiments.

-Note that such an experiment cannot work without an EPR channel, or, one or more entangled particles.
It's exactly this feature which will see to it, that Particle 3 will immediately respond (with a collapse),
on a measurement far away (in our case: the measurement of Alice on particles 1 & 2).

-Also note that we need a classical way to transfer bits, which encode the state of Particle 1, so that Bob
is able to reconstruct the state of Particle 3 into the former state of Partcle 1.
This can only work using a classical signal, thus QT does NOT breach Einstein's laws.

-Also note that the "no cloning" theorem was also proven here, since just before Bob was able to
reconstruct the state of Particle 1 onto Particle 3, the state of the original partice (particle 1)
was "destroyed" in Alice's measurement.

-Again, note that both a classical- and a nonclassical (EPR) channel, are required for QT to work.

Chapter 2, simply "fits" quite well in the central theme of this note. It seems that QT does not work
without "strong" entanglement between member particles.
Whether this is really proof of "non-locality"..., it seems quite likely.

Chapter 3. A nano introduction to QM, with regards to operations and notations.

I think that it might be useful to provide for a "nano-introduction" into QM.

So, only some of the very basic subjects will be shown in this chapter.
Nevertheless, using the whole of this text, including this chapter, might help if you like
to try the professional articles on EPR and related subjects.

I will start with describing the "wavefunction", which was the orginal form to represent entities
like particles having position and momentum. This is still heavily used today.
However, Dirac's "ket" vector notation "fits" vector spaces (Hilbert spaces) rather well,
and using it, seems to be the favourite practice among physicists (since quite some time).

3.1 The Wavefunction and Probabilities.

3.1.1 Description of the Wavefunction:

Near the end of the 1800's and the early 1900's, some amazing experiments were performed.

While the "classical" theories (electrodynamics, classical mechanics), made a clear distincion between particles
and "waves", some experiments pointed towards a more dualistic character of entities.

For example, particles that were beamed through a "double slit", created an interference pattern,
a phenomenon of which it was formerly thougth, that it could only be produced by "waves" like electromagnetic radiation.

As another example, the photoelectric effect showed that light, at certain circumstances, behaved like particles,
like transferring momentum to a real particle (such as an electron).

So, at certain observations, particles could behave like waves, and the other way around, what was traditionally seen
as waves, could behave like a particle.

Planck and Einstein found in such observations, that radiation thus exibited a "corpusculair" character.
Also was discovered that radiation seemed to be emitted (or absorbed) in "quanta" (discrete energy packets).
And, as already noted, it seemed that those quanta possessed "momentum" too, as was observed in certain experiments.

Many other observations, and theoretical considerations, lead to the start of a new theory, which was able
to accurately describe microscopic entities and events. Indeed, this was the primordial form of Quantum Mechanics.

Around 1924, 'De Broglie' showed, that there exists a "relation" between momentum (p) and wavelength ( λ),
in a "universal" way. In fact, it's a rather simple equation (if you see it), but with rather large consequences. It's this:

p = ℎ / λ (equation 9)

where h is Planck's constant. Now, "momentum", at that time, was considered to be a true 'particle-like' property,
while 'wavelength' was understood to be a typical 'wave-like' property, which for example stuff like
light and radio waves have.
The formula of De Broglie, is quite amazing really. The conseqence is thus, for example, if you have a particle
like an electron flying around, you can associate a "wavelenght" to it.
So, what's going on here? Do we have a "matterwave" or something?
Intuitively, when we now think of a particle's position, we might "visualize" a certain spreading in that position.

This and many more considerations, lead some people to introduce the "wavefunction" Ψ(r,t) , to describe observables
of an entity. More specifically, it can be used particularly well to describe the position of a particle.

Or formulated in a more general way: a wave function in QM is a description of the quantum state of a system.

Around the twenties of the former century, more and more physicists started to describe "states" in terms of
the wave function.
What fitted the bill quite well, was that around 1926, Erwin Schrodinger published a mathematical equation
about the evolution, that a quantum system undergoes with respect to time, when that system happen
to be in some sort of force field.
So, a mathematical description was sought, to find a wave-like equation, that could for example describe the
position of a particle, and which also adheres to Schrodinger's equation.

A flat plane wave, like:

Ψ(r,t) = e^i(kr-ωt) (equation 10)

does not normalize, since it's just a flat front expanding in space and time.

By the way, equation 10 very much resembles a classical flat wave equation.

But a superposition of multiple modes, can normalize, if the modes (defined by "k") are also multiplied by
a sort of gaussian distribution function. The result then is a "wave packet", localized around some maximum,
and quickly lowering "amplitude" of we observe positions away from that maximum.

The addition of waves to the packet, is like adding harmonics with a mode "k" (related to the frequency), each
with a relative amplitude "g(k)". The more wave components you would add, the more the particle gets localized.
In the limit, the summation would become an integral. In the example below, we use one dimension "x" only,
and we also consider the time-independent solution as well.

Ψ(x) = 1/√ (2π ℏ) ∫_-∞^∞ g(k) e^ikx dk (equation 11)

If you would really use an infinite summation, thus really an a integral, the equation above would represent a very
localized position "x". If it is a summation, then a spreading around a maximum, would be expected.
A physical interpretation of this, is that any wavefunction φ(x) can be expressed as a superposition
of states e^ikx/ℏ

In case of a limited superpostion, and discrete "k", we then can write equation 10 as:

Ψ(x) = 1/√ (2π ℏ) Σ A_n e^ikx (equation 12)

Maybe, in using the original QM language, it's better to say that a particle does not have classical properties
like "position" or "momentum". It might be better to say, that the wavefunction must be used to calculate
probabilities of such observables. That's probably right! So now we are going to see how to do that.

3.1.2. Probabilities:

It's often called the "Born's rule", since Max Born was the first who proposed how to calculate probabilities
when using the wavefunction.
Specifically, it can be used (and easily be visualized) when discussing the position of a particle
represented by the wavefunction Ψ(r,t).

Although the Dirac notation is more often often used, the orthodox wavefunction representation is rather
equivalent to Dirac's notation, although Dirac's "Bra Ket" vectors, nicely and directly relate to
mathematical vectorspaces.

Now, how can we use Ψ(r,t) to calculate probabilities?

It can be made plausible, by going back for a while to traditional electrodynamics.
It's really true, that if you would have a classical wave φ, then the energy density of that wave
is proportional to |Ψ|². So, the energy (per unit volume) of a light wave is:

E ∽ |Ψ|²

Born applied that same reasoning to the QM wavefunction probability density. However, the wavefunction
is a complex valued function, so the QM probability density is:

ρ(r,t) = Ψ(r,t)^*Ψ(r,t) = |Ψ(r,t)|² (equation 13)

where Ψ(r,t)^* is the socalled "complex conjugate". Now, from complex number theory,
we know that "z^* z = |z|²", so then equation 13 is correct.

This knowledge, must imply that, the probability P to find a particle in a certain region is:

P(somewhere on x-axis) = ∫_-∞^∞ |Ψ(x,t)|² dx = 1 (equation 14)

P(x in [a,b]) = ∫_a^b |Ψ(x,t)|² dx (equation 15)

where I only used one dimension "x", instead of 3 dimensional space r.
Equation 14 is logical, since the probability to find the particle "somewhere" on the x-axis, must be "1" (100%).

Rewriting that for R³ would then result in:

P(somewhere in Space) = ∫_-∞^∞ |Ψ(r,t)|² dr = 1 (equation 16)

3.2 Schrodingers equation.

We now know what the wavefunction represents. Ofcourse, it's gearded towards "position" and "momentum",
of an entity like a particle. In short:

The wavefunction Ψ(r,t) represents the probability distribution of finding the particle.
But how does it behave in time?

The evolution of the wavefunction, is described by Schrodinger's equation (1925). It sort of describes the "movement"
of the wave, due to a force acting on it. This is the case if the particle has a potential, due to some field.
It's a differential equation, relating "change in time" (movement) to "change in Energy" (change in potential).
It's often notated as:

iℏ

∂
--
∂t

Ψ(r,t)

Ĥ Ψ(r,t)

(equation 17)

where Ĥ is the "Hamiltonian", which is the operator (or expression) corresponding to the total energy
of the quantum system Ψ(r,t).

Or, written explicitly in the Kinetic Energy and Potential Energy components:

iℏ

∂
--
∂t

Ψ(r,t)

ℏ²
--
2m

Δ Ψ(r,t)

V(r) Ψ(r,t)

(equation 18)

You can read it as: "Total Energy term = Kinetic Energy term + Potental Energy term".

If you would take time for it, and like to try some math, you would see that if you would try equation 10
in Schrodinger's equation, you would see that it indeed is a "solution".
You can put equation 10 into the differential equation, and it works.

Therefore, a superposition of equations like equation 10, is a solution too!

Formally, the differential equation above, is also called the "time dependent" Schrodinger equation.

A simpler variant is the time-independent Schrodinger equation, since it effectively does not take
the differential in time, into consideration.

Sometimes this is indeed allowed, if a system does not fundamentally change in time.
For example, the "particle in a box" problem can be solved using a time-independent Schrodinger equation.
The particle in a box problem, is, viewed from a timeline perspective, a stationary problem.
However, you can always simply start (or try) using the time-dependent Schrodinger equation.

We are going to try that as an example.

Example: The "particle in a box" problem (1 dimensional, x-axis).

For about the "math" in this eaxmple, I think it is not relevant at all.
But, it would be really nice, if you just follow the general argument here.

Just interpret the problem as a wavefunction in a 2 dimensional "box", that is, a horizontal x-axis
of length L, with vertical "boundaries" at x=0 and x=L.
Along the y-axis, we may visualize the Amplitude of the wavefunction.

As a little abstraction of the situation, let's suppose that:

-at the vertical boundaries of the box, the potential "V" is ∞.
-for 0 < x < L, the potential "V" is 0.

For a wavefunction "trapped" between the two boundaries, we might say that we are dealing with
a "standing wave", just as we know this effect from classical wave theory.
Between the boundaries, several modes (frequencies) of Ψ(x); will fit, so we introduce
the subscript "n", for Ψ(x);, so we will express the solutions as Ψ_N(x).

Precisely at the barriers, the potential barrier is ∞, which means that at those
particular positions, Ψ=0.

Between the barriers, thus for 0 < x < L, and V=0, the Schrodinger equation becomes:

E_N Ψ_N(x)

ℏ² ∂²
--------
2m ∂ x²

Ψ_N(x)

(equation 19)

Since, if we look at equation 18, here we simply have "V=0", and Δ means ∂² / ∂ x²,
in a one dimensional situation.

Although it's not important to follow the "math" here, it's really possible to rewrite equation 19 into:

∂²
---
∂ x²

Ψ_N(x)

k_N² Ψ_N(x)

= 0

(equation 20)

where k_N² = 2m E_N / ℏ²

The equation above, is actually a well-known differential equation, for which mathematicians and physicists
immediately have general solutions.

Again, this note is not a math textbook, and here it is not important at all, so I will simply say
that a solution can be:

Ψ_N(x) = C e^ik_Nx

Or, you might also say that you have found a "classical-like", harmonic solution as

Ψ_N(x) = A cosk_Nx + B cosk_Nx

If a mathematician or physicists, use the "boundary conditions" in a proper way, a good solution then is:

Ψ_N(x) = P/L sin (Nπx/L)

where P is some constant, and L is the interval on the x-axis where "V=0".

Remember too, that k_N² = 2m E_N / ℏ²

What we see here, is that we have an integer "N", determining the "eigenstates" of the equation, or in other words,
we have found "sine-like" wavefunctions, each respectively with a higher frequency, and a higher energy E_N too.
They all obey the Schrodinger equation, as expressed in equation 19, for this particular situation.

3.3 Common operations and notations.

In order to be able to understand professional articles, you need to know at least
the meaning of the operations and notations listed below.

This section will be very short, and very informal, with the sole intention to provide
for an intuitive understanding of such common operations and descriptions.

If we consider, for a moment, "vectors" with real components (instead of complex numbers), some notions
can be easily introduced, and get accessible to literally everyone.

As a basic assumption, we take following representation as an example of a "state" |φ> = Σ a_i|u_i>,
like for example |φ> = a₁|u₁ > + a₂|u₂ > + a₃|u₃ >.

Especially, I like to give a plausible meaning to the following notations:

(1): < A|B > : The "inproduct", or inner product, of vectors, usually interpreted as
the projection of |B > on |A > (or rather the ket |B> "casted" on the bra < A|).

