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By: Albert van der Sel

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Quantum Mechanics (QM) is a real working horse in the arena of

But countless physicsts, philosophers (and who knows what else) broke their heads on the "interpretations" of QM.

Indeed, the stochastic character of QM, and the "fuzzyness" at times, is sometimes hard to "link" with "reality" (whatever

Maybe modern insights may have shed lights on some effects, which were formerly not well understood (like "decoherence" replaces the "collapse"),

but plenty effects remain which needs much more study.

Let's try to find out what "Quantum Teleportation" actually is, and, let's try a very simple approach.

I suppose that I

Let's start. But, I do not aim for a short note anymore. I'am afraid it will be rather lengthy (but not too long...)

QT is the "transfer" of the Quantum state of an observable of a Quantum System, like the "spin" of a qubit.

What a qubit is, will be explained later in this note, but for now, you may think of a particle like an electron,

which has a "spin" which is the superpostion (sum) of two basis states ("up" and "down"), simultaneously.

So, the "state" of such a qubit "q

remote from location "A".

Since only a "state" is transferred, QT has

As we will see later, this is not a "cloning" process, since we will see that the state of "q

One prerequisite is the existence of a socalled non-classical "EPR" channel between entangled particles. To actually transfer the information

(bits) of the state of a qubit, a

So, since a classical channel is required, no "laws" from Einstein's relativity theories are challenged or broken.

As a matter of fact, you will see that QT is not that terribly "mysterious", after all. However, the "EPR" channel, up to this day, still is.

Note:

Although I consider this humble note to be "reasonable" for it's content, it has to be labeled to be "fun stuff", since creating a (hopefully)

entertaining note, is actually my main motive. But, I tried to get the facts straight..., so..., I hope it all works out alright.

Contents:

Chapter 1. Just a few milestones, to get to the notion of the "wavefuction"..

1.1 Schrödinger's equation

1.2 Quanta.

1.3 Double slit experiment: interference patterns.

1.4 Superpositions, and wave-

Chapter 2. Hilbert space/Vector space. Or, the representation of a "state".

2.1 General representation of a "state vector"

2.2 Collapse of the State Vector, or Decoherence, or MWI, or something else...

2.3 Example of a 2 statevector: representation of a "qubit".

Chapter 3. Quantum Information versus Classical Information.

Chapter 4. Collapse of the statevector replaced by Decoherence.

Chapter 5. Some remarks on Commuting/Non commuting observables & Heisenberg & measurements.

5.1 Commuting and Non commuting observables.

5.2 Heisenberg uncertainty principle.

Chapter 6. Product states and Quantum Entanglement.

6.1 Simple "pure states" and simple "seperable states".

6.2 Quantum Entangled states.

Chapter 7. Quantum Teleportation.

7.1 Original setup.

I could not find a real good one. Then, when a looked at the older literature on Quantum Mechanics (QM), like books from the '70s. They often start

with Schrödingers equation, and then see how it works out on physical systems like the Hydrogen atom.

More modern literature often starts out with describing Hilbert spaces,and Dirac notation.

Still others, start with some great examples of experimental observations from the late 1800s, early 1900s.

So, finally, I got an idea: why not just make a start, using a nice 'soup' of all of those sort of intro's? Great!

Let's do it in "example" form. But, keep it short.

But please note this: If you see "math" you don't like,

In fact, it's a rather simple equation (if you see it), but with rather large consequences. It's this: p = h / λ (where h is Planck's constant).

Now, "momentum", at that time, was considered to be a true

which stuff like radio waves have. For example, a rolling bowling ball has quite some "momentum", and ultraviolet light has a certain "wavelength".

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?

It didn't fully came out of the blue, ofcourse. A few years before, scientists (most notably Planck and Einstein) independently found clues for the opposite, so to speak.

Namely, that

Planck and Einstein found in some observations, that radiation exibited a "corpusculair" character, like that radiation seemed to be emitted in "quanta" (discrete energy packets).

But not only that, it seemed that those quanta possessed "momentum" too, as was observed in certain experiments.

So, in a way, both sides were covered now. It was puzzling. People started to talk about the "particle-wave"

Inspired by this and other stuff, around 1926, Erwin Schrödinger published a mathematical equation about the

undergoes with respect to time, when that system happens to be in some sort of force field.

This quantum system is represented by a "wave function" Ψ (r,t), which will be explained in a minute. The equation goes like this:

The "ih" can be considered to be just a constant, so we dont worry about it. What the equation really says is this:

the change of the system in time (notated by ∂ Ψ (r,t)/ ∂ t), is the effect of the Hamiltionian "Η",

where the Hamiltonian "H" just stands for all forces and fields operating on the system.

It should make sense, doesn't it? A system

Note: ∂ Ψ (r,t)/ ∂ t means that small delta's in time occurs, and the cooresponding change on the position (r) of the system, is a result

of the acting Energy on it..., or the Hamiltonian on it..., or the effects due to kinetic and potential Energy...

Usually, it is supposed to be applied to microscopic systems, like for example elementary particles etc..

