The simple answer to your question is that you cannot understand entanglement in the normal sense of the word "understand". We just have to accept that nature behaves in this way. We can, however, conclude that entanglement follows from the laws of nature and that it can be observed. The world record of terms of distance for an experimental demonstration of entanglement of two photons is 144 km. It was achieved in an experiment in which photons were exchanged between two Canary Islands and was performed by Anton Zeilinger and his group.

Thank you for your answer. It seems really complicated.

First of all, it is easier to think about a state as your knowledge about the system, not some "physical entity". This makes things easier to understand.

An entangled state is a global state of two particles. In fact the property that when you measure one particle then you know result of measurement on the other is not the key problem: the same thing you can get with two classical spinning balls that originated from one source that ensures that global angular momentum of them is 0.

The key thing about entanglement is that this state cannot be expressed as a product state. In classical probability theory, when you have two subsystems described by some probability measure (a state), then the compound system is described by product measure. And any measure on compound system can be expressed as a product of marginal probability distributions.

This is not the case if both algebras of observables are non-commuting, i.e. in the case of so-called quantum probability. Then there are measures on compound system (tensor product), which do not arise as "products". These are entangled states. And that's what Bo Thidé means by saying that "we have to accept that". It's mathematics.

Finally, I would like to emphasize that, there is no dynamical action here. Nothing changes for the second party if you measure the first one. This is simply non-Kolmogorovian probability that describes the outcomes of experiment, and this, at least for some, is counter-intuitive.

Thanks so much for the answers from Mr. Tylec and Prof. Thidé! Your answers make me move forward to the the laws of nature. The entangled state should not be understood from the "physical entity" point of view but from the "probability" stand (perhaps Einstein might not believe in this). Although the entangled state is quantum A can be known as soon as the state of B is measured, it is useless for communication because we don't know the result of measurements. That's why the superluminal communication can not be achieved. I'm I right? What a amazing world!

I liked Tomas Tylec's answer. Perhaps as an addition to this answer it might be said that the problem would become even more comprehensible if we would stop entertaining an interpretation of quantum mechanics in which the wave function is thought to yield a description of an individual particle. This would be more in line with Einstein's idea of incompleteness of quantum mechanics, the wave function being interpreted as a description of an ensemble (of identically prepared individual particles). Then projection of the wave function can be seen as a transition toward a sub-ensemble, which does not need any interaction because it can be realized by means of selection on the basis of (actually obtained) measurement results on

the other particle.

@Zheng-shi Yu: yes, state has a lot to do with probability. In fact state of classical system is nothing else as probability measure on a phase space. When this measure is Dirac's measure, i.e. point measure, we call it a pure state and identify it with a specific point of a phase space. Observables are "good" functions on a phase space.

In fact, this exactly fits Kolmogorov's axioms of probability: we have a sample space (a phase space) and a probability measure (a state) and random variables (observables, i.e. functions on phase space).

This analogy between probability and classical physics can be used to generalise notion of probability. In fact probability theory was considered by Hilbert in his famous problem list as a part of physics.

In quantum mechanics we know that we don't have a phase space, so we must forget about sample space. Fortunately, in classical probability we can also forget about sample space. It is enough to have the algebra of random variables and the set probability measures (which are a certain class of functionals on the algebra of random variables).

So we take operators on Hilbert space as generalised random variables and quantum states (inc. mixed ones) as generalised probability measures. This is how we define quantum probability models.

Because it is more general, some properties of classical probability are not satisfied. Particularly, these connected with entanglement.

Three comments:

First, entanglement is not necessary for quantum-secured communications (the original BB84 algorithm does not use entanglement).

Second, You are correct in that no information is transmitted to the other side when one party measured the state, and therefore the is no violation of special relativity.

Third, to illustrate the difference between bipartite entanglement and classical bipartite correlations, consider the above example of two spinning objects, and measure whether they are spinning in the same direction, not only along the z-axis, but also along the x- and y-axes. For classical spinning objects, you'll find full correlations only along one axis. For a fully entangled spin pair, full correlations will be measured on all three axes (Note that by "measured" here I mean that for every axis separately, many measurements of the spinning directions are taken, each time on a newly prepared pair, and the results are averaged).

I suggest you look at this short note by Artur Ekert and Alastair Kay how the (two-qubit) CNOT gate creates entanglement, that an unknown quantum state cannot be cloned/copied but can be teleported and that CNOT together with single qubit gates are universal for quantum computation.

http://www.arturekert.org/sandvox/note3-2.pdf

An excellent reference is

Nielsen, Michael A. & Chuang, Isaac L. (2000). Quantum Computation and Quantum Information. Cambridge University Press.