(2): |B > < A| : usually corresponds to a matrix or linear operator.

(3): |φ > < φ| : corresponds to the "Density" matrix of a pure state.

(4): < φ| O |φ >: corresponds to the expectation value of the observable O.

(5): The "Trace" of an Operator Tr(O) = Σ < i| O |i >

Proposition 1: (1): < A|B > : is the inner product, of vectors or Kets.

-Inner product of two kets < A|B >

If we indeed use the oversimplification in R³, then a (regular) vector or Ket) |B > can be viewed as a column vector:

|B >

┌ b₁ ┐
│ b₂ │
└ b₃ ┘

Note the elements "b_i" of such a vector.
We know that we can represent a vector as a "row" vector too. In QM, it has a special meaning, called a "Bra", as to be
the row vector with "complex conjugate" elements "b_i^*". Let's not worry about the term "complex conjugate",
since you may view it as a sort of "mirrored" number. And if such an element would be a real number,
then the complex conjugate would be the same number anyway.

The Bra < B| can be viewed as a row vector:

< B| = [b₁^*,b₂^*,b₃^*]

The "inner product", as we know it from linear algebra, operates in QM too. It works the same way.
The inner product of the kets |A > and |B > (as denoted by Dirac) is then notated as <A|B>.
From basic linear algebra, we usually write it as A · B, or sometimes also as (a,b).
However, we stick to the "braket" notation:

< A|B >

[a^*₁,a^*₂,a^*₃]

┌ b₁ ┐
│ b₂ │
└ b₃ ┘

a^*₁b₁ + a^*₂b₂ + a^*₃b₃

Which is a number, as we also know from elementary vector calculus.

Usually, as an interpretation, < A|B > can be viewed as the length of the projection of |B> on |A>.
Or, since any vector can be represented by a superposition of basis vectors, then <φ_i|Φ >
represents the probability that Φ collapses (or "projects" or "change state") to the state φ_i.

-Inner product of a ket, with a basivector: <u_i|φ>

As another nice thing to know is, is that if you calculate the inner product of a (pure) state, like:

|φ> = a₁|u₁> + a₂|u₂> + a₃|u₃>

with one of it's basis vectors, say for example |u₂> (and this set of basis vectors is orthonormal),
then:

<u₂|φ> = a₁<u₂|u₁> + a₂<u₂|u₂> + a₃<u₂|u₃> = a₂

since <u₂|u₁> =0 and <u₂|u₃>= 0 and <u₂|u₂>=1, because our basis vectors are orthonormal.

-Operators: The operator "O", as in O|B >=|C >, meaning O operating on ket |B >, produces the ket |C >

Ofcourse "Operators" (mappings) are defined too in Hilbert spaces. Here, they operate on Kets.
Indeed, linear mappings, or linear operators, can be associated with matrices.
This is no different from what you probably know of "vector calculus", or "linear algebra".

=> Here is an example. Suppose we have the mapping "O", and ket |B>.
Then in many cases the mapping actually performs the following:

O |B> = |C>

meaning that the columnvector (ket) |B> is mapped to columnvector |C>.
Or, simply said, the operator "O" maps the ket |B> to ket |C>

We can write that as:

┌ a₁₁ a₁₂ a₁₃ ┐
│ a₂₁ a₂₂ a₂₃ │
└ a₃₁ a₃₂ a₃₃┘

┌ B₁ ┐
│ B₂ │
└ B₃ ┘

┌ a₁₁*B₁+a₁₂*B₂+a₁₃*B₃ ┐
│ a₂₁*B₁+a₂₂*B₂+a₂₃*B₃ │
└ a₃₁*B₁+a₃₂*B₂+a₃₃*B₃ ┘

┌ C₁ ┐
│ C₂ │
└ C₃ ┘

=> Operators can "work" on a "bra" too, resulting in another "bra":

<A| O = <C|

where, in our simple model, B and C are row vectors.

We can write that as (using real numbers only, as an example):

[ A₁ A₂ A₃ ]

┌ a₁₁ a₁₂ a₁₃ ┐
│ a₂₁ a₂₂ a₂₃ │
└ a₃₁ a₃₂ a₃₃ ┘

[ C₁ C₂ C₃ ]

The above is interresting by itself, but I used it here for a start of proposition 2.

Proposition 2: |B > < A| : usually corresponds to a matrix or linear operator.

Let's try this:

|B > < A|

┌ B₁ ┐
│ B₂ │
└ B₃ ┘

[ A₁^* A₂^* A₃^* ]

┌ B₁A₁^*+B₁A₂^*+B₁A₃^* ┐
│ B₂A₁^*+B₂A₂^*+B₂A₃^* │
└ B₃A₁^*+B₃A₂^*+B₃A₃^* ┘

Above, we see an example of how to multiply a column vector with a row vector, which is a
common operation in linear algebra. It simply takes the syntax and outcome as you see above.
So, proposition 2 seems to be plausible, since it follows that |B > < A| is a matrix.

Proposition 3: |φ > < φ| : corresponds to what is called the "Density" matrix of a pure state.

In proposition 2, we have seen that |B > < A| usually produces a "matrix".

Now, if we take a ket |φ > and "multiply" it with it's dual vector, the bra < φ|,
as in |φ > < φ|, then ofcourse it is to be expected we get a matrix again.

However, the elements of that matrix are a bit special here, since the elements tell us something
about the probability to find that pure state in one of it's basis states.

In a given basis, the diagonal elements of that matrix, will always represent the probabilities that
the state will be found in one of the corresponding basis states.

In it's most simple form, where we for example have that |φ > = |u₁ > + |u₂ > ,
the density matrix would be:

┌ ½ 0 ┐
└ 0 ½ ┘

The density matrix is more important, as a description, when talking about mixed states.

Proposition 4: < φ| O |φ >: corresponds to the expectation value of the observable O.

-Trying to make that plausible using simple vectors:

We can make that plausible in the following way:

We have associated a certain observable (such as momentum, position etc..) with a linear operator "O".
Now suppose for a moment that we have diagonalized the operator, so the only diagonal elements of the matrix, are not "0",
and represent the eigenvalues. Then we may use an argument like so:

Suppose |φ> = a₁|u₁ > + a₂|u₂ > + a₃|u₃ >.

where |u_i > are basis vectors. We can write it as a columnvector too (in our simplification):

|φ>

┌ a₁ ┐
│ a₂ │
└ a₃ ┘

We are going to show that < φ| O |φ > is the expectation value of O, by making it plausible
for a simple case, thereby hoping that you will agree that it is true in general as well.

Now suppose O is represented by the matrix:

┌ 0 0 0 ┐
│ 0 0 0 │
└ 0 0 1 ┘

Then:

O|φ>

┌ 0 0 0 ┐
│ 0 0 0 │
└ 0 0 1 ┘

┌ a₁ ┐
│ a₂ │
└ a₃ ┘

┌ 0 ┐
│ 0 │
└ a₃┘

Which "looks" as if we have projected |φ> on it's eigenvector (basisvector) |u₃ >,
with a corresponding eigenvalue of a₃.

Now, suppose all diagonal elements of O are not null (thus having some value), and all others are "0".
Then an expression like:

< φ| O |φ >

can be written as:

[a^*₁,a^*₂,a^*₃]

┌ a₁₁ 0 0 ┐
│ 0 a₂₂ 0 │
└ 0 0 a₃₃ ┘

┌ a₁ ┐
│ a₂ │
└ a₃ ┘

= (a^*₁a₁) a₁₁ + (a^*₂a₂) a₂₂ + (a^*₃a₃) a₃₃

which result can be read as the "weighted average" of the eigenvalues. Thus we say that it's the "expectation value" of O.
I hope you can see some logic in this. Proposition 4 is however, valid for the general case too.

-Using the usual argumentation in QM:

We know that for the usual wavefunction (over "x", not considering "t"), the probability distribution (density),
can be expressesd as:

ρ(x) = Ψ^*(x)Ψ(x)

which resembles equation 13 (but in this example for one dimension only).

For the expectation value <x>, that is, the best average of finding Ψ to be at a certain position "x",
considering all possible x (all x Space), then may be expressed as:

< x > = ∫_-∞^∞ Ψ^*(x)Ψ(x) dx

If we now have a certain observable "Q" of Ψ, and we like to know the expectation value < Q >
of that observable, then we can use the integral shown above, for the determination of the average of Q:

< Q > = ∫_-∞^∞ Ψ^*(x) Q(x) Ψ(x) dx (equation 21)

We need to multiply the probability distribution ρ(x) of Ψ, with Q(x), and integrate that over
all space (here we only have considered one dimension).

Proposition 5: The "Trace" of an Operator is: Tr(O) = Σ < u_i|O|u_i >.

The "trace" of an Operator, or matrix, is the sum of the "diagonal" elements.
With respect to pure- and mixed states, it has a different outcome (namely 1 or < 1 respectively).

To give an very simple example on how to obtain it, let's consider vectors
in R³ (thus real numners only).

In R³, we can have the following set of orthonormal basisvectors:

┌ 1 ┐
│ 0 │
└ 0 ┘

┌ 0 ┐
│ 1 │
└ 0 ┘

┌ 0 ┐
│ 0 │
└ 1 ┘

You may say that those basis vectors corresponds to |u₁>, |u₂>, |u₃>,
like in our usual ket notation.

If we consider the rightside of the expression "Σ < u_i|O|u_i >", then we have "O|u_i >".
We can interpret this as that O operates on a basisvector |u_i>.
Suppose that i=1, meaning that it is our first basis vectors, just like the set of basisvectors of R³,
as was listed above.

Let's operate our matrix of O, operate on our basisvectors. I will do this only for the (1,0,0) basisvector (i=1).
For the other two, the same principle applies. So, this will yield:

┌ a b c ┐
│ d e f │
└ g h i ┘

┌ 1 ┐
│ 0 │
└ 0 ┘

┌ a*1+b*0+c*0 ┐
│ d*1+e*0+f*0 │
└ g*1+h*0+i*0 ┘

┌ a ┐
│ d │
└ g ┘

Well, this turns out to be the first column vector of the matrix O. Let's call that the vector "|A >" (†).

Next, let's see what happens if we perform the leftside of Σ < u_i|O|u_i >".
We already had found that the vector |A > corresponds with "O|u_i >".
Using the leftside, we have < u_i|A >. This is an inner product, like:

┌ 1 ┐
│ 0 │
└ 0 ┘

┌ a ┐
│ d │
└ g ┘

Note that this number "a", is the "top left" element of the matrix O.

Since Tr(O) = Σ < i|O|i >, it means that we repeat a similar calculation using all basisvectors,
and add up al results. Hopefully you see that this then is the sum of the diagonal elements.
I already proved it for the first diagonal element (a), using the first basis vector.
The 2 vectors remaining, to be used for a similar calculation, will then produce "b" and "c".

In this simple example, we then have Tr(O) = a + b + c.

Note that in general O|u_i> produced the i^th column of O (see †) above.

In the exceptional case where O|u_i> produces "a|u_i>", thus
a scalar coefficient "a" times a basisvector, thus "O|u_i=a |u_i>", then in our simple example
in R³, we would have:

O|u₁> = a |u₁>
O|u₂> = b |u₂>
O|u₃> = c |u₃>

And, keeping in mind that O|u_i> the i^th column of O (see †), then we would
have a matrix with only the diagonal elements which are not null, and all others (off diagonal elements),
which would then be nul.

In such case, it is often said that the elements a, b, and c are the "eigenvalues" of the operator O.

It's absolutely formulated in "Jip & Janneke" language, but I hope you get the picture.

3.4 A few words on Dirac's notation

3.4.1 Some general observations:

In general, we may write a "state vector", or a "ket", as expanded (as a superposition) of basis states |u_i>:

|φ> = Σ c_i|u_i>

where

c_i= < u_i|φ >

Thus each number c_i is the inner product of |φ > and |u_i>

|φ> represents a quantum system to be in the state φ and is called the state vector.

Operations on |φ>, or it's basis states, goes very similar as to what you may know from "vector calculus".
Indeed, for example, in section 3.3 we have seen the "inner product" of two kets.

And the following are kets too, like |x> or |p>, representing the position and momentum respectively.
In some (exceptional) cases, you might even use "definitive" kets, like |x=2>, meaning the position of
the quantum system (like a particle), is at "x=2".

It's true that we often treat kets, like a statevector, where we speak of a certain probability of finding
the statevector to be in the basis state (eigenstate) "|u_i>", after some measurement is done.
That is true, but it can also be that the state is in a certain value (or certain basis state), for example,
after an observation has been performed.