However, in our macroscopic world there are similarities too: if you apply a force to a ball, it starts to roll.

Now, here is

So, believe it or not, such solutions are thus indeed wave equations.

Actually, the Schrödinger equation applies to a "quantum system", which usually is percieved as a particle. Now, this too suggests that a particle is sort of

"smeared out" in space. However, such a view was never very satisfactory.

After many other publications, conferences etc.., during the late twenties, and thirties of the former century, most physicists gradually

adopted the view that the

This actually means that it is not as much viewed as a "true wave", but should be better viewed as the likelyhood (chance) of finding the particle

at a particular place.

For example, the solutions of Schrödinger's equation applied to the Hydrogen atom, shows "electron orbitals", which are "distributions" of the electron

around the atomic nucleus (instead of circular orbits). Maybe you like to "google" on that further.

One of the many experiments that shows the quantum nature of light (or in general,

The experimental setup, is rather simple. But it's fair to say that the results of such experiments were a large stimulus for Quantum Mechanics.

These sorts of experiments were performed in the late 1800's and early 1900's, and clearly showed the "breakdown" of the classical field theories

like ElectroDynamics from Maxwell.

In short it is this: of you shine light on a sheet of metal, under certain conditions, electrons get's freed from that metal.

If the frequency is too low, it will not happen. No matter

Only if you use a very precise frequency, electrons will be emitted. That happens even when the intensity

Such a observation is not in accordance with classical theories: if you shine light with a very high intensity (very bright), a lot

of energy is poured in, and it should expell electrons of the metal. But none is freed.

Now, if the intensity is very low, but of a certain freqency, electrons

It was percieved as really strange (at that time). It was not in accordance with "continuous wave theory" like the one from Maxwell.

Around 1905, Einstein came with a good explanation (partly based on thoughts of Planck). Instead of a "continuous wave", Einstein proposed

discrete packages, or

frequency of the radiation.

So, if such a quantum has an Energy that is equal or higher than the "binding" energy of outer orbit electrons, it is able to knock it out

from the metal.

Einstein's reasoning explained why the energy of freed electrons was dependent

Simulation:

I recommend to download a small 'simulation' program (from colorado.edu), which illustrates the 'photo electric effect' brilliantly.

If you go here you can download it, and run it on your workstatstion.

But you need java (since the program is a .jar file). If you have it, then use for example "sodium", and vary the frequency from red to violet,

while at each colour (freqency) also varying the "intensity" of that light.

While this example has no direct link to "wave function", it highlights another aspect of QM, namely that some entities, or properties,

come in "discrete" steps. This does not mean that QM deals with "discrete" properties or entities only. Oh no!

However, it's an example of how QM can differ from "continuous classical theories".

But, generally speaking, the experiment is a strong "plus" for wave-particle duality, as we will see.

Please take a look at figure 1. The picture in the middle, looks like how Thomas Young performed his experiment, in the early nineteenth century.

He used an ordinary light source, and a screen with two narrow slits.

he light emitted, will pass through those two narrow slits. Then, on a screen in the back, an "interference pattern" is shown, that is,

bands of lighter and darker regions can be observed.

At later times, these experiments were repeated, however this time using "true" particles, like electrons or neutrons.

Amazingly, a similar "interference pattern" is then observed.

Fig 1. The double slit experiment.

At this stage, we have to bypass certain angles to approach stuff, like using Heisenberg theorems, or viewpoints from "weak" measurements.

However, what we are going to see in a minute, is already spectacular enough. But let's start with an easy explanation.

Using light, and the classical wave interpretation, the results ("interference pattern") is as expected.

When light passes the two slits, at such a slit, it will be "defracted", and a spherical wavefront travels to the screen.

But keep in mind that there are two slits. Now, the waves are "sinusoidal" and have a certain fixed wavelength.

Even without calculations whatsoever, you can imaging that those waves travel different lengths, before they reach the screen in the back.

Now, take a look at the right image of figure 1. In some cases, waves may "cancel out", because at a certain point at that screen,

it holds that the wave from slit 1 happens to have a max amplitude, while the wave from slit 2 is exactly the opposite. Therefore the sum

of those waves cancels out (the centre of a dark region).

At other points on the screen, it holds that the wave from slit 1 happens to have a max amplitude, while the wave from slit 2 is max too.

This will then be the case for the centre of the bright bands on the screen.

The whole effect is just due, by the fact that the waves from slit 1 and 2, have travelled different distances, where at some points,

it will be n x λ (n as an integer number), and thus they amplify (the center of the bright bands).

At other places it will be ½ x n x λ, so the maxs and mins of the waves, cancel out (the center of the dark bands).

And ofcourse, there are regions where the waves partly positively or negatively add up (regions between the bright and dark bands).

Here, it gets interesting if we lower the intensity. So, suppose we have a situation where occasionally photons are emitted.

Next, we observe the screen in the back. Now, we can see the small "impacts" which at first sight, are randomly distributed on the screen.

However, as time goes by, we see that an interference pattern builds up, exactly as we saw above.