Otherwise, there are many much more advanced accounts of entanglement like with algebraic geometry

http://xxx.lanl.gov/abs/1204.6375

By definition an entangled state (let's say between Alice and Bob) cannot be factorized (between a state available at Alice location and a state available at Bob's location). But an entangled state can be shared by mutually commuting operators such as XX,YY and ZZ, where X, Y and Z are Pauli spin matrices and the product is the tensor product, i.e. XX= X \otimes X.

@Karol Gietka: existence of non-separable states (i.e. entangled) is a feature of non-commutative probability theory. This is mathematical notion. The fact, that you can explain the outcomes of your experiments using them, means that you need non-commutative theory to explain your experiment.

By the state I mean (non-commutative) probability measure. I agree that it might not have always straightforward "fundamental" physical interpretation. But neither "qubit physics" is fundamental: it's only a model which nicely fits some class of experiments.

Recall that you cannot realise Heisenberg commutation relations with operators on finite dimensional Hilbert space, which undoubtably is fundamental axiom of quantum mechanics. Neither you have bosons.

Maybe there is a very easy and intuitive way to understand entanglement: It is just that two particles have one common property. e.g. two photons from an spontaneous emission have total spin of zero...

The problem with von Klitzing's answer is that it is applicable to two different statistical states, one being entangled while the other is not entangled.

Gentlemen, there are no entanglement of far separated particles.If you deal with Hilbert state, the wave function of two separated particles cannot be a product, nor sum of products, because the product goes to zero with separation. You discuss nothing.

I mean Hilbert space

Sir, if I understand your right, then you are wrong. In your picture, a pure state of two (separated) particles is described on the tensor product of two Hilbert spaces. For simplicity let's focus on particles on line. Then you assign to each particle Hilbert space L^2(R). The two-particle system is described by tensor product L^2(R)\otimes L^2(R) = L^2(R\times R). Now, if e.g. the support of wave function of both particles is a unit ball centred at (1, 10), then particles are clearly "separated". Moreover, if this wave function can be written as a product: h(x_1, x_2) = f(x_1) g(x_2) (note, that we have two different arguments), then the state is separable. Otherwise, it's entangled.

Moreover, not always the Hilbert space can be interpreted as L^2(R^n). E.g. take Fock representation. Or any representation associated to finite temperature equilibrium state. Or finite dimensional Hilbert spaces so useful in various models.

The image of an entangled system as two (or more) subsystems sharing a common property (namely, total spin zero) is not enough.

One could think of pairs of green and red balls, made of metal or wood and whose weight is 0.5 or 1 Kg. Each time we measure the color of one ball (let's say green), then the color of the other is automatically determined (red), and the same with fabric (metal or wood) and weight (0.5 or 1Kg). Bob knows that the choice of measurement apparatus (let's say, flashlight, fire or scales) by Alice on one member of the pair would never determine his results (color, fabric and weight) on the other member of the pair, as long as no information is exchanged between them. This is because macroscopic (classical) objects (like balls) have defined values of their physical properties (color, fabric and weight) before they are measured. However, this is not true for subatomic (quantum) entities, where all possible values can coexist in a quantum superposition.

I am attaching a divulgative article by myself where you can find this discussion (sections 4 and 5).

Hello to all.

I just started a thread on entanglement. See

https://www.researchgate.net/post/Is_quantum_entanglement_any_better_than_classical_entanglement

With most cordial regards,

Daniel Crespin

One of the many possible ways to understand the quantum entity of entanglement is to examine excitons. These are quasi-particles which are formed from two paired fermions: an electron and hole. An exciton behaves likes a boson so long as the electron and hole are entangled as highly correlated fermions. Once the entanglement effect diminishes, each hole and electron assume greater degree of carrier identities, in other words possess more fermionic features and the exciton becomes less of a boson....or coboson. Entanglement effects overcomes the shortcomings of classical forces, as one cannot use classical models to examine bosonic features of excitons.

1. The state of a system can be represented by a vector in a Hilbert space.This is an axiom of QM but can be proven for a large class of models used to describe systems.