Some notations we already are familiar with, can be written in "a slightly" different (but equal) form.
Just take a look at this. We already know our qubit:

|φ> = a|1> + b|0>

Now suppose we have a 2 dimensional Hilbert space. An arbitrary vector can be expressed as a linear combination
of the unit vectors or basis vectors. Suppose the vectors |i> and |j> form such a basis.
An arbitrary vector "v", or ket, might be written as:

|v > = c₁ |i > + c₂ |j > = < i |v > |i > + < j |v > |j > = |i> < i |v > + |j> < j |v >

It's a slightly different format, but it's really the same as "c₁ |i > + c₂ |j >".

Since |v > = |v >, the above equation also means that |i> < i |v > + |j> < j |v > = (|i> < i| + |j> < j|) |v >. Thus:

|i > < i| + |j > < j| = 1

You can also express it in a matrix, since we already know from the former section that, in general,
|A > <B| is a matrix. So:

┌ 1 0┐
└ 0 1┘

|i > < i| + |j > < j|

= I

Where I is the "identity matrix" or "identity operator".

Above, we considered the 2 dimensional case. In general, in dimension n, we may say that:

Σ |u_i > < u_i| = I

If u_i > represents a complet set of orthonormal basis states, of that n dimensional space.

We know that:

|φ> = Σ c_i|u_i>

But we may thus also write:

|φ > = Σ |u_i > < u_i|φ >

3.4.2 Interpretations of the Bra:

We know what a "ket" represents, which is not too hard to visualize. It's simply a vector.
But how can we interpret a "bra"?

1. Informal interpretation:

This interpretation is "almost" true, but it can be used as a very close, and good pictorial interpretation.

In "proposition 1", of the former section, we saw that if we represent |A > as a column vector, then
< A| is the corresponding row vector, with "complex conjugate" elements.
If needed, take a look at proposition 1 again, of section 3.3.

So, it's actually a nice interpretation (I think). If you have the ket |φ>,
then < φ| represents the same state, but as a mathematical object, it's the transpose of |φ>,
that is, this time a row vector. Since the elements are complex numbers, < φ| uses "complex conjugate" elements.

And what's really quite the same, note that:

< φ| = ( |φ > )^†

Where the dagger symbol "†", stands for "conjugate transpose" or "Hermitian transpose".
Here it means reversing the column elements to row elements (or vice versa), and taking the complex conjugate
of those elements.

2. More abstract (and formal) interpretation:

It "fits" mathematically well, to say that "bra's" are elements of the dual Hilbert space.
Or, the bra's are vectors of that dual Hilbert space.

What this also means, is that "bra's" are functionals on the ket, which produce a (complex) number.
That's fine, since we already know that for example:

< u_i|φ > = c_i

Thus, here we can view it as the functional < u_i|, operating on |φ >,
producing the number "c_i".

But if we consider a certain ket, |φ >, then it's corresponding bra, is unique < φ|.
There is always a one-to-one correspondence from a certain ket, to it's unique bra.

3. The "bra" as the projection, or final state:

If we look at (2) again, above, we can also say that < u_i|φ > represents the projection
of |φ >, on < u_i|.

Note that we here talk about a different bra, and a different ket (not a certain ket with it's unique corresponding bra).

So, if in an observation (or experiment), it turns out that we find the value "c_i", then we can also
say (or interpret it), as that |φ > was projected (or "cast") on < u_i|, or you may also say
that |φ > was projected (or "cast") on |u_i >.

3.4.3 Some well known states you may see in articles:

- Qubit:

|Ψ> = a|1> + b|0>

- Qutrit:

|Ψ> = a|0> + b|1> + c|2>

- Product state, or "outer product" of two qubits:

|Ψ> = |φ₁> ⊗ |φ₂> = a₀₀ |00> + a₀₁ |01> + a₁₀ |10> + a₁₁ |11>

- General Product state, or "outer product" of three qubits:

|Ψ> = a₁ |000> + a₂ |001> + .... + a₈ |111>

-Specific example of the product state of three qubits:

|Ψ> = 1/√8 |000> + 1/√8 |001> + 1/√8 |010> + 1/√8 |100> + 1/√8 |110> + 1/√8 |011> + 1/√8 |101> + 1/√8 |111>

-Bell entangled states of 2 qubits:

|Ψ¹> = 1/√2 ( |00> + |11> )
|Ψ²> = 1/√2 ( |00> - |11> )
|Ψ³> = 1/√2 ( |01> + |10> )
|Ψ⁴> = 1/√2 ( |01> - |10> )

- (Entangled) Singlet state:

|Ψ> = 1/√2 ( |01> - |10> )

-A Greenberger-Horne-Zeilinger (GHZ) entangled state, of three qubits:

|GHZ> = 1/√2 ( |000> - |111> )

Often, this is the entangled 3 qubit state which is used in experiments or theoretical conjectures, quite similar
to our familiar bi-particle Bell states, or the Singlet state.

3.4.4 Some more words on Operators, Matrices, Observables:

It's probably really neccessary, I believe, that you have seen section 3.3, before
turning to this one.
I (like to) think that 3.3 is a quite "gently" introduction on how we use matrices in QM.

(1). One of the rules in QM is that with each measurable "observable" of a quantum system,
is associated a quantum mechanical (linear) Operator.

If you read that literally, then the Operator defines (so to speak), the observable.

We know that the wavefunction, or statevector, represents the probability amplitude of finding
the system in a certain state of the observable.

(2) You may also rephrase (1) to this: A wavefunction (Schrodinger like) or statevector (like Dirac formulated it),
describes the observable quantity, while the Operator acts on the wavefunction (or statevector).

It is true that a "measurement" can be formulated as an Operator acting on the wavefunction (or statevector).
This often then means a projection on one of the eigenstates, with a certain associated probability
of finding that particular eigenstate.

With this view in mind, we can indeed say that the "Operator" is associated with the "observable".

Formally, in the sense of "regular" vector calculus, the Operator "A" working on the ket |Ψ>,
produces another ket |φ>. So:

A |Ψ> = |φ> (equation 22)

Equation 22 is very general. That is, the ket |Ψ> is "mapped" to |φ>, just as we know from
regular vector calculus. However, if we would have:

A |Ψ> = a |Ψ> (equation 23)

then |Ψ> is called an "eigenstate" or "eigenket" of the operator A, and "a" is called an "eigenvalue".

As another thing, since QM is about physical systems, for A it is also required that:

< φ | = < Ψ| A^† (equation 24)

where "†", stands for "conjugate transpose" or "Hermitian transpose".
Remember that it is formally defined, that we have a dual Hilbert space of bra's (H^*), associated with
every ket of the regular Hilbert space H.

Don't worry about these (possibly confusing) statements. It simply means that there are certain
requirements on such Operators. We know that an Operator can be viewed as a matrix, especially when
using the Dirac notation. So, there are certain requirements on those matrices too.

Taking the "conjugate transpose" of a matrix A, means that we switch to rows and colums, and take
the complex conjugate of all the matrix elements.

It's also rather neccessary to have

A = A^† (equation 25)

An important reason is, that the expectation value of A, meaning < φ| A |φ >,
must produce a real number, and not a complex number.
Yes indeed ! In real experiments we simply must find real numbers, although QM entities uses complex numbers.

If you like, you can write it out, using the examples of section 3.3, and see that this requirement
really is rather acceptable.

When A = A^†, then we talk about "self adjoined" Operators. So it simply means that
taking the "conjugate transpose" of a matrix A, results into the same matrix.

Note also that in section 3.3, proposition 2, it was made plausible that:

|B > < A|

represents a matrix, as a very general statement, for two kets |A > and |B >.
If needed, you can take a look again at section 3.3, to verify that statement, and also
some example operations by some "Operator" O, on kets.

Stated in some more formal terms:

Given vectors and dual vectors, we can define operators O (i.e., linear mappings from H to H),
in the format of:

O = | φ>< Ψ|

As already said above, we usually want Hermitian or self-adjoined Operators, garanteeing that
the expectation value of the Operator, is a real (and not complex) number.

Example 1:

Consider the following Operator (and matrix) "A":

A =	┌ 0 -i┐ └ i 0 ┘

Is this a "self adjoined" Operator?

Here it is true that A = A^†, since if you take the "conjugate transpose", that is,
switch the rows and columns, and take the complex conjugate of each matrix element, you will see
that this is indeed true.

Note:
- the complex conjugate of a real number, like "1", is the same real number again ("1").
- the complex conjugate of an imaginary number like "i", is "-i" and vice versa.

Maybe you have still some questions right now. If so, take a look at section 3.3 again.
It will show some foundations of matrices, which may help.

Chapter 4. Meaning of pure states and mixed states.

4.1 A few words on "pure states":

While you might think that a completely defined state as |0 >, is pure, it holds in general for our
well known superpositions.

An example of an "superpositional" state, can be this:

|φ> = a₁|u₁> + a₂|u₂> + a₃|u₃>

You may also view a "pure state" as a single state vector, as opposed to a mixed state.
So, even at this stage, we already may suspect what a mixed state is.
Thus "pure states": We have seen them before in this note, sofar.

A mixed state is a statistical mixture of pure states, while superposition refers to a state
carrying some other states simultaneously.

Although it can be confusing, the term "superposition" is sort of reserved for pure states.

So, our well-known qubit is a pure state too: |φ> = a |0 > + b |1 >

Or as a more general equation, we can write:

|φ> = Σ a_i|u_i> (equation 26)

This is a shorthand notation. Then i runs from "1" to "N", or the upper bound might even be infinite.

Usually, such a single state vector |φ>, is thus represented by a vector or "ket" (|>) notation,
and is identified as a certain unknown observable of a single entity, as a single particle.

So, a pure state is like a vector (called "ket"), and this vector be associated with a state of one particle.
A pure state is a superposition of eigenstates, like shown in equation 29.

Other notes on pure states:

Such vectors are also normalized, that is, for the coefficients (a₁, a₂, etc..), it holds that
|a₁|² + |a₁|² + ... =1

It's also often said that a pure state can deliver you "all there is to know" about the quantum system,
because the system's evolution in time can be calculated, and Operators on pure states work as "Projection" operators.

In sections above, we have also seen that the coefficient a_i can be associated with the probability
of finding the state to be in the |ua_i> eigenstate (or basisvector) after a measurement has been performed.

In general, an often used interpretation of |φ>, is that it is in a superposition
of the basis states simultaneously.Then, the "keyword" here is "simultaneously".
However, this interpretation depends on your view of QM, since many "interpretations" of QM exist.
But "superpostion" will always hold, and is a key term of a pure state (like equation 29).

When you would insist on the qualifying phrase "a pure state gives us all there is to know", then probabily
(or maybe) known coefficients are required too, like for example with:

|φ> = 1/√2 ( |0 > + |1 > )

Note that some authors treat it that way.
But in general, undetermined coefficients are OK too. As long as we can talk of a "ket", we have a pure state.

Furthermore, it is required that the inner product of |φ> with it's associated "bra", is normalized,
that is, has a "unit length". That is, the inner product returns the value "1". Thus:

< φ |φ> = 1

4.2 A few words on mixed states:

A "mixed state", is a mix of pure states.
Or formulated a little better: a probability distribution of pure states, is a mixed state.

It's an "entity" that you cannot really describe, using a regular Ket statevector.
You must use a density matrix to represent a mixed state.

Another good description might be, that it is a "statistical ensemble" of pure states.

So we can think of mixed state as a collection of pure states |φ_i>, each with associated probability
density ρ_i, where 0 ≤ ρ_i ≤ 1 and Σ ρ_i = 1.

It cannot be stressed enough, that a linear superposition is not a mixture.

Mixed states are more commonly used in experiments.

For example, when particles are emitted from some source, they might differ in state.
In such a case, for one such particle, you can write down the state vector (the Ket).
But for a statistical mix of two or more particles, you cannot.

The particles are not really connected, and they might individually differ in their (pure) states.
What one might do, is create a "statitistical mix", what actually boils down in devising
the density matrix.
The statistical mix, is an ensemble of copies of similar systems, or even an ensemble with
respect to time, of similar quantum systems

So, you can only write down the "density matrix" of such an ensemble.

In equation 3, we have seen a "product state" of two kets. That's not a statistical mix, as we have here
with a "mixed state".

In a certain sense, a "mixed state" looks like a classical statistical description, of two pure states.