This is not so easy at all. There was

are emitted

The following may seem unbelievable (at first). Many physicists say that the photons are

to produce the interference pattern.

So, you might be tempted to suppose that a quantum goes through one slit. However, the best view is that such a quanta goes through both slits simultaneously.

This is difficult to reconsile with a "matter/quantum - like" view of radiation.

But there is no good alternative to explain the "interference pattern", if we use photons which are emitted one after the other.

So, it really looks like, in some strange way, that each photon is interfering with itself.

But hold on. Now let's see what happens if we do not use light, but "true" particles like electrons.

This completely resembles what we have seen in (2). So, using a very low intensity, where electons one by one are emitted to the slits,

after a while, an "interference pattern" is build up.

Again, we must say that an electron goes through both slits simultaneously.

Unfortunately, at this stage, my choice of words is not very "optimal". This is so since there are some "intracies" with the "art of measurements",

like for example if we would use Heisenberg's theorems, or see what we have with commuting- and non commuting observables, and using "weak" measurements.

We have to leave that for a later moment.

However, what we discussed above, is really true, and at this point, quite amazing.

The discussion above partly reinforces the idea "particle-wave" dualisme. However, what it really shows is that we must "give up"

the idea of a photon or an electron

You see that? We cannot even fully "escape" using a "probability distribution". The entity

slits. Here we get to a point where, even up to this day, physicists, philosophers, and many others, break their heads

on the results of this "seemingly" simple experiment.

Again, let's talk about such an electron again. It goes through both slits, so it acts like a wave? Thus, if so, you might then say

that it makes no sense to ask at which slit the electron passed through. Ok, so the electron behaves like a "wave" here.

Great stuff..., Yes?...No?...Yes!!!

If you don't fully understand it, Richard Feynman (more or less) said:

quantum description of the event. If you say,

interference patterns emerge, which can be explained by the fact that the electron interfered with itself...

But,

But there must be some connection with "probabilities" too.

The Schrödinger equation from section 1.1, might help us too, understanding the principle of "superposition".

It's just a "partial linear differential" equation, and there is nothing "fuzzy" about it at all.

Solutions for it, for several traditional "problems" have been worked out (like H-atom, particle in a box, a potential "well" etc...),

and it seems a nice starting point for QM.

Since the equation is

Ask any mathematician which is nearest to you: he/she will absolutely confirm that !

So, if Ψ

Now, a solution like:

Because, if we calculate this:

over

This is why physicists introduced the "Wave Packet". 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. So, the following is a good solution:

as you move "away" from the packet's center.

However, theoretically, even at large distances, the amplitude is not "0", but it sort of asymptotically nears "0" the further you go.

It's very important to realize, that such a

Often, it is called "superposition of states", where each state represent a certain "probability" that the particle actually resides in.

Those waves (states) of the packet are

that is, sinusoidal-like waves in "shape".

Note:

That the component waves (states) of the packet are

However, a wave packet in general. "naturally" spreads because it contains waves of different momenta and hence different velocities.

Remember the single electron that was used in the double slit experiment (1.3)? Here there was interference "with itself" meaning

interference from those superimposed waves.

Fig 2. Illustrating a Flat wave versus a wavepacket.

Let's now turn to a good way to "notate" or "describe" Quantum Systems (like a particle, photon, atom etc..), using the Dirac way

of handling stuff...

the original framework of QM, grossly in the period 1900-1950.

Originally, to work with QM, a "calculus" type of mathematics was used initially (like the partial differential equation of Schrödinger).

Somewhat later (in that same period), a "vector/matrix" type of approach to QM, was introduced. In many ways, that was in large thanks to Dirac.

This indeed made operations in QM much more uniform, and way better to understand (I think).

However, new work/discoveries as from, say, 1950 up to now (2015), were enormous. Also in the field of

For about the latter: we certainly will see some stuff about "Collapse of the state vector", "Decoherence", "Many World Interpretation" (MWI),

and the "no-nonsense" interpretation (as some new physicists apparently seem to view QM lately).

Now, here is a little about

Essentially, it's a "vector type of calculus". Ofcourse, if you dive into the professional literature, you will learn about

formal definitions on Hilbert spaces, complex spaces, conjugates, bras & kets, all of them with lots of theorems and collaries (and proofs thereof).

But we keep it simple ! We only want to crack

need the info to get there.

What is a "ket" (Dirac) or "vector" anyway? It's often defined as an entity with both a magnitude and direction. However, visualizing a vector

in 2 dimensional space (a plane), or 3 dimensional space, is really easy.

Take a look at figure 3, where we see a picture of a point in space (1,2,3), and a vector "pointing", from the origin "O",

Fig 3. A vector in R

In figure 3, you see the vector (1,2,3) "going" from the origin "O" to the point (1,2,3) in space.

Most interesting is the fact, that we can say that to reach (1,2,3), we need to do "1 unit step" in the x direction, then "2 unit steps" in the y direction,

followed by "3 unit steps" in the z direction.