2. If you have two systems S,S' which interact you can proceed along two ways. The first is ta represent each system with the variables specific to the system say X (X') and add the variables accounting for the interactions say Z (Z'). But it can be proven that, under some logical rules, it can be done quivalently by taking the tensorial product of X and X'. Then we have a single system, with the tensorial product of the initial Hilbert spaces. The dimension of the tensorial product is the product of the dimensions, meanwhile the dimension of the simple product of vector spaces is the sum of the domensions. So one can store more information in the tensorial product, which is necessary to account for the interactions (Z,Z' do not appear any longer). But this representation has important consequences.

3. The only pertinent vector, which stores all the information, is the a tensor. A tensor which can be expressed as the tensor product of vectors is said to be separable, But if any tensor can be expressed as the sum of separable tensors, there is no canonical way to extract two vectors from a tensor. So usually the state of the interacting systems can no longer be seen as if each system had a specific state.

4. Because this is a bit disturbing, it is natural to find a way to get back X,X' from the tensor. A simple algebra shows that usually this is not possible, so the solution is to use a probabilist approach, similar to what is done when you try to estimate a matrix as the product of two vectors. This is the way probability comes into the picture : what one induces is a "proxy" for rhe states of each system, which gets back as best as possible the tensor. Moreover, as tensors can be assed, one can imagine that we have a superposition of such states given by separable tensors. Of course this is mathematically legitimate, but this superposition has only the physical meaning that one is ready to assume.

What is true for two systems holds for many systems, and when they are similar, indistinguishable, the result is then special because it must be a representation of the symmetric group (antisymmetric if we have a spin). The states of the system are quantized.

Notice that this is pure mathematics and logic, it holds whatever the scale (it the model meets some general criteria). Moreover there is no assumption about the way the systems interact. The physicist has to be consistent with his choice : if he (or she) assumes that there are interactions, they must appear somewhere, either explicitely, of in a tensor product. He has to live with his choice : he cannot choose a model with interactions some times, and forget them after.

I think that entanglement is quite tricky and perhaps not fully understood for its implications in Physics.

Most research in Quantum Information "admits" that entanglement is a "valuable" resource (see the famous book of by Nielsen & Chuang : Quantum Computation & Quantum Information 2000) this has motivated an enormous amount of research (Quantum Cryptography, Quantum Computing).

Now experiments show that entanglement is much more widespread than initially believed (see for example the paper from Vlatko Vedral "Quantifying entanglement in macroscopic systems" Nature June 2008) it is becoming "macroscopic" and observed in many materials having for example magnetic properties (the field of many-body Physics).

Perhaps the qu-bit approach (Alice and Bob...) is not the only way to produce entanglement there are measures (entanglement witnesses) which can" evaluate" entanglement of an unknown quantum system. And I believe that the recent debate on the the D-Wave computer has something to do with this...

Thanks to Bell's Inequality and a huge amount of experimental work, entanglement is neither some abstract concept nor a mathematical illusion, but an experimentally verified and quite useful feature of how our universe behaves.

To understand entanglement you need to keep two principles in mind.

The first principle is that quantum mechanics is the physics of phenomena whose specific history (sequence of events and outcomes) has not yet been classically determined. In the mathematically precise language of quantum electrodynamics (QED), “no specific history” translates into something called the integral of all possible histories. This region of historical uncertainty is more commonly known as the wave function of the phenomenon.

The second principle can be a bit surprising: It is that the conservation laws of physics always stay true, even if the outcomes appear to violate the speed of light.

Put those two principles together and what you get is this: If you poke some part of a wave function (e.g. “the electron went through this slit”), you force that wave function it to give up some of its historical ambiguity.

Calculating such outcomes is old hat for physics, but a more subtle point is that when you actually measure a specific outcome, the set of histories that led to that result become the only ones that still accurately describe how the event unfolded. Think about that carefully: It means that the measurement created not just an outcome, but an entire back story for how that event could have come about. For particles that have been traveling, say, though interstellar space, that back story can reach very deep into time.

But what if your wave function involves more than one particle, e.g. two particles with opposite and fully canceling angular momentum (spin)?

You’ve got it: You end up creating the entire back story for your local result, right back to point (possibly in the very distant past) when the two particles first separated. That back history must respect and be shaped by every applicable conservation law of physics, including conservation of angular momentum.

While the very idea of creating such a back history sounds as though it would result in flagrant violations of causality, such violations turn out rather trivially to be impossible because by the definition the outcome of that quantum wave function has not yet impacted any part of the universe.

But what the newly created back history can and does impact is the other particle within the wave function. It gives that other particle a new and subtly different wave function. That is entanglement.