When particles are send out by some source, say at some interval, or even sort of continuously,
it's even possible to write down the equation (density matrix) of two such particles which were emitted
at different times. This should illustrate that the component pure states, do not belong to the same
wave function, or Ket description.

You might see a "bra ket-like" equation for a mixed state, but then it must have terms
like |φ > < ϕ|, which indicate that we are dealing with a "density matrix".

In general, the density matrix (or state operator) of a (totally) mixed state, should have a format like:

ρ = Σ ρ_i |u_i > < u_i|

Hopefully, you can "see" something that "looks" like a statistical mixture here.

An example:

Here is an example that "describes" some mix of two pure states |a > and |b >:

ρ = 1/4 |a > < a| + 3/4 |b > < b| (equation 27)

Note that this not an equation like that of a pure state.

Ofcourse, some ket equations can be rather complex, so not all terms perse need to have to be
in the form |φ > <φ|. Especially "intermediate" results can be quite confusing.
Then also: by no means this text is complete. That's obvious ofcourse. For example, partial mixed systems
exist too, adding to the difficulties in reckognizing states.

A certain class of states are the socalled pseudo-pure families of states.
This refers to states formed by mixing any pure state, with the totally mixed state.
So, please do not view the discussion above, as comprehensive description of "pure" and "mixed" states,
which is certainly not the case here.

4.3 What about our entangled two partice system?:

Equation 4, which described an entangled bipartice system, is repeated here again:

|Ψ> = 1/√2 . ( |01> + |10> )

Note that this is a "normal" ket equation, and it is also a superposition.
We do not see the characteristic "| > < |" terms which we would expect to see in a mixed state.

Therefore, it's a pure state !

There are several perculiar things with such entangled states. We already have seen some in section 1.2,
where Alice and Bob performed measurements on the member particles, in their own seperate Lab's.

Another perculiar thing is this: I will not illustrate it further, but using some mathematical techniques,
it's possible to "trace out" the state of one particle, from a two-particle system.

-For example, if you would have a "normal product" state like equation 3, then tracing out particle,
like particle 2, just gives the right equation for particle 1. This was probably to be expected, since the
product state is "seperable".

- If you would do the same for an entangled system, then if you try to trace out a particle,
then you end up with a "mixed state", even though the original state is pure.
That's is really quite remarkable. Later more in this.

For now, let's go to the next chapter.

Chapter 5. The inequalities of Bell, or Bell's theorem.

5.1 The original formulation.

The famous Bell inequalities (1964), in principle, would make it possible to test if a "local realistic theory",
like the Local Hidden Variables (LHV) theory, could produce the same results as QM.

Or, in stated somewhat differently:

No theory of local Hidden Variables can ever reproduce all of the predictions of Quantum Mechanics.

Or again stated differently:

There is no underlying classical interpretation of quantum mechanics.

For about the latter statement, I would like to make a "small" (really small) reservation, since, say from 2008 (or so),
newer "parallel universe" theories have been developed. Although many don't buy them, the mathematical frameworks and ideas
are impressive. In chapter 8, I really like to touch upon a few of them.

The Bell theorem was revised at a later moment, by John Clauser, Michael Horne, Abner Shimony and R. A. Holt,
which surnames were used in labeling this revision to the "CHSH inequality".
The "CHSH inequality" can be viewed as a generalization of the Bell inequalities.

Probability, and hidden variables.

To a high degree, QM boils down to calculating probabilities of certain outcomes of events.
Most physicist, say that QM is intrinsically probabilistic.

This weirdness is even enhanced due to remarkable experiments, like the one as decribed in section 1.2.
It is true that the effects described in section 1.2, are in conflict with local realism, unless factors play
a role of which we are still fully unaware of, like "hidden variables".

We may say that Einstein's view of a more complete specification of reality, related to QM,
is our "ignorance" of local pre-existing, but unknown, variables. Once these unknown hidden variables are known,
the "pieces fall together", and the strange probabilistic behaviour can be explained.
This then includes an explanation of the strange case as described in section 1.2 (also called the "EPR paradox").

This is why a possible test between local realism, and the essential ideas of QM, is of enormous importance.
It seems that Bell indeed formulated a theoretical basis for such test, based on stochastic principles.
I have to say that almost all physicist agree on Bell's formulation, and real experiments have been executed,
all in favour of QM, and against (local) hidden variables theories.

What is the essence of the Bell inequalities?

In his original paper (Physics Vol. 1, No. 3, pp. 195-290, 1964), Bell starts with a short and accurate description
of the problem, and how he wants to approach it.
It's really a great intro, declaring exactly what he is planning to do. I advise you to read
the secions I and II of his original paper (or read it completely, ofcourse). You can find it here:

ON THE EINSTEIN PODOLSKY ROSEN PARADOX (cern.ch)

Bell's Theorem, or more accurately, the "CHSH inequality", has been put to the test, and also many theoretical work has been done,
for example, on "n-particle" systems, and other more complex forms of entanglement.

On the Internet, you can find many (relatively) easy explanations of Bell's Theorem. However, the original paper
has the additional "charm" that it explicitly uses local variables, like λ, which stands model for one
or more (possibly a continuous range) of variables. His mathematics then explicitly uses λ in all derivations,
and ultimately, it leads to his inequalities.

If we consider our experimental setup of section 1.2 again, where Alice and Bob (both in remote Labs), perform
measurements again on their member particles, then one important assumption of local realism is, that the result for particle 2
does not depend on any settings (e.g. on the measurement device) in the Lab of particle 1, or the other way around.
In both Labs, the measurement should be a "local" process. Any statistical "illusion" would then be due,
to the distribution of λ, in the respective Labs, as prescribed by a Local Hidden variable theory.

The Bell inequalities provide a means to statistically test LHV, against pure QM.
In effect, experimental tests which "violate" the Bells inequalities, are supportive for QM non-locality.
Sofar, this is indeed what the tests have delivered.

Some folks see the discussion in the light of two large "believes": or you believe that signalling is not
limited by "c", or you believe in "super determinism".
Super determinism then refers to the situation where any evolution of any entity or process is fully determined.
So to speak, as of the "birth of the Universe", from where particles and fields "snowed" from the false vacuum.
Interestingly, all particles and other stuff, indeed have a sort of common origin, and thus may have given rise to a
super entanglement of all stuff in the Universe. Still unkown variables have then "sort of" fixed everything, thus
a sort of "super determinism" follows.
Personally, I don't buy it. And it seems too narrow too. There are also some newer theories (Chapters 7 and 8) which do
not directly support "super determinism".

5.2 Newer insights on the Bell inequalities and LHV's.

-Simultaneous measurements vs non-Simultaneous measurements.

Since the second half of the 90's (or so), it seems that newer insights have emerged on Bell's Theorem,
or at least some questions are asked, or additional remarks are made.

One such thought is on how to "integrate" the Heisenberg relations into the Theorem, and the test results.
Here is a good example of such an article:

Bell's Theorem: a new Derivation that Preserves Heisenberg and Locality

The authors state that "near" simultaneously measurements, implicitly relies on the Heisenberg uncertainty relations.

This is indeed true, since if Alice measures the spin along the z-direction and if she finds "up", then we may say that if
Bob would also measure his member particle along the z-direction too, then he will certainly find "down".
Therefore, the full experiment will use (also) axes for Alice and Bob which do not align, but have a variety of different angles.
Then, afterwards, all records are collected, and correlations are established, and then using Bell's inequalities,
we try to see if those inequalities are violated (in which case LHV gets a blow, and QM seems to "win").

The point of the authors is however, that the measurements will occur at the same time.
If now a time element is introduced in the derivation of Bell's theorem, a weakening of the upper bound of the Theorem
is found. As the main cause of this, the authors make it clear that second-order Broglie-Bohm type of wavefunctions
may work as local operators in the Labs of Alice and Bob.

I personally can't really find "mistakes", apart from the fact that Broglie-Bohm is actually another interpretation of QM,
which might not have a place in the argument. However, I am not sure at all.
By the way, the Broglie-Bohm pilot wave interpretation, is a very serious interpretation of QM, with many
supporting physicists.

However, the main point is that the traditional Bell inequalities (or "CHSH inequality") in combination with
the experimental setup, is not "unchallenged" (as good physics should indeed operate).

-Werner states.

Amazingly, as was discovered by Werner, there exist certain entangled states that likely
will not violate any Bell inequality, since such states allows a local hidden variable (LHV) model.

His treatment (1989) is a theoretical argument, where he first considers the act of preparing states,
which are not correlated, thus not entangled, like the example in equation 3 which is a seperable product state.

Next, he considers two preparing devices, which have a certain "random generator", which makes it possible to generate
states where the joint Expectation value, is no longer "seperable" or factorizable.
His artice is from 1989, where at that time it was hold that systems which are not classically correlated,
then they are EPR correlated.

Using a certain mathematical argumentation, he makes it quite plausible to have a semi-entangled state,
or Werner state, which has "the look and feel" of entanglement, and where a LHV can operate.
He admits it's indeed a "model", but it has triggered several authors to explore this idea in a more general setting.

The significance is ofcourse, to have non seperable systems, using a LHV.

If you are interested, take a look at his original paper:

Quantum states with Einstein-Podolsky-Rosen correlations admitting a hidden-variable model (Werner, 1989)

-Countless other pro's and contra's:

There are many articles, (somewhat) pro- or contra Bell's Theorem.
Many different arguments are used in the "battle". You can found them easily, for example,
if you Google with the terms "criticism Bell's theorem arxiv", where the "arxiv" will produce the
free, uneditorial, scientific papers.

Here is one that makes a strong point against LHV, and is very much pro QM:

A Bell Theorem Without Inequalities for Two Particles, Using Efficient Detectors

This article is great, since it uses a model of 2 entangled particles without a common origin, and thus
this system is very problematic for any type of classical or LHV related theory.

I am not suggesting that you should read the complete artice. Contrary, often only the introduction of such articles
is good enough, since then the authors outline their intentions and arguments.

The next article uses a truly different perspective. According to this author, we do not need "non-locality"
and all strange observed effects, are simply due (under the hood) to the superpostion principle.
Furhermore, he makes a case that QM simply does not give a complete view on reality, just like Einstein said.

You can find that article here.

However, the EPR experiments, and what we have seen in Quantum Teleportation, is probably hard to understand
just by "superposition" alone, and not thinking in terms of non-locality.

So, what do we have up to now?

Taken all together, you cannot say that there exists a full consensus among physicist on how to exactly
interpret the EPR experiments, together with Bell's inequalities.
However, a majority of physicists still thinks in terms of non-locality, to explain the experimental results,
and see sufficient backing of theoretical considerations, for their position.

Sofar, what we have seen in section 1.2 (EPR entangled bi-particle experiment), and chapter 2, (Quantum Teleportation),
is that "something" that behaves like "an immediate action at a distance", seems to be at work.
This does not suggest that any form of signalling/communication exist, that surpasses the speed of light.
As said in section 1.2, the "no communication theorem" states exactly that.
However, not all folks would agree on this.

By the way, the QT effect we saw in section 1.4, simply also needed a classical channel in order to "transport"
the state of particle 1, to particle 3 at Bob's place.
That is also supporting the view, that true information transfer does not go faster than "c".

There exists a number of interpretations of QM, like e.g. the Broglie-Bohm "pilot-wave" interpretation.
Rather recently, also newer parallel universe models were proposed, with a radical different view on QM.
For about the latter: you might find that strange, but some models are pretty strong.

The most commonly used interpretation, is the one that naturally uses superpositions of states.
That model works, and is used all over the World. For example, most articles have no problem at all in writing
a state (Ket) as a superposition of basis states, like in a "pure state", as we have seen in section 2.1.
In fact, once describing QM in the framework of Hilbertspaces (which are vectorspaces), superpostion is then sort of "imposed"
or un-avoidable.
But ofcourse, the very first descriptions using wave-functions to describe particles and quantum systems in general,
is very much the same type of formulation.
And this vector formulation, fits the original postulates of QM, quite well.

But it seems quite fair to say that it is actually just this principle of superposition, that has put us
in this rather weird situation, where we still cannot fully and satisfactory understand, exactly why we see what think we see
as was described in section 1.2 (or the lightgreen text above).

Not all physicist like the "non-locality" stuff as displayed in the lightgreen text above.
For quite a few, a Hidden Variable theory (or similar theory) is not dead at all.
Although the experimental evidence using the Bell tests seems rather convincing, there still seems to exist
quite some of counter arguments.

For now, we stay on the pure QM path (superpositions, EPR non-locality, probabilities, Operators, Projectors etc..),
and how most people then nowadays interpret "Quantum Steering", "Entanglement", and "Bell non-locality".