With a little imagination, you see the "unit vectors" (1,0,0), (0,1,0) and (0,0,1) positioned along the x, y, and z directions, respectively.

So, the vector (1,2,3) is equal to:

(1,2,3) = 1 x (1,0,0) + 2 x (0,1,0) + 3 x (0,0,1).

Indeed, the quantum system (like a particle) is in een superpostion of such basis states.

Vector calculus provides a natural way to describe a quantum system in that way. Note that we now go to one of the

of the original (1900-1950) formulation of Quantum Mechanics!

So, if we let the states that all together form the superposition,

or statevector, of a quantum system, in the following way:

Instead of fully writing down all unit vectors of the superposition, often the Greek ∑ "summation" symbol is used, to denote exactly that.

Those basis vectors are often callen

So, a quantum system then, written in vector notation (the State vector), generally is a superposition of those "eigen states".

This is truly a core concept in the framework of Quantum Mechanics. Another one is the "measurement problem", which I will illustrate

using an example of what we have learned in this section, namely using the description of a

But, at a measurement, "something" happens.

As we talk about pure quantum systems here, if we do not observe it, we actually do not know the state of that system,

only that it is in a superposition of (possibly) many states

Remember the electron we used with the double slit experiment of section 1.3? It seemed that it passed both slits.

We even had to give up our notion of "clear path" or "clear trajectory" here.

Now, the following might not strike you hard, initially, but it's really something!

Suppose I place a small detector at one of the slits, say slit 1. Maybe I see a reading from my detector, meaning I have detected

the electron.

There won't be ever a interference pattern at the screen in the back.

Since I have located (or measured) the position of the electron, the "superposition" of all possible locations of the wavepacket

suddenly

There are many other examples. The next one is a bit "blown up", but I want to make a point clear, even with an exeggerated example.

Suppose we send out a photon. We should regard it (for now) as a sperical wave, since we do not know anything of it's location.

Now, somewere in space, I have located a detector. If I find a reading, meaning I found (detected) the photon,

all posible locations "collapsed" into that single point.

As a more quantitative and realistic example:

Suppose I have the following 2 state quantum system:

The coefficents "a" and "b" should determine the actual state of the system. However..,

QM tells us that we

And we have not observed, or measured, anything yet!

Now, just as was the case with the electron, if we perform a measurement on the system, then we always find

the system to be in:

state

This might be percieved as quite weird.

Actually, some folks formulated it this way: our measurement "destroyed" the former quantum system. It (the quantum system) is something else now.

This is very close to the famous "measurement problem" in QM. It seems that our measurement was quite "disturbing" to the system.

Note: many people also call it a "strong measurement", or "perturbative measurement".

Did you noticed the

The stuff clearly shows us that QM is probabilistic in nature.

Moreover, experiments have shown that those coefficients (a and b) relate to the probability of finding the system in state |a> or in state |b>.

Then, since the probability of finding the state (after measurement) to be in |a> or |b>, must be "1" (or 100%).

Ofcourse, we can only find |a> or |b>, so the total chance added up, must be 100%.

However, the chance to find |a>, or to find it in |b>, is less than 1. It could be (for example) 30% and 70% respectively.

This can only be effectively determined after many experiments and simply count how often you have found |a> or |b>.

The only thing we really can say is this: all probabilities added, must be "1" (or 100%), and mathematically this equates to:

|a|

The interpretation of this phenomenon has long been a true issue for physicists and other scientists like philosophers.

It is still not fully understood, however, the socalled "Decoherence theory" provided us with a somewhat simpler way to digest it.

I like to touch on "Decoherence", a bit later on.

What we have seen here, is often called

It should be clear, that once a quantum system "collapsed" (or decohered) into an eigen vector, and you would immediately perform

a measurement again, then you will find that same eigen vector again. At least, that would be obvious from the theory presented thusfar.

Maybe all this stuff was a bit "abstract", so let's take a look at a

since "qubits" are used in the new technique of "Quantum Computing".

Here is a general representation:

Now, since qubits are used in quantum computing, the |a> and |b> vectors are often "rewritten" as |0> and |1>.

There are two reasons for that.

One is to emphasis the "computing" element, since in traditional computing, ordinary "bits" (0,1) are the most fundamental units, ofcourse.

Secondly, often particles with "spin" (we will come to that in a minute) are used, as the physical entities, and such a spin

can be "up" or "down", which is often expressed as |0> and |1>, or sometimes also as |↑> and |↓>

There is ofcourse an enormous difference between classical bits (used in regular computing) and "qbits".

- A regular bit can only be "0" or "1".

- A quantum bit, or qubit, is a superpostion of |0> and |1>, and those "span", in principle, an infinite number of resultant states.

You might still be amazed. Well, the different combinations, which two basis vectors can "span" (with coefficents where |a

can be visualized as a circle. Please take a look at figure 4.

Fig 4. Ψ as all possible combinations of |0> and |1>: the Bloch sphere.

In figure 4, you see the two example Ψ Statevectors, in green. But these are

can "span" a Ψ statevector. That is, a

Just keep in mind that a

Since we have two independent eigen vectors (|0> and |1>), it automatically defines this vectorspace as 2 dimensional (or a plane).