The new wave function will ensure that whatever conserved quantities were detected on the first particle will now be exactly balanced by those of the other particle. If for example the first particle was an electron whose spin axis was determined by the measurement to point to galactic north, then the wave function for the other electron will “point” to galactic south, in the sense that 100% of such electrons would be detected by a device pointed in exactly that direction. If the detectors are more randomly aligned the math gets a bit more complicated; that is what Bell’s inequality is all about.

Dear Terry,

My interpretation of entanglement relies on pure mathematics, clear and precise assumptions, classic theorems and demonstrations. I do not make any physical assumption about interactions, waves functions or esoteric feature of the real world. Whenever you model a system according to some mathematical formalism, this formalism entails some features, such as Hilbert space, observables, eigen vectors, probability, and tensorial product (and actually it is also true besides physics). So, from basic and clear assumptions about the formalism, you get eventually most of the results which are aknowledged by physicists through QM. The starting point is not the same, but the outcome is essentially the same.

So physicists can use this formalism, or they can stay with the usual interpretation of a quantic world (because it is clear that what is true at some scale is not true on the macroscopic world), invoke a wave function (that in any book is said to be a mystery without any physical meaning), accept violation of physical laws, interaction obserever / particles, minimal rule substitution (what a trick!),... Personally I really do not care : it is a matter of personal belief. I feel sorry for the millions of students who try to understand these mysteries. But mysteries have their charms.

It reminds me Galileo. The question "Is the Earth circling the sun, or the Sun moving around the Earth" is, for a physicist, pointless : motion is relative, so both answers are right. For 2000 years the Ptolemaic system has provided computations which were precise enough for the needs of the time. The astronmers could rightly say that their observations sustained their interpretation. The difference was that the Galleo solution was simpler, and gave the crucial step for the gravitation laws by Newton. But simplicity is bad for the business. Experts get their power from the mysteries of their craft. Science is no exception.

Abdel-Haleem is perfectly right. Entanglement is related to the difference of classical and quantum mechanics, the crucial difference being that in quantum mechanics, contrary to classical mechanics, incompatible observables exist.

Willem,

Can you give me a precise, complete, physical definition of an observable ? and the difference between compatible and incompatible observable ? And what makes that the difference between classical and QM observables ?

Jean Claude,

Please consult any textbook of quantum mechanics for a mathematical definition of a quantum mechanical observable. There is quite a bit more to tell if you would be interested in the physical meaning of an observable. I suggest that you have a look at my website http://www.phys.tue.nl/ktn/Wim/muynck.htm , where an extensive discussion is given on this subject. If any questions are left, please ask me either here (if you think they are interesting for participants of this thread), or directly via my email adress (which can be found on the website).

I know the books. Actually there are two kinds of definitions : the phsysical one (the process which gives a measure), the abstract one, based on von Numann algebras which actually takes as granted the description of a system by a Hilbert space.

Let me give you an example. Let us consider the motion of a body (it can be proton, a car, a galaxy) followed by its coordinates x function of the time t from 0 to T. The ssytem is fully known (its state) if one knwos the map x:[0,T]t->x(t). Usually physicists are not too demanding about the properties of such maps, one can safely assume that it is square integrable. The space of square integrable maps is a separable Hilbert space, so the state of the system if fully defined by such a vector ; the value of the map x. Now if we want to measure x, it is defined by countably infinite components. So any physical measure, based on a finite number of data, will give an estimation of x.Tthe issue is not the precision of the measure, but the specification that we take for the estimation (a straight line, a circle,...). The simplest estimation is just to take a finite number of components. Then the observable is the map which from x gives this estimation. This is a linear self adjoint map; an observable. It is not too difficult to see that the result will always be some eigen vector of the observable. Uusually one does not bother to identify a basis. Then the choice of a specification is random, and it is easy to see that the probability to measure any value is proportional to the square of the vector.

So from the simple mathematical formalism we get the axioms of QM. They can be fully proven, without any assumption of the "real world". There is no scale involved. The result stands at a macroscopic scale but of course usually we do noy bother with these considerations.

Of course, because only the eigen values will ever appear, one can say that the body has a random behavior. But this is an assumption whch is in no way necessary.

Of course this can be extended to more general models, the proofs stand under general, but precise conditions. And the results about interacting systems are similar..

The discrepancy between classic and quantum mechanics come mainly from the fact that the modellisation of particles and fields require more complex tools, for which these effects become signficant.