Let's go to the next section.

Chapter 6. Steering, Entanglement, and Bell non-locality.

6.1 Some descriptions:

Let's first try to describe "steering":

Quantum Steering:

Quantum steering is the ability of Alice to perform a measurements on her local member
of an entangled system, with different outcomes, and that leads to different states for the remote part of
that entangled system (at Bob's Lab), independend of any distance between them.

How did I came up with such a nice description? Here you can find an article of the man who
used such text for the first time (Schrodinger, 1935), as a response to Einstein's EPR paper:

DISCUSSION OF PROBABILITY RELATIONS BETWEEN SEPARATED SYSTEMS. BY E. SCHRODINGER (1935)

(in the document, of the url above, you might scroll down a bit, to view the article)

If I may quote a nice paragraph from that article:

...
(when he is dicussing two remote members of an entangled system, or entanglement in general...)
...
It is rather discomforting that the theory should allow a system to be steered or piloted
into one or the other type of state at the experimenter's mercy in spite of his having no access to it.
This paper does not aim at a solution of the paradox, it rather adds to it, if possible. A hint as regards
the presumed obstacle will be found at the end.

Schrodinger already considered (or suspected) the case (as described in section 1.2), that the result
that Alice measures, instantaneously "steers" what Bob will find.

Althoug in section 1.2 we saw "steering" at work, I also like to try to discuss a modern "test" too,
involving steering, and this all under the "operational" definitions as listed below.

Many questions are left open at this point, among which are:

- Can Alice steer Bob?
- Can Bob steer Alice?
- Does "two-way" steering exists?
- What is the difference when pure systems and mixed systems are considered?
- Does all types of entangled systems, enable steering?

We are not too far of from possible answers. Let's next try to describe "entanglement" and "Bell non-locality".

Entanglement:

When 2 or more particles can be described as a "product state" (like equation 3), they are "seperable".
A measurement of an observable on one particle, is independent of the other particles.
You can always "seperate" the original ket (of a certain particle) from the "product state".

In many cases however, two or more particles are fully "intertwined" (with respect to some observable),
in such way, that you cannot seperate one particle from the other(s).
A measurement on one particle, effects the other particle(s) too.

A "state" as for example in equation 4, describes both particles (together in SpaceTime).
They truly have a common (inseperable) state.

A sort of "key" definition then seems to be: If you cannot write a combined system as a full product state,
then it's an entangled system.

But then still various forms exist, like partially entangled systems (partly seperable), or maximally
entangled systems (not seperable at all).

Bell non-locality:

This seems to apply for any situation, for which QM violates the Bell inequalities.
So, it seems to be a "very broad" description.

You might say that "entangled states" as in sections 1.2 and 1.4, "fall" under the non-locality description.
How about steering? Seems that this too, as a subset, is smaller than the notion of non-locality. But this is not correct.

The exact difference, or applicability, between steering, entanglement, and Bell nonlocality,
was not so much of a very "hard" issue in the minds of physicists, so it seems.
We have to admit that steering, entanglement and Bell nonlocality, seemed to have much overlap in their meanings.
Well, it proved to be not entirely true.

Then, in 2006, the following article appeared:

Steering, Entanglement, Nonlocality, and the EPR Paradox (2006)

by Wiseman, Jones, and Doherty.

They gave a pretty solid description for Steering, Entanglement, Nonlocality, in the sense of
when such term applies.
As the authors say themselves: they provided (sort of) operational definitions.

The statements above with respect to the relative "place" (as subsets or supersets) of steering, entanglement, and nonlocality,
were not corrects.
As the article points out:

Proposition 1:
-We need entanglement to enable quantum steering.
-But not all entangled systems provide "conditions" for quantum steering.

The above sounds rather logical, since quantum steering, or EPR steering, is pretty much involved,
and just seems to be a rather hard quality for true a non-classical phenomenon.
The authors formulate it this way: Steerable states are a strict subset of the entangled states.
So, if you would regard this from the perspective of Venn diagrams, then "Steerable states" lie within "entangled states".

Or, in other words: the existence of entanglement is necessary but not sufficient for steering.
Thus: "steering" is deeper than "just" entanglement, although entanglement is required.

Proposition 2:
-Steering is a strict superset of the states that can exhibit Bell-nonlocality.

This thus would imply that steering could happen in a local setting, which might be percieved as quite amazing.
In other words: in a Bell local setting (thus NOT nonlocal), steering is possible too.
Or, and this is important, some steerable states do not violate the Bell inequalities.

As we shall see a while later, if we would only consider pure states, the original equivalence holds
to a large extent. But considering mixed states too, leads to the propositions above.

I recommend to read (at least) the first page of this article. True, all these sorts of scientific papers are rather spicy,
but already on page one, the authors are able to explain what they want to achieve.

6.2 Entanglement Sudden Death:

Maybe the following contributes to evaluating entanglement..., or not.
However, it's an effect that has been observed (as of 2006) in certain situations.

"Early-stage disentanglement" or ESD, is often called "Entanglement Sudden Death" in order
to stress the rapid decay of entanglement of systems.

It does not involve "perse" all types of quantum systems, which are entangled. Ofcourse, any sort of state
will interact with the environment in time, and decoherence has traditionally been viewed as a "threat",
in for example, Quantum Computing.

ESD however, involves the very rapid decay of the "entangled" pairs of particles, that is, the entanglement
itself seem to dissipate very fast, maybe due to classical and/or quantum noise.
But the fast rate itself, which indeed has been measured for some systems, has surprised many physicists
working in the Quantum field.

Ofcourse, it is known that any system will at some time (one way or the other) interact with the environment.
Indeed, a general phenomenon as "decoherence" is almost unavoidable. It's simply not possible to fully
isolate a quantum system from the environment.
This even holds for a system in Vacuum. Even intrinsic quantum fluctuations has been suggested as a source for ESD.
However, many see as the source for the fast decay, the rather "normal" local noise, as e.g. background radiation.

Yu and Eberly have produced quite a few articles on the subject.

The sudden loss of entanglement between subsystems may be even explained in terms of how the environment
seems to select a "preferred basis" for the system, thus in effect aborting the entanglement.
Just like decoherence, ESD might also play a role in a newer interpretation of the measurement process.

Whether it is "noise" or something else, it's reported "quick rate" is still not fully understood.

A good overview (but not very simple) can be found in the following article:

Sudden Death of Entanglement (2009, Ting Yu, J. H. Eberly)

To make it still more mysterious, an "entanglement decay" might be followed by an "entanglement re-birth",
in systems, observed in some experimental setups, with the purpose of studying ESD.
A re-birth might happen in case of applying random noise, or when both systems are considered to be embedded
in a "bath" of noise or other sort of thermal background.

Many studies have been performed, including pure theoretical and experimental studies.
A more recent article, describing the behaviour of entanglement under random noise, can be found below:

Sudden death and rebirth of Entanglement for Different Dimensional Systems driven by a Classical Random External Field. (2015)

As usual, I am not suggesting that you read the complete article. This time, I invite you to go to the "Conclusion"
in the article, just to get a taste of the remarkable results.

6.3 Types of entanglement:

Ofcourse, this whole text is pretty much "lightweight", so if I can't find something,
it does not mean a lot.

So far, as I am able to observe, there is no "complete" method to truly systematically group entangled states
into clear categories.

There probably exist two main perspectives here.

The perspective of formal "Quantum Information Theory", in which, more than just occasionally,
the physics is abstracted away. This is not a black-and-white statement ofcourse.
Pure physics, that is, theoretical- and experimental research.

Both sciences deliver a wealth of knowledge, and often must "overlap", and often also are complementary
in initiating idea's and concepts.

So what types of entanglement, physicists have seen, or theoretici have conjectured?

How much the points below contribute to the understanding of "entanglement", I do not dare to say.
However, those point constitute "knowledge", so at least they must have something to say to us....
Anyway, let's see what this is about:

1. Pure- and mixed states can be entangled.

For pure states, a general statement is, that an entangled state is one that is not a product state.
Rather equivalent, is the statement: a state is called entangled if it is not separable.

Mixed states can be entangled too. This is somewhat more complex, and in section 5.4 I will try do
a lightweight discussion.

2. The REE "distance", or strength of entanglement.

"Relative Entropy of Entanglement" (REE) is based on the distance of the state to the closest separable state.
It is not really a distance, but the relative entropy of entanglement, E_R compared to the entropy of
the nearest, or most similar separable state.

In Physics Letters A, december 1999, Matthew J. Donalda, and Michal Horodecki, found that
if two states are "close" to each other, then so are their entanglements per particle pair,
if indeed they were going to be entangled.
Over the years after, the idea was getting more and more refined, leading to the notion of REE.

So, it's an abstract measure of the strength of entanglement. It's an area of active research.

Intuitively, it's not too hard to imagine that for nonentagled states, E_R = 0,
and for strong entangled states E_R -> 1.
So, in general, one might say that 0 ≤ E_R ≤ 1.

You could find arguments that this is a way, to classify entangled states.

3. Bi-particle and Multi-particle entanglement.

By itself, the distinction between a n=2 particle system, and a n > 2 system, is a way to classify
or to distinguish between types of entanglement.
Indeed, point 1 above, does not fully apply to multiparticle entanglement.
In a "n > 2" system we can have fully separable states ofcourse, and also fully entangled states
However, there also exists the notion of partially separable states.
In ket notation, you might think of an equation like this:

|Ψ> = |φ₁> ⊗ |ϕ_2,3>

and suppose we cannot seperate |ϕ_2,3> any further, then |Ψ>, which then is only separeated
in the factors |φ₁> and |ϕ_2,3>, is a partially separable state.

4. Classification according to polytopes.

When the number of particles (or entities) in a quantum system increases, the way entanglement might be
organized, is getting very complex. While with n=2 and n=3 systems, it's still quite manageble,
with n > 3, the complexity of possible entangled states, can get enourmous (exponentially with "n").

In 2012, an article appeared, in which the authors explicitly target multi-particle systems,
which can expose a large number of different forms of entanglement.

The authors showed that entanglement information of the system as a whole, can be obtained
from a single member particle.

The key is the following: The quantum correlation of the whole system "N", affects the single- or local particle
density matrices ρ(1),...,ρ(N) which relate to the reduced density matrices of the global quantum state.
Thus using information from one member alone, delivers information about the entanglement
of the global quantum state.

From the the reduced density matrices, which thus also correspond to the density matrices ρ(1),...,ρ(N)
of one member particle, the eigenvalues λ_N can be obtained.
Amazingly, using the relative sizes of λ_N, a geometric polyhedron can be constructed
which corresponds to an entanglement class. From this different geometric polyhedrons (visually like trapeziums)
at least stronger and weaker entanglement classes can be calculated.

Using a local member this way, you might say that this single member acts like "a witness"
to the global quantum state.

If you like more information, you might want to take a look at the original article
of the authors Walter, Doran, Gross, and Christandl:

Entanglement Polytopes: Multiparticle Entanglement from Single-Particle Information (2012)

Chapter 7. A few words on the measurement problem.

This will be very short section. But I hope to say something useful on this extensive subject.
Certainly, due to chapter 8, the role of the observer and the measurement problem, simply must be addressed.

It's in fact a very difficult subject, and many physicists and philosophers broke their heads on this stuff.

It's not really about "in-accuracies" in instruments and devices. One of core problems is the intrinscic probability
in QM, and certain rules which have proven to be in effect, such as the Heisenberg uncertainty relations.

And indeed, on top of this, is the problem of the exact role of the observer.
Do not underestimate the importance of that last statement.
There exists a fairly large number of (established) interpretations of QM, and the role of the observer varies
rather dramatically over some interpretations.

Whatever one's vision on QM is, it's rather unlikely that it is possible to detach the observer completely
from certain QM events and related observations, although undoubtly some people do believe so.

So, I think there are at least five or six points to consider:

Intrinscic probability of QM.
Uncertainty relations, and non-commuting observables.
Role of the observer.
Disturbive (strong) measurements vs "weak" measurements.
The quantum description of the measurement process.
Decoherence and pre-selection of states "near/in" the measuring device.

The problem is intrinsic to QM, and in many ways "un-classical".

7.1 Decoherence::

For example point 6, if you are familiar with "decoherence", then you know that a quantum system
will always interact with the enivironment. And in particular, at or near your measurement device,
decoherence takes place, and a process like "pre-selection" of states may take place.