So, many folks visualize state |0> as the unit vector (1,0) (along the x-axis), and |1> as the unit vector (0,1) (along the y-axis).

In accordance with the theory of section 2.2, if you would perform a "measurement" on the qubit, you will find |0> or |1>

since the State vector will "collapse" or "decoheres" into one of those states.

Notes:

=> (1) What physical system can be a Qubit?:

Researchers might consider

electrons, or any other particle or system. Thus, In principle,

eigen states, could be a "candidate" for a qubit. Again, unmeasured, the system would be in a superposition of those states.

Often, a spin 1/2 particle, is a good choice, like a "trapped" electron (quantum dot).

The big "enemy" of researchers is decoherence with the "environment", since that will alter the state of the qubit.

=> (2) Limitations:

As note before, this is not a "book" on Quantum Mechanics. Lot's of stuff is omitted in this note, and the concepts that

are simplified to a high level. For example, some "States" cannot be represented by by a "ket" or Statevector.

To narrow this down a bit, let's compare information of qubits to "classical bits".

Maybe not all of the items below will be immediately clear, but don't worry about it.

If you would consider a special case of

then some obvious observations can be made.

For example, the state of a computer register, is completely "known" at a certain moment. It might contain a bitstring like "1001 1101",

and this can be read, copied to tape, or disk, or another computer, or multiple disks etc..

=> (1) For Classical information, it is generally true that:

- you can read a register, or memory location, without altering it. It's "state" will not be
**altered**. - you can
**"clone"**classical bits without any problem. - you can
**"broadcast"**classical bits to many destinations without any problem. - If "power keeps on" and futher you do nothing at all, a memory location keeps that same bit string for years.
- if you
**don't know**the state (content) of a register (or memory location) you can**still copy it**.

- You don't know the "value" of a qubit. It's "unknown". You only know it can be represented by a superposition of states.
- If try to "read" it, you interact with it, which "alters" it state.
- You cannot "copy" a qubit, since you cannot read or measure it precisely. It's a superposition, unless you "collaps" or "project" it.
- In general, if you "measure" a superposition of states, you get different outcomes all the time. At best you can get (over time) an expectation value.

- In general, if you "measure" a Quantum System, you get "a value "which you might consider as "classical" but it does not represent the former (fuzzy) Quantum state. That is lost.
- Quantum Information cannot be fully converted
*precisely*to classical information, and the other way around.

Before we go to "Teleportation", we need need some basic information about a

Then, we need just a

Then, in chapter 7, we will finally deal with the true subject of this note: Quantum Teleportation.

=> Collapse of the Statevector / Copenhagen Interpretation:

In chapter 2, we have seen a few examples of the socalled "collapse of the Statevector". If an actual measurement on a quantum system is done,

out of many possible results, just one is "selected" in "some magical way".

An observable, initially in a superposition of different eigenstates, appears to reduce to a single value of the states,

after interaction with an observer (that is: if it's being measured).

This idea was much debated in the late 20s and 30s of the former century. Eventually the idea evolved into the socalled "Copenhagen Interpretation",

and while you could not say that it had true "advocates", it's a fact that many just accepted it as a "workable solution" for the Theory and observations.

(You may note that I completely bypass any thoughts about Hidden Variables, as an alternative for the collapse).

=> Decoherence:

In in the late 70s, 80s and early 90s of the former century, research got a renewed momentum to find a good "internal explanation" for the presumed "collapse".

This is no to say that much earlier, some folks (like Von Neumann in 1932) already investigated idea's that are quite close to "decoherence".

However, during the "renewed investigations", it was finally argued that a process called "decoherence" provides for a reasonable explanation

for just an

During that time, it was more and more realized that a quantum system is generally "embedded" in the environment. This is even true if you do not

measure anything at all. But it is especially true, if you let the quantum system interact with a "measuring device".

In QM, each state that contributes to the superposition of the State vector, needs to be in a coherent state.

Remember from section 1.4, that the waves (states) of the "wave packet" are

You can see them as solutions to a "harmonic occilator", that is, sinusoidal waves in "shape".

The theory of Decoherence states, that while the quantum system "nears" the measuring device, the different components entangles more and more with the many quantum systems

of the measuring device, (or in general: the environment), and a process called "einselection" takes place. It means that most entangled waves (or "coupled" waves) leakes out

to the environment, until a pure state is left over. This process is thus responsible for an apparent collapse.

Nowadays, it's "almost" fully accepted that "measurements" involve entanglement of the quantum system with the environment,

and the former idea of the sudden "reduction", or "collapse", of the statevector, is replaced by decoherence.

Especially Zurek created nice articles that go into depth in the theory of Decoherence, while it's still nice reading.

I can recommend this article (arxiv.org), although it's quice "spicy" (technical).

Other great stuff can be found here: Quantum Decoherence (www.ipod.org.uk)

If you look at a wave packet (of section 1.4), a lot of waves are superposed, with a sort of gaussian probability distribution.