In a somewhat exaggerated formulation: the specific environment of your lab, may "unravel"
your quantum system in a certain way, and dissipates certain other substates.
Could that vary over different measurement devices, and with different environments?
What does that say, in general, about experimental results?
The subject is still somewhat controversial.

A nice article is the following one. It's really quite large, but reading the first few pages
already gives a good taste on the subject. You can also search for articles of Zurek and co-workers.

Decoherence, the measurement problem, and interpretations of quantum mechanics (2003)

7.2 Role of the Observer:

And ofcourse, the famous or infamous problem of the role of the observer.
For example, are you and the measurement apparatus "connected" in some way? Maybe that sounds somewhat "hazy",
but some folks even study the psychological- and physiological state of the observer, with respect
to measurements. I don't dare to say anything on such studies, but you should not dismiss them.
A comprehensive analysis of making an observation, and certain choices, is very complex, and
is probably not fully undestood.

But there exists more factual and mathematical considerations. For example, what is generally understood
with the "Heisenberg uncertainty relations"?

7.3 A few words on the Heisenberg uncertainty relations:

Today, among physicists, a large degree of consensus exists on the following:

The Heisenberg uncertainty relations, have nothing to do with disturbing effects of the art
of making a measurement.
Here too, these relations are truly intrinsic in QM, independent of any measurement.

So why then a few words on the Heisenberg uncertainty relations?
In the early days of QM, especially Schrodinger’s approach, was popular, which essentially is integral
and differential calculus. Dirac's vector Ket (bra ket) notation emerged somewhat later.

The initial way to describe a quantum system, was by using "wave equations".
Some time before, it was realized that even particles could exhibit a wave-like character, and that
electromagnetic radiation, proved to have a particle-like character at certain observations.

Since a particle is quite localized, but not exactly localized, is why physicists introduced
the "Wave Packet" φ(r,t).
It's a superpostition too, but for a free particle, the components add up in such a way,
that the packet is grossly localized, like some sort of "gaussian" distribution,
with a maximum, and it fastly diminishes at the fringes.

The addition of waves to the packet, is like adding harmonics with a mode (related to the frequency) "p", each
with a relative amplitude "g(p)". The more wave components you would add, the more
the particle gets localized.
In the limit, the summation would become an integral like: (one dimension "x" only, time-independent)

φ(x) = 1/√ (2π ℏ) ∫_-∞^∞ g(p) e^ipx/ℏ dp (equation 28)

Now, the more "modes" you add, the "narrower" in position "x", the gaussian bell shaped wave packet becomes.
When the number of modes is really large, we get the integral "over" p, and a very localized "x" (position).

This "p" is related to the frequency of such mode, and represents the "momentum" of that mode.

So, with just a low number of p's, we have a very dispersed (wide) packet, in the "x" position.
If we have a large number of p's, we have a very narrow packet, in the "x" position.

In other words:
With a large variation in "p", we have a narrow "x".
With a large variation in "x", we have a narrow "p".

Using Operators, or just wave functions and Fourier transforms, it can be formalized.
But even in this stage, we can hopefully understand how Heisenberg came to his famous uncertainty principle
with respect to "momentum" and "position":

Δ p_x Δx ≥ ℏ/2 (only for x-dimension)

and in general:

Δ p Δ r ≥ ℏ/2

For these observables "p" and "r", it must hold that no matter how small you try to make the variation,
(the delta's), the product must always be larger than a certain number.

This result is simply a consequence of the QM wave approach to the position and momentum
of a particle.

Using the notion of Observables, and corresponding Operators/Matrices, it can be shown that
for non commuting observables (or Operators), it is not possible to measure them
simulteneously with unlimited precision.
For example, if you would pinpoint the position very precisely, the momentum is very in-precise,
and vice versa.

The principle applies to all non-commuting Observables, like Energy "E" and time "t",
and for example also for the spin components along the x and z directions,
usually notated as σ_x and σ_z.

This is why the principle can be of relevance in any discussion about measurements:
The principle is intrinsic to QM, however, the consequence (or collary) is that
it is not possible to measure non-commuting observables simulteneously, with unlimited precision.

7.4 The strange case of section 1.2, and the role of measurements:

The "strange case" of section 1.2, or the EPR paradox, represents one of the most fundamental,
and strangest scientific debates for about 80 years (or so), since 1935.

We can indeed read all about Bell, and experiments, and "steering/entanglement/non-locality", and many more
theoretical arguments, but the truth is..., there still is no full consensus reached among physicists.

Still to do...

Chapter 8. Some "Many Worlds" interpretations in QM.

Some rather remarkable idea's have been published rather recently on new interpretations in QM.
Although you cannot truly call them "theories", it's also true that you cannot call them "speculations" either,
since the authors provide a conceptual and mathematical framework.
So, in this text, I will simply call them "theories".

I like to spend a few words on a few of those "rather remarkable "theories", but I am afraid I cannot
classify this in any other way than saying that it resides in my "hobby sphere".
So, if there was any reader at all..., this stuff is one of my hobbies, so to speak.

In this chapter, I like to spend a few words on rather new "parallel universe" theories,
and, in chapter 8, a few words on recent resarch on micro blackholes/wormholes, at the Planck scale.

However, I will start with the "Grandfather" theory of Many Worlds in QM, that is, Everett's MWI theory,
which was published in 1957. Much later, around 2010 and 2014, newer models were introduced.

8.1 Some important Interpretations of QM:

I should start, with a "sort of" listing of established interpretations.

There are quite a few of them, and a few represent a rather dramatic deviation from the (more or less)
standard Copenhagen interpretation. Here are a few of them:

The Copenhagen interpretation (or in a new jacket).
QM with Hidden variables and local realism.
QM with non-locality.
Decoherence as a successor to the specific Copenhagen "collapse".
Everett's Many Worlds Interpretation (MWI).
The Many Minds Interpretation.
Poirier's "Waveless" Classical Interpretation (MIW).
Quantum hydrodynamics and Trajectory analysis (e.g. Madelung, Holland).
The transactional interpretation.
Time symmetric theories.
de Broglie - Bohm "Pilot Wave" Interpretation (PWI).
possibly also the "Two State Vector Formalism".

I would not dare to say that this listing is complete. Furthermore, some of them quite overlap,
but at certain issues's, there are fundamental differences (like nonlocality, and local realism).

There are also quite subtle differences. For example, a Ket vector, or wavefunction, can be interpreted as
"just" a "working vehicle", but it's true existence might be doubted. Some other interpretations see them as a
real representation of entities and properties.

There are no "polls" of who likes which the most, but as I observe it, it seems as if
a lot of folks are attracted by some features of the Broglie - Bohm "Pilot Wave" Interpretation.

8.2 The "Grandfather" theory of Parallel universes in QM: Everett's MWI (1957).

It's impossible not to spend a few words on the original Parallel universe theory in QM.
It was formulated by Everett, in 1957, although Schrodinger in 1952 already hinted
to such an idea, in the form of simultaneous existing outcomes.

Everett published his theory as a Phd thesis, called the "Relative State Formulation of Quantum Mechanics",
under a certain degree of guidance by his mentor Wheeler.
Slightly later, also due to the promotional work of DeWitt, it became know as the "Many Worlds Interpretation" or MWI.

In this theory, the wavefunction is a real existing entity, and forms the basis for all entities.
The key of his idea is the following. Although in the simple example below, a ket vector notation is used,
it's rather equivalent to a wave function setting.

If you would consider the following state (rather equivalent to a wave function interpretation):

|φ> = a₁|u₁> + a₂|u₂> + a₃|u₃>

and in an observation, you will find the state |u₂>, then in the Copenhagen interpretation
it is said that the state |φ> collapsed (or was projected) to the state |u₂>, with a probability
that relate to "a₂".
It's quite possible then, that you could have found, for example, |u₃>, with a probability
that relate to "a₃". However, you found one outcome, and the former state is destroyed.

In Everett's theory, all 3 outcomes are realized. So, if you performed the measurement, then 3 different
states "forks off" and undergo their further evolution.

So, following the example above, the quantum superposition of the combined "observer-state" and
"observed object-state" wavefunction, will resolve into three "relative states", completely independed
from each other.
Hence the notion of "branched off" worlds, or "Many Worlds".

If you want to know more, then here is a (new version) pdf of Everett's original article:

Relative State Formulation of Quantum Mechanics (1957)

Indeed, it's not a strange framework at all ! Although his idea did not find any support at first, it is true that
as from the late '60's, up to the '90's, it became quite popular, and many physicists considered it
to be a viable interpretation. Some even considered it to be the best interpretation thus far.

Note also that Everett's theory is a no-collapse formulation of QM, quite unlike the Copenhagen interpretation.

-However, the popularity declined over later years, and quite a few articles expressed to have found
some (supposedly) inconsistencies in MWI. As is rather usual in science, not all physicist turned around,
and still some are very sceptical on those (supposedly) inconsistencies in MWI.
Below I provide some links to arxiv articles, pro- and contra MWI, which will demonstrate some of
those inconsistencies.

-New theoretical paths in general in physics, and in QM specifically, probably did not helped MWI much.
For example, around the '80's, the theoretical principle of "decoherence" was discovered (Zeh, Zurek),
which provided a new way on how a wavefunction would interact with it's environment, like a measuring device,
or the "environment" in general.
But it must be said that "decoherence" shows some important similarities with MWI, but then ofcourse
without the branching into other Worlds.

But even up to this very day, we have physicist who publish arguments in favour of MWI.
It's still a valid interpretation of QM, and as usual, some folks like it, and some don't.

I highly recommend to invest some time to explore some great articles in MWI.

The following short article tells us about some inconsistensies in MWI, which ofcourse, you don't need to take
for granted. But this is an exceptionally nice and sometimes humouristic text, and I am sure you like it.
By the way, you will absolutely learn a lot of MWI, by reading this article.

THE INTERPRETATION OF QUANTUM MECHANICS: MANY WORLDS OR MANY WORDS? (Tegmark)

However, the same author, at another moment, produces an article which you can call rather pro MWI.
Since this one explains MWI rather well, I like to list it here too:

Many lives in many worlds (Tegmark)

The next article is from Vaidman, and is rather against QM non-locality, and the article expresses lot's
of motivation, why MWI should the theory of choice.

Quantum Theory and Determinism (Vaidman)

8.3 Modern Parallel Universe theories in QM: beyond 2005

In 2004, a rather remarkable article appeared, from P. Holland, titled:

Computing the wavefunction from trajectories: particle and wave pictures in quantum mechanics and their relation.

It looks quite "non-quantum" to talk about precisely defined spacetime trajectories. At least, this is a common
view among some people in the field. Also, the common interpretation of the "wavefunction", works against
such a view too.
However, plenty of physicists would not not agree to such statements.

Ofcourse, it also depends on which particular "Interpretation" of QM, you favour the most. For example, the Broglie-Bohm
"Pilot Wave" Interpretation seems to re-introduce some classical features again. Not literally, but essentially
the theory states that we have a real particle, with an accompanying pilot wave, which can be regarded
as a "velocity field" guiding the particle.
The pilot wave then (sort of) has "the look and feel" of the Copenhagen wavefunction.
One remarkable thing is, that if you would have the initial conditions, you can further calculate
future positions precisely.

Hollands article is not part of my intended discussion, but ofcourse the article is great.

In short, his article seem to deal about the following:

Holland notes that in QM, in certain cases, certain results of hydrodynamics and fluid mechanics, can help in
calculating particle positions. Such an idea was first pointed out by Madelung.
In such approaches, the QM probability density then looks similar to the fluid density.
Then he works out all sorts of calculations, using classical Euler-Lagrange equations
to calculate the trajectories of the fluid particles as functions of their initial coordinates,
as if you would regard such initial "ensemble" as the initial wavefunction.

In 2010, Poirier published a remarkable article too, called Bohmian mechanics without pilot waves.
It was first published in Chemical Physics, Volume 370, May 2010.

An online copy can be found here (www.researchgate.net).

The amazing thing is, that Poirier has arguments to "strip away" the Pilot wave, from the particle,
when considering Bohemian mechanics (Broglie-Bohm Pilot wave mechanics).

As he says: can a "trajectory ensemble itself", be possible too, in the sense that it obeys Schrodinger's equation,
and therby making no explicit or implicit reference to an external pilot wave?

Then, using kinematics and rather classical looking differential equations, and using results from
(especially) Holland's work (I think), he arrives at the remarkable proposition:
It seems possible to replace the wavefunction by a real-valued trajectory "field", furher defined
by a "trajectory density weighting" function. The orthodox complex valued wavefunction is unneccessary.

Note this line of thinking seems like a reduction of QM, to a semi-classical statistical Mechanics.

From this on, he is not so far away anymore from his final MIW interpretation, that is,
the "Many Interacting Worlds" (MIW) interpretation.

The "Many Interacting Worlds" (MIW) interpretation.

It seems that Poirier and Schiff were first (2012) in devising a true MIW model.
However, in 2014, Wiseman, Hall, and Deckert used a slightly other approach to reach
(I think) a better model, since they can avoid a continuum of "many interacting worlds".

Generally, MIW essentially means this.

The proposition is essentially, to have "n" Classical parallel "universes", where in one such Universe,
an entity (like a particle), has well defined (sharp) observables (like position), all the time.
However, there exists a variation over such observables, in the various different Universes,
in such a way, that an observer experiences the illusion of QM, like the implicit
spreading of the wavefunction (which leads to an uncertainty in e.g. the position).

So, each "World" it has it's own "private" sharp values for that observable, but since there exists variation
between "Worlds", we percieve the illusion of a composed State vector where the same
experiment might show different values all the time.
In fact, in MIW, there is no state vector, no ket's, no wavefunctions.

This is a remarkable different view, compared to Everett's MWI.
Most notably, the differences are:

In Everett's MWI, the wave function is real and plays a central role. In MIW, the wave function
does not exist, and in fact MIW is a sort of QM "without the wavefunction".
In Everett's MWI, universes branches off due to the fact that superimposed waves each interact
with the environment.
In Poirier's MIW, there is no branching, no wavefunctions, but the many worlds may "interact".
Everett's MWI uses standard Quantum Mechanics (wavefunctions, probabilities etc..), however Poirier
uses established rules from many parts of physics, whereas many are simply familiar classical, like trajectories,
position/impuls, Lagrangian etc.., where paths over different trajectories relate to the
Many Interacting Worlds.

A short survey of one of the MIW theories:

I think the approach of Hall, Deckert, and Wiseman is the best one, up to this day,
for "constructing" a MIW theory.

Their original article, published in 2014, is the following:

Quantum phenomena modelled by interactions between many classical worlds (2014).

As usual, I advice to read the whole article. However, even only reading the first 2 pages (the Introduction),
will tell you what Poirier's MIW theory tried to prove, and what the plan of the authors is, to provide
for a better alternative derivation, in order to arrive to an (almost) equivalent MIW theory.

In this paragraph, I will try to say something useful on this improved MIW theory.

Essentially, it goes like this:

If you would have "n" particles, then you might define a (single) "configuration space" for that set of particles, which space
can be described by the vector:

Q = {q¹,q²,...q^K}

The number "k"depends on which number of dimensions, and which attributes, you would take into consideration.
The vector Q, then describes the set of particles, for example, with respect to position.
The authors also see this vector as to be equal to the "configuration space" of the set of particles, and also
called the "world space" for that set.

For Q, at a certain "t", you could define a position as Q_R= {r¹,r²,...,r^K}.

Sofar, there is nothing too special here.

As an ïnterim object" it's possible to define a Probability Density P(Q_R) = |φ(Q_R)|
This would then express the distribution of the positions of the particles {q¹,q²,...q^K}.

Next, they make the assumption that "N" of such worldspaces, thus Q₁.. Q_N could exist,
or if you like, could be postulated.

If we would find that a credible assumption, we then would have "N" configuration spaces for those "n" particles,
expressed as the collection {Q₁,..., Q_N}.
"N" is not aboslutely fixed, and might even go to ïnfinity.
The remarkable thing then is, and what is not showed here (see the article above), that a sort of local repulsive force
can be derived, between those worldspaces Q_i. A very important aspect of that local force is,
that it is only of relevance when Q_l and Q_m are "close", with respect φ(Q_R).

If the worldspaces Q_i are now viewed as seperate "Universes", which might be percieved as a rather bold proposition,
then at least the force between "close" Universes would be repulsive.
The authors then apply this theory to e.g. The Ehrenfest theorem, spreading of the wave packet, tunneling,
and a few other QM effects. Remarkably, they indeed seem to succeed in their examples.

For example, the spreading of a particle over "configuration spaces" (the Universes),
gives the illusion of a wave packet.
However, the Universes themselves, are fully classical, and all observables have "sharp" values at all times.

Other Multiple Worlds theories in the scope of QM:

- Many Minds theory:

Everett's MWI, and Poirier MIW, are not the only propositions of Many Worlds, in the scope of QM.
For example, the "Many Minds" theory of Dieter Zeh, is one important line of thinking.
This theory, actively integrates the observer (or the mind of the observer) in the process of observation,
and interpretation.
It maybe of interest of people working in the field of psychology, or any sort of neural science.

Other Multiple Worlds theories not directly in the scope of QM:

Outside the immediate scope of QM, some other "Parallel Universe" models were deviced.
Here, you might think of "Hubble Volumes", or "Chaotic Inflation" and some other models.
Some time back, I tried to make an overview in some sort of note...
I you want to try it, you can use this link.

Even if you do not buy these classes of theories..., they certainly gives us a different perspective on QM,
and that's valuable anyway.

Chapter 9. A few words on entropy.

9.1 What is entropy?

Entropy is a physical quantity, which is especially used in statistical mechanics, and thermodynamics.
Originally, you might say that it especially useful in macroscopic systems, having many entities,
like gasmolecules in a room, or cilinder etc..

In ordinary statistical mechanics, the "entropy" (S) provides us a pointer to the measure of the
multiplicity of microstates that "sits" behind a particular macrostate.

For example, you have a large number of gas atoms, where all of them are somehow magically placed
in one single corner of a vacuum room, and you release the "ban", you would observe that immediately the gas will
be distributed all over that volume of that room.

In this example, we can talk about on how many different ways the atoms or molecules can be arranged.
Just after the "ban" was released, the number of arrangements increased enormously.
When energy is increased into a system, or by "adding" new arrangements (microstates), entropy is increased.

The second law of thermodynamics says that the entropy of a closed system, will never decrease.

In fact, that's a remarkable law, and it even may suggest that Nature wants to promote disorder
over order, in an isolated system. Ofcourse, we need to be very careful using such sort of statements.

A mathematical relation has been found that relates the Entropy "S", to the number of microstates "W":

S ∽ ln(W) (equation 29)

In thermodynamics, chemistry, the usual equation is:

S = k_B ln(W) (equation 30)

where k_B is Boltzmann constant.

So, entropy in general seems to be a macroscopic quantity...?

True, but it's applicability has turned out to be extremely general. For example, just think of
the all of the subwaves that contribute (or sum up) to a "wavefunction" or "wavepacket".
So, entropy can be used here as well !
And it uses the same "natural logarithm", that is, the "ln" function.

Note:

Note that equations 29, or 30, have a "certain" resemblence to Shannon's Law (1948) of data communication
technologies, and related sciences:

S = B log₂ (1 + S/N) (equation 31)

where C is the maximum attainable error-free data speed in bps that can be handled by a communication channel,
B is the bandwidth of the channel in Hz, and S/N is the signal-to-noise ratio of that communication channel.
Also note that "log_N(x)" is a logarithmic function, based on "N", while "ln" is the natural
logarithmic function, based on "e".

Although the above relation stems from 1948, modern theories consider variables, symbols, and distribitions.
Indeed, "entropy" has a very central meaning as well in Information Sciences.

9.2 Why entropy can be expressed as a "ln(W)" function.

Many processes in Nature can be expressed in the form of an "e^x" function, or a "ln(x)" function.
Both are indeed quite remarkable.

-In the figure below, you see "e^x", which starts out rather slow, but as "x" increases,
it almost explosively starts to climb "upwards".

- The inverse function of "e^x", is "ln(x)". This one is exactly the "mirrored" function of
"e^x", with respect to the line y=x. The function "ln(x)" starts out climbing extremely rapidly,
but as "x" increases, it climbs less and less steeply, until it almost (but not quite) reaches a nearly horizontal slope.

The function ln(x) is "natural". For example, if you have a certain amount of radiactive material,
then it will decay over time.
The amount of active material that's left after some period "t", can be expressed as a function of
of the "half-live" or "mean lifetime" τ, that is "ln(τ)".

How can we make plausible, that ln(W) is indeed related to the entropy S of a system?

Suppose you add microstates to a system. Suppose you originally had 10 states. Adding for example 3 additional states,
is a rather big increase in entropy. Indeed 10+3 is a relevant change. However, once you reach the number of 1000 states,
and again add 3 microstates, then 1000+3 is not a large increase in entropy anymore. This behaviour reflects precisely
the graph of an ln(W) function. The slope gets less and less, if W increases.

9.3 Entropy of a quantum system.

The text below, perhaps, may strike you as a bit weird.

By now, we know what we must understand about a pure- or mixed state. As it will turn out, the entropy
of a pure state is zero. That's remarkable, since we know that, in general, it is a superposition.

First, in general, about the concept of entropy which is involved here, is the Von Neumann entropy,
which is shown below. This framework was, and still is, accepted, since it works.

Secondly, considering a pure state describing superposition, we talk about quantum superpostion !
Indeed, we can write a ket (statevector) as a sum of eigenstates (basis vectors), but we don't know
really what it is in terms of classical physics. Now, you might say that this is an incorrect statement, since
we can attain a probability of finding a certain eigenstate.
That is still true, however, a quantum superpostion is still something very different from
a statistical ensemble, which a mixed state is supposed to describe.
Indeed, in case of a statistical ensemble, we have physical states, on which we can apply
true classical statistics, since that statistical ensemble "looks" exactly like an ordinary statistical ensemble.
Remember from chapter 4, that a mixed state is a statistical mixture of pure states, while superposition
refers to a state carrying some other states simultaneously.

It might be relatively hard to understand (or not). Let's also take a look at the statements below:

-Statement:
Entropy, as a quantity that in general relates to distinguishable microstates, applies well to mixed states.
It simply looks like a classical statistical ensemble, on which we can apply the term "entropy".

-Statement:
Entropy, as applied to quantum superposition, as is meant in a pure state, should indeed return "0",
also since we cannot say anything definite of the state (unless we observe it).
If we observe it, the former state is lost. After measurement, we simply have one eigenstate.

-Statement:
Since 1948, entropy started to be used in terms of "mathematical communication theory", as information entropy.
Folks started to think that the physical degree of distinguishable states of a system (statistical entropy),
is related to its "information" (information entropy).

For a pure state in superposition, we only know that |Ψ> = |Ψ>.
If we observe it (Copenhagen talk), we are left with a certain eigenstate.

A mixed state is a true statistical ensemble.

Hopefully, we are in the clear now, for how we must interpret "entropy" for pure- and mixed states.

Let's see how this part of QM works.

Chapter 10. ER=EPR models.

Personally, I find these theories (or hypotheses) quite appealing.

For example, do you remember the Inflationary Universe theorie(s)?

Those are regarded as the most plausible theories today, to explain (or describe) the origin and evolution
of the Universe. At the tiniest fragment of the start of time, at the earliest phase (maybe around 10^-42 sec),
a Quantum Fluctuation (according to the theory), gave rise to a pre-form of SpaceTime and a precursor of Gravity.
Then, for very short period, an exponential inflation of SpaceTime took place.
In case you are not familiar to Inflationary Universe theorie(s), you might want to do a websearch first on that topic.

A rather bold assumption might be made:

An "embedded" relation, even today, between SpaceTime "components" and for example entanglement,
might be possible, according to these lines of thought.

Thus the current theories try to establish plausible models for SpaceTime, Gravity, and indeed, QM effects,
like entanglement.

-Be warned though, that most physicists seem (or probably really are), very weary or sceptical on those models.
It's true that only a rather select company of theoretical physicists are actively working on these models.

-Be double warned, since the entities they try to study (from a theoretical framework), are on the smallest scale possible,
that is, typically in the order of the "Planck length". This scale is fully "out-of-reach" for direct experimental work,
and the present day particle accelerators, lack many billions of orders of magnitude of Energy, to probe such small scales.
Since this is really a fact, all work done up to now, is purely "theoretical".

-Or, be not warned at all, since science is just simply "always in progress" and sometimes
we have an established theory, which does not hold up anymore to new experimental results, or new, sufficiently backed,
theoretical considerations, especially if established theories fail in certain domains.

10.1 General overview.

Let's start with a sort of overview, of which sort of "ideas" emerged, and when.
In the second section, I will try to go somewhat deeper into the theories, but for now,
having this sort of overview helps to put stuff in perspective.