It might look "fuzzy", but it's just a superposition, and we can even capture it in an equation.

So is it really fuzzy? Probably not, but what might look "fuzzy" is the "probability distribution".

Such statements are not really exact. Besides that, different folks may have different

Please consider again the formalism of section 2.2, where we represented the Statevector (of a quantum system) as a superposition of eigenstates.

In such a case, if you observe the quantum system (or perform a measurement), then we find a eigen state with a certain probability.

That's not fuzzy, since you can find

is that this is associated with a "probability". Indeed QM, is intrinsically "stochastic".

However, some say that the theory of "decoherence" removed the "stochastic" character of QM. That's not true, since the original superpositions

still are part of the theory. But, again, different folks may have different

And, what we have not seen here much, is that at many places in QM, discrete "quantum numbers" play an important role.

Now, I need to make a little "refinement" on what I have presented sofar:

Actually, when we observe a quantum system, an

may have multiple observable "properties", like

Question is: if we measure for example the "spin" of a particle, then the statevector for that observable will collapse/decoheres to an eigen state.

So, did we now collapsed the "whole of the particle", or just that particular property? In general, this is not so easy to answer.

However, in some cases, it's not difficult. Suppose you measure the orientation of the spin of an electron, and you find the state "up" (|↑>),

then it's still an electron! However, that specific observable decohered (collapsed).

In some other cases, it's not so easy. If you "measure" the position of a photon, then it's "gone" (meaning that maybe some electron in the material of a screen,

absorbed the energy quantum, and was freed, or went into a higher atomic orbital..).

Only in the very most simplest case, an observable defines the state of of a Quantum system.

But in general, more observables are needed to "define" or "pin down", a quantum system. Just take a look at a particle. It has a

position in space (wave packet), a momentum, and possibly it may have a "spin" too.

Up to now, we have "acted" as if one observable defines a quantum system. Now, we know that in general that is not true.

Measurements on observables "A" and "E", can be interpreted as having

get a similar naming as the observables (with an additional token like a circumflex, hyphen etc.., but that depends a bit really).

1. Commuting observables (operators):

It is said that operators (or the associated observables)

Â Ê Ψ = Ê Â Ψ

In the equation above, you see that in this case the order of taking measurements does not matter!

We are not going to mathematically prove it, but the following may sound plausible.

It can be true if both observables (State vectors) can be expanded (or written) in a

Often, people say that A and E have a "mutual eigenbasis".

Actually, it's a better way to describe that two observables "commute", if both can be measured simultaneously, with definite values

and thus both (sort of) more "pinned down" the state of the quantum system.

It's also often said that the observables are "compatible", which is the same as "commute".

Unfortunately, it's all pretty abstract for now. But if you believe that both observables have a "mutual eigenbasis", then both

can be measured and definite values can be found at the same time.

You can also say this:

- This is only so if Ψ is an eigenvector of both operators.

- Two observables can be known simultaneously only if they have a common set of eigen vectors.

2. Non commuting observables (operators):

There are also Non commuting observables (or Operators). It's also often said that those onservables are "incompatible".

Two incompatible observables cannot have a common set of eigenstates (no "mutual eigenbasis").

Say we have A and E again. It can be proven that it's simply not possible to find a well defined value for A,

for E at the same time (simultaneously).

Furthermore, the order of performing measurements now

Rephrasing Non commuting operators in "brute force language":

Gaining knowledge of one observable through a measurement, destroys information about the other.

3. Further explanation:

If Ψ really would be an eigenvector of both operators (observables), so both observables have a mutual set of eigenvectors

then you can hopefully see that

Since finding an eigenvector was already a matter of probability, but the system stayed in the same vectorspace.

So, the second measurement will produce again an eigenvector with a certain probability.

But, it boils down to the fact that you can find precise values for A and E, even at the same time.

If the observables are incompatible, both have

so much, that the uncertainty in E increases highly. A perturbative measurement on E disrupts everything so much, that the uncertainty in A increases highly.

You cannot have precise values for both observables.

I realize that this is not a good attempt to try to explain the differences between commuting and non-commuting operators.

Using math, it could be explained way better. But I try to avoid math as much as possible.

Some examples:

- Adding numbers commute, since in any addition you can reposition terms as you like, e.g. a+b+c=a+c+b
- Rotating an object over two axes (thus two operations), does not commute. If you turn 90% along the x-direction,

and then 90% along the z-direction, will produce a different orientation of the object than the other way around. - Position and momentum does not commute. If you measure precisely the position of a photon, the momentum is destroyed.

Some historians have researched where the "principle" actually first originated.

They have reasons to believe that actually Pauli was first to express ideas of such content, while others see evidence

in papers of Wiener to believe that actually "Wiener was first...."

Sometimes the

That is not a good view, really.

Here I am not saying that there does not exists a "measurement problem" in QM. There absolutely exists a "measurement problem" in QM,

which constitutes one of the hearts of QM.