Entanglement seems to have (or might have) a rather large area of application. At least, that is how many
theoretical physicists look at it nowadays, especially since the 2000's, and even more so since 2013.

But long time ago, but after QM stood firmly in the physics books, science "went on", ofcourse.
All those years during the 50's, 60's etc.., in the former century, enormous progress was made
in astronomy, particle physics, theoretical physics etc..

It's impossible to say anything useful here, ofcourse, unless one is planning to write a book on the
theme of "progress in physics, during the '50's up to today...

It's seems fair to say that string theory (since the '80s), AdS/CFS, Quantum field theories, Quantum Gravity,
SpaceTime models, Cosmological models etc..., kept people busy for a long time.
I can't say that those theories, ultimately, came together, but certain results from all of them,
created an atmosphere (so to speak), to bring in entanglement into the picture.

Holographic principle and Entanglement Entropy.

Here is an article (2006) from Shinsei Ryu and Tadashi Takayanagi, where the authors link "entanglement"
from a holographic perspective on entropy from AdS/CFT:

Holographic Derivation of Entanglement Entropy from AdS/CFT (2006)

This link above, is for illustrational purposes only. You can read it ofcourse, but it's rather involved.
I simply only like to create a (although on a nano-scale) small historical perspective too.
That's why I listed the article above.

AdS/CFT is a specific SpaceTime model, and in some respects compatible with string theory, in the sense
of a certain correspondence.
The article seems to succeed in deriving entanglement entropy from minimal surfaces (one dimension less),
in some form of AdS space of certain dimensions.
It's all highly theoretical, but it's getting quite concrete in using entanglement in SpaceTime models.

Entanglement and the creation of SpaceTime geometries.

More importantly is the following article. And this time, you are encouraged to read (or browse through) it.
Possibly, it's the first concrete article, postulating "entanglement" as the cement in the SpaceTime fabric.
It's the classic article from M. van Raamsdonk (2010), and you can find it here:

Building up spacetime with quantum entanglement (2010).

Ofcourse, the article is partly inspired from former work, like e.g. articles of Maldacena, but nevertheless,
as far as I know, it explicitly uses quantum entanglement to build geometries of SpaceTimes.

Ofcourse, a various idea's on the dicreteness of SpaceTime, spin-networks, loops, already circulated quite some
time before (e.g. some idea's of Penrose and others).
But finding entanglement as the fundamental sculpter of the geometry of SpaceTime, is quite new (or "new").

Essentially, using the methods from Ryu and Takayanagi, Raamsdonk shows that if you would slowly tear down
entanglement from a certain AdS model of SpaceTime, then when entanglement is finally reduced to nothing,
this SpaceTime will be no more than fully disjoint parts of SpaceTime.
Hereby, making plausible that the principle of entanglement, creates SpaceTime.

Remember, this was only 2010. The methods of Ryu and Takayanagi stems around 2006. Some fundamental ideas
of Malcedena were from (about) 1997.
For (about) 2010 to today, the ideas are really alive throughout the community
of theoretical physicists, and many refinements and explorations were made.

Entangled Black Holes.

As a slightly other line of thought, theoretical explorations were done on the subject of "entangled black holes".

Although Einstein's Relativity theory allows for the principle of Wormholes (also called the "Einstein-Rosen" bridge,
or the ER bridge), Juan Maldacena and Leonard Susskind, introduced the idea of applying entanglement on pairs
of Black Holes.

One of their articles is the following:

Cool horizons for entangled black holes (2013).

In this facinating article, the authors explore the idea that an "Einstein Rosen bridge" between two black holes,
might be very similar to the EPR-like correlations as seen in many applications and experiments in QM.
In other words: entangled Black Holes.

In the same article, the authors say that it's "tempting" to suspect that any correlation through entanglement,
has it's roots in ER bridges (or wormholes) on a microscopic scale.
Hereby, they "rooted" the idea of ER=EPR, which made quite a few folks enthousiastic for that concept.

The article is quite spicy, if not at least some core ideas are introduced.
That's my challenge for the sections below.
I am afraid that it all will be a bit lengthy....

EPR and Steering in Quantum Mechanics (QM)...

Chapter 1. Introduction to the EPR paradox.

1.1 Introduction and Some background information.

|φ> = a|1> + b|0> (equation 1)

|φ> = a|0> + b|1> + c|2> (equation 2)

|Ψ> = |φ1> ⊗ |φ2> = a00 |00> + a01 |01> + a10 |10> + a11 |11> (equation 3)

|Ψ> = 1/√2 . ( |01> + |10> ) (equation 4)

1.2. Describing "the apparent strange case" of the EPR paradox.

- One (temporary) explanation with a certain consensus among physicists:

- A Modern understanding (without full consensus among physicists):

Another, but related, puzzle:

ρ = ½ ( |0 > < 0| + |1 > < 1| ) (equation 5)

-Other views:

1.3 EPR and possible alternatives.

Chapter 2: a few words on the first article on Quantum Teleportation (QT) (1993).

What is it, and how does an experimental might look like?

| θ123> = |φ1> ⊗ |Ψ2,3 > (equation 6)

Chapter 3. A nano introduction to QM, with regards to operations and notations.

3.1 The Wavefunction and Probabilities.

3.1.1 Description of the Wavefunction:

p = ℎ / λ (equation 9)

Ψ(r,t) = ei(kr-ωt) (equation 10)

Ψ(x) = 1/√ (2π ℏ) ∫-∞∞ g(k) eikx dk (equation 11)

Ψ(x) = 1/√ (2π ℏ) Σ An eikx (equation 12)

3.1.2. Probabilities:

E ∽ |Ψ|2

ρ(r,t) = Ψ(r,t)*Ψ(r,t) = |Ψ(r,t)|2 (equation 13)

P(somewhere on x-axis) = ∫-∞∞ |Ψ(x,t)|2 dx = 1 (equation 14) P(x in [a,b]) = ∫ab |Ψ(x,t)|2 dx (equation 15)

P(somewhere in Space) = ∫-∞∞ |Ψ(r,t)|2 dr = 1 (equation 16)

3.2 Schrodingers equation.

ΨN(x) = C eikNx

3.3 Common operations and notations.

Proposition 1: (1): < A|B > : is the inner product, of vectors or Kets.

-Inner product of two kets < A|B >

-Inner product of a ket, with a basivector: <ui|φ>

|φ> = a1|u1> + a2|u2> + a3|u3>

<u2|φ> = a1<u2|u1> + a2<u2|u2> + a3<u2|u3> = a2

-Operators: The operator "O", as in O|B >=|C >, meaning O operating on ket |B >, produces the ket |C >

Proposition 2: |B > < A| : usually corresponds to a matrix or linear operator.

Proposition 3: |φ > < φ| : corresponds to what is called the "Density" matrix of a pure state.

Proposition 4: < φ| O |φ >: corresponds to the expectation value of the observable O.

ρ(x) = Ψ*(x)Ψ(x)

< x > = ∫-∞∞ Ψ*(x)Ψ(x) dx

< Q > = ∫-∞∞ Ψ*(x) Q(x) Ψ(x) dx (equation 21)

Proposition 5: The "Trace" of an Operator is: Tr(O) = Σ < ui|O|ui >.

3.4 A few words on Dirac's notation

3.4.1 Some general observations:

|φ> = Σ ci|ui>

ci= < ui|φ >

|v > = c1 |i > + c2 |j > = < i |v > |i > + < j |v > |j > = |i> < i |v > + |j> < j |v >

|i > < i| + |j > < j| = 1

Σ |ui > < ui| = I

|φ> = Σ ci|ui>

|φ > = Σ |ui > < ui|φ >

3.4.2 Interpretations of the Bra:

< ui|φ > = ci

3.4.3 Some well known states you may see in articles:

|Ψ> = a|1> + b|0>

|Ψ> = a|0> + b|1> + c|2>

|Ψ> = |φ1> ⊗ |φ2> = a00 |00> + a01 |01> + a10 |10> + a11 |11>

|Ψ> = a1 |000> + a2 |001> + .... + a8 |111>

|Ψ> = 1/√8 |000> + 1/√8 |001> + 1/√8 |010> + 1/√8 |100> + 1/√8 |110> + 1/√8 |011> + 1/√8 |101> + 1/√8 |111>

|Ψ1> = 1/√2 ( |00> + |11> ) |Ψ2> = 1/√2 ( |00> - |11> ) |Ψ3> = 1/√2 ( |01> + |10> ) |Ψ4> = 1/√2 ( |01> - |10> )

|Ψ> = 1/√2 ( |01> - |10> )

|GHZ> = 1/√2 ( |000> - |111> )

3.4.4 Some more words on Operators, Matrices, Observables:

A |Ψ> = |φ> (equation 22)

A |Ψ> = a |Ψ> (equation 23)

< φ | = < Ψ| A† (equation 24)

A = A† (equation 25)

Chapter 4. Meaning of pure states and mixed states.

4.1 A few words on "pure states":

|φ> = Σ ai|ui> (equation 26)

4.2 A few words on mixed states:

ρ = 1/4 |a > < a| + 3/4 |b > < b| (equation 27)

4.3 What about our entangled two partice system?:

|Ψ> = 1/√2 . ( |01> + |10> )

Chapter 5. The inequalities of Bell, or Bell's theorem.

5.1 The original formulation.

5.2 Newer insights on the Bell inequalities and LHV's.

|Ψ> = |φ₁> ⊗ |φ₂> = a₀₀ |00> + a₀₁ |01> + a₁₀ |10> + a₁₁ |11> (equation 3)

| θ₁₂₃> = |φ₁> ⊗ |Ψ_2,3 > (equation 6)

Ψ(r,t) = e^i(kr-ωt) (equation 10)

Ψ(x) = 1/√ (2π ℏ) ∫_-∞^∞ g(k) e^ikx dk (equation 11)

Ψ(x) = 1/√ (2π ℏ) Σ A_n e^ikx (equation 12)

E ∽ |Ψ|²

ρ(r,t) = Ψ(r,t)^*Ψ(r,t) = |Ψ(r,t)|² (equation 13)

P(somewhere on x-axis) = ∫_-∞^∞ |Ψ(x,t)|² dx = 1 (equation 14)

P(x in [a,b]) = ∫_a^b |Ψ(x,t)|² dx (equation 15)

P(somewhere in Space) = ∫_-∞^∞ |Ψ(r,t)|² dr = 1 (equation 16)

Ψ_N(x) = C e^ik_Nx

-Inner product of a ket, with a basivector: <u_i|φ>

|φ> = a₁|u₁> + a₂|u₂> + a₃|u₃>

<u₂|φ> = a₁<u₂|u₁> + a₂<u₂|u₂> + a₃<u₂|u₃> = a₂

ρ(x) = Ψ^*(x)Ψ(x)

< x > = ∫_-∞^∞ Ψ^*(x)Ψ(x) dx

< Q > = ∫_-∞^∞ Ψ^*(x) Q(x) Ψ(x) dx (equation 21)

Proposition 5: The "Trace" of an Operator is: Tr(O) = Σ < u_i|O|u_i >.

|φ> = Σ c_i|u_i>

c_i= < u_i|φ >

|v > = c₁ |i > + c₂ |j > = < i |v > |i > + < j |v > |j > = |i> < i |v > + |j> < j |v >

Σ |u_i > < u_i| = I

|φ> = Σ c_i|u_i>

|φ > = Σ |u_i > < u_i|φ >

< u_i|φ > = c_i

|Ψ> = |φ₁> ⊗ |φ₂> = a₀₀ |00> + a₀₁ |01> + a₁₀ |10> + a₁₁ |11>

|Ψ> = a₁ |000> + a₂ |001> + .... + a₈ |111>

|Ψ¹> = 1/√2 ( |00> + |11> )
|Ψ²> = 1/√2 ( |00> - |11> )
|Ψ³> = 1/√2 ( |01> + |10> )
|Ψ⁴> = 1/√2 ( |01> - |10> )

< φ | = < Ψ| A^† (equation 24)

A = A^† (equation 25)

|φ> = Σ a_i|u_i> (equation 26)

|Ψ> = |φ₁> ⊗ |ϕ_2,3>

φ(x) = 1/√ (2π ℏ) ∫_-∞^∞ g(p) e^ipx/ℏ dp (equation 28)

Δ p_x Δx ≥ ℏ/2 (only for x-dimension)

Q = {q¹,q²,...q^K}

S = k_B ln(W) (equation 30)

S = B log₂ (1 + S/N) (equation 31)