A remark on the measurement problem in QM:

If you would measure the temperature of a large swimming pool, your device will not (really) have any relevant influence

on the swimming pool. In fact, the measuring device is so small compared to that large body of water, that it is absolutely

correct to say that your measurement did not "disturbed" the swimming pool at all. It was not a "perturbative" measurement.

The large swimming pool was not changed by your measurement.

However, QM is used in the world of elementary particles, photons, atoms etc.. These quantum systems are very small,

and any measurement you perform, will be "perturbative" to some extend.

In fact, your measurement will change the state of the quantum system. And even one of the "core" descriptions in QM, namely

that a State vector "collapse" or "decoheres" to an eigen state when a measurement is performed, actually "sort of" confirms

that your measurement is "perturbative".

It's a core problem, and even up to this day, different perspectives are published in a whole range of scientific articles.

Note: "weak" measurements.

A "perturbative" measurement is also often referred to as a "strong" measurement.

Contrary to such "disturbing" measurements, in 1988 a couple of physicists published an article on "weak" measurements,

in which they stated that in principle, extremely weakly coupled measurements could be performed, which, run over a longer period,

could give information about the "undisturbed" quantum system.

This idea raised quite some debate in the scientific community, which is not over yet.

A remark on the Heisenberg uncertainty principle:

The Heisenberg uncertainty principle is simply

you "get it for free". In fact, it's really not directly related to the "measurement problem".

One of the most used example of the Heisenberg uncertainty principle, is when you consider the pair of "non commuting" observables

position (x) and momentum (p).

In fact,

will be build in automatically. It arises from the wave properties which are inherent in the QM description of nature. Really !

The Heisenberg inequality relation for position (x) and momentum (p) is:

Δ x . Δ p > ℏ / 2

It means that the more precisely the position of some particle is determined, the less precisely its momentum can be known,

and the other way around.

It has been proven, theoretically and experimentally, for many quantum systems.

For example, if you go to the following wikipedia article, and go to the "particle in a box" problem, the inequality will be derived in a fairly simple way.

And it's just based on the wave mechanics of QM ! So, as please see this example (wikipedia)

Please note that the derivation has nothing (!) to do with "precision" of measurements. It's just build deep in the description of QM.

Next, a discussion on "Quantum Entanglement" is on the list of "things to do".

Technically, QM distinguishes between "pure states" and "mixed states" (and even others), which can be quite confusing, really.

Since this note is very simple, I don't go into that. What we have seen thusfar, are

a "State vector" as linear combination of eigenvectors,

I want to keep that image, so to speak, as not to introduce too much complexity.

A mixed state is often a statistical mixture of pure states, where a more complex interference might be present,

and it has certain complexities in interpreting coefficients as probabilities.

Mixed states introduce an additional layer of complexity, and is often used in "ensembles" studies.

However, some say that "mixed states" can only be described using density matrices.

Here we keep it simple (and thus less general...).

- A state vector with a basis of two eigenstates (a qubit):

- A state vector with a basis of three eigenstates:

|Ψ > = c

- A state vector with a basis of n eigenstates:

|Ψ > = c

2. A few example of "product states":

We will first take a look at the normal "product state". If we have two systems, and we want to describe their "joint state",

an outer product is used to accomplish that.

Suppose we have two qubits like:

|φ

|φ

then:

|Ψ> = |φ

ac|00> + bd|11> + ad|01> + bc|10>

The equation above, is a way to describe the combined state of both qubits.

It's just a

Note that for example |00> is no more than a short way to mean |0>|0>.

Actually, what we have seen above, is the "Quantum Way" for the following simple mathematical equation:

(a + b) x (c + d) = ac + ad + bc + db

Whenever you have a situation where it is possible to write:

|Ψ> = |φ

and |φ

since their joint state is a simple product of both states.

It's really just the same as the simple mathematical analogy, that:

"ac + ad + bc + db" is seperable into "(a + b) x (c + d)".

Here again, we describe two or more quantum systems (like particles, photons), with a sort of common state vector, like we

have seen above. However, this time, the systems are

That is, we

It means that we cannot "seperate" the systems "just like that"!.

Their "intertwinement"is so high, that we speak of "correlated" systems, with respect to the "observable" (often spin, polarization).

Automatically, since we cannot seperate the individual states anymore,

That is the "crux" of "entanglement". Don't think lightly about that. Although it's a common feature in Nature, as we know now,

it has important consequences, and produces lot's of brain fire-crackers for "philosophers" and others who want to understand nature.

Just take a look at this. Supose two correlated particles are in close vincinity. Now, we move one of those system further and further away

from it's partner particle. It's commonly accepted that still holds that:

Now, if the distance is very large, even so large, that some sort of "signal" between the partners would violate the speed of light barrier,

then we might be very puzzled.

Indeed, it's experimentally established that

than some "magical" signal would need timewise. So, the partners cannot "inform" each other about a change in state.

This is the effect that Einstein called

Noteworthy is the article

by A. Einstein, B. Podolsky, and N. Rosen which resulted in what later was called

As the authors argue, a number of results of QM, seem to be inadequate to describe physical reality. As we can interpreted it now,

most of their arguments actually refer to the effects of entanglement, collapse of the state vector, and other aspects of QM.

It's certainly worth it, to perform a web search, and browse through some articles which explain the original article.

It's certain that the last words has not been spoken yet on entanglement, and many suggestions have been forwarded why we

still not fully comprehend it, like for example:

- We do not correctly describe the entangled state, and/or interpret it wrong.

- SpaceTime issues related to "Entanglement" are not (yet) understood.

- Strange "hidden variables" are at work here, and sadly, we don't known of those yet...

- Nature sometimes just works "non-local" in certain events. This is the

There are some interesting studies to the "nature" of "what" exactly sort of "binds" the partners of an entangled system.

The "non-locality" principle is widely accepted. However, at this moment it does not deliver the "mechanics" of the strange binding.

In chapter 7, we will see that it is often called "The EPR channel" as a tribute to the famous EPR article.

Some further studies try to explain the EPR channels using further maths on Hilbert spaces, and the resulting "channel maps" thereof.

One consequence seems to be, that "time reversals" are neccessary to keep the theory consistent.

Needless to say, that if true, it would be very spectacular.

As an example, you might browse through this article (arxiv.org), but it's quite hard to read.

Other studies try a lattice approach. Still others, use any other concievable way to to pursue the EPR channel further.

One other amazing path, is the

at the ultimate smallest scale of SpaceTime. The EPR wormholes then, supposedly connect entangled particles.

However, many physicists are sceptical, and like much more "body" on those models. Many articles have appeared, pro and contra,

since the original ideas were published.

As an example, you might browse through this article (arxiv.org), but here too, it's not easy to read.

One thing is for sure though: physicists do not accept any "faster than light" signalling, to explain it.

(Note: Even in the case of Planck-scale wormholes, the theory uses "non-traversable" wormholes, or maybe, the theory leads

to non-traversable wormholes. This should evade superluminous signalling.)

Here is a well-known example, namely a singlet state of two particles (particle 1 and particle 2) with respect to their spin.

So, the wave function above, describes the "common" state of such an entangled system with respect to their spin.

Do you see that it is a superposition of the states "|↑

If a measurement is done, and you find particle 1 to be "up" (↑

Since "obviously", the wavefunction "collapsed" into state "|↑

that particle 1 is "up" and particle 2 is "down".

But both spins might take any value along a particular axis, as long as you have not measured it. This strange "thing" is a hard part of QM

to understand.

An entangled system may "exist" for some time, but often, quite quickly, a particle interacts with the environment, at which decoherence

will end the (two-system) entanglement. So, it's not easy to seperate the members of a singlet state, for example, hunderds of meters, or a few km's (or further) apart.

People working in the field of "Quantum Computing" often spend quite some time investigating how to "battle" Decoherence so that

they can preserve their qubits and gates for a longer period of time.

So, what do we have "qubit wise" now?:

|Ψ> = a|0> + b|1>

The general state of a "two qubit system" is given by the quantum state:

|Ψ> = a

Here, the system exists in the states |00>, |11>, |01> and |01>, at the same time (simultaneously).

We know that it is a "product state" of two single qubits, and it's "seperable".

Two qubits (or more qubits) can be in an "entangled" state.

There are different possibilities here. Especially, the socalled

You know that the "parts" of an entangled system might be seperated, in different regions, for example in different corners of a Lab,

or different rooms, or maybe 500m apart. Usually, folks use characters like Alice and Bob as two

imaginary persons, each holding one part the entangled system. That's why often a subscript "A" and "B" is added to a substate of the

superposition, to denote "where it is" (at Alice or at Bob). Be warned though, many folks just leave it out just as easily.

Here they are:

The most important to remember here is, for example, if you look at "|0>

if the system would collapse into that state, it means Alice found |0> (down), and Bob must have found that too.

Notes:

Such "pure" maximally entangled states, could be used as a

However, the "sender" and "receiver" might only be able to share a "mixed" entangled state (

In most articles on entanglement and qubits, you might often see representations like we already know, and I mean like this example:

Keep this one in mind.

Maybe we can try Quantum Teleportation now...

For example, we don't need to explain "qubits", "the state vector", "entanglement" etc..., anymore.

Quantum Teleportation is not about the "teleportation" of matter, like for example a particle.

It's about teleporting the

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

It might seem rather bizar, since in the classical world, many examples exists where you can simply

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

meaning that in fact

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.

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 3 above.

It's very important to realize, that we need this equation (equation 3) to describe

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:

|Ψ

|Ψ

|Ψ

|Ψ

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

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

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

|φ

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

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:

| θ

=

½ x [ Φ

=

½ x Φ

Note the factor "Φ

We have managed to "factor out" the state of particles 1 and 2 into the "Φ

the state of particle 3 "looks like" a superpostion of four qubit states.Indeed. Actually,

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 "Φ

This will result in the fact that "Φ

It will

of the four qubit states as we have seen in equations 5 and 6 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 |φ

We only smartly rearranged equation 4 into equation 5 or 6, 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

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.

Hope you liked it!