All these explanations are purely mathematical. ''If we represent the state AND interactions as a tensor ...'' . But nature is not the mathematics we use to describe it. The moon is not the word 'moon'. Stones do not fall down as a pure expression space = one half acceletation time squared, etc. So if we want to EXPLAIN entaglement we cannot resort to puere mathematics. If we understand entanglement, we should be able to explain it in words that any housewife grandma can understand.

Jean Claude,

Since you evidently think this discussion to be interesting for this thread

(and you are certainly right on this score) I will try here to summarize my views on the subject, which are more extensively discussed on my website

http://www.phys.tue.nl/ktn/Wim/muynck.htm

as well as in the papers that can be consulted there. In my view an observable should be defined as a `symbolic representation of a measurement procedure labeled by a certain mathematical entity'. Measurement results yielded by the measurement procedures are in general not seen as directly referring to properties of the measured object, but to pointer positions of measuring instruments applied in the measurement procedures.

This definition may be used both in classical and quantum mechanics, the only difference being that the mathematical label is taken from a different physical theory. The crucial difference between classical and quantum mechanics is the possibility of incompatibility of observables in the latter theory, and its absence in the former. For the classical case this definition may perhaps be felt as too weak, because, of course, measurement is carried out to obtain information on the object, the measuring instrument just being a tool intermediate between object and observer. However, even in classical mechanics there are examples where a distinction between properties of the object and properties of the measurement procedure/measuring instrument should be taken into account (think about a stick that is partially immersed in water, which appears to be broken, but which in reality does not have this property). As already stressed by Bohr and Heisenberg, in quantum mechanics measurement plays a crucial role.

This role is investigated in the quantum mechanical theory of quantum measurement in which the interaction between object and measuring instrument is studied. A result of this investigation is a theory of nonideal measurement clarifying Heisenberg's idea of mutual disturbance of measurement results in a simultaneous measurement of incompatible observables. During a long time such measurements were thought to be impossible, and indeed they are if mutual disturbance is not taken into account. However, nowadays such measurements are standard practice, showing that the broken stick effect is rather common in quantum mechanical measurement. This issue plays an important role in an assessment of the physical meaning of entanglement.

Willem de Muynck

Dear Williem,

Thanks you for your patience. Since Aristotle and Descartes we know the difference between the perception of a physical phenomenon, and the concept that we associate to this perception.This is the same in experimental sciences : we rely on data, figures corresponding to phenomena measured by specified processes. These mesures are associated to variables, formal matheùatical quantities which are used 1. to organize the collection of data, and 2. in any formal theory to proceed to computations. A system is represented in any theory (not necessarily in physics) by a set of variables. Whenever these variables, as mathematical objects, meet certain conditions,which are quite general, one gets all the basic axioms of QM. This can be mathematically proven, does not depend on the scale, or the physical phenomenon involved.

To use, in a simpler and safer way, the usual tricks of QM one does not need to invoke some special behavior of the physical world. I believe that in sciences we should adopt a conservative point of view : do not invent new concepts when this is not necessary, and certainly not assume bizarre features of the physical world. I cannot accept the usual "nobody understands QM". For me this is the negation of science.

The issue with "compatible observables" in particular seems to me a wrong basis for the foundations of QM. One cannot measure simultaneously the speed and location of a particle, and so what ? When I see the beautiful pictures taken at the exit of a collider, I guess that the the physicists can attribute both speed and location to particles when they collide, this is what matters. In almost all practical case the form of the trajectory is assumed, and the speed follows simple rules. The knowledge of a phenomenon (what it is really) on one hand, and the use of a conept and its measure on the other hand, are different things. I believe that the purpose of physics is to give an efficient way to represent the reality, in order to act upon it. The "true knowledge" of reality (what is really a particle ?) is for me a matter of philosphy. (and the philosophers do not agree between themselves on this topic).

Of course there are aspects specifics to the world of particles and fields, they come from the fact that their kinetic properties must be defined properly (mostly because of the relativist effects which can rarely be omitted). And the axioms of QM are useful to choose the right representation.

See the idea is pretty simple to grasp although I agree it is very counter-intutive.

Imagine two systems that interact initially and then move far apart after interaction. Since they initially interacted, they are described a same quantum state. Now if you measure one system with an observable, then it instantaneuosly affects the other system. For example, if I am measuring one system with a particular basis, then that measurement automatically affects the other system and cause it to collapse in one of the measurement basis states that was used to meausre system 1. This is just awesome.

But the catch is that if the two systems are held by two persons, then the person of system 2 does not know that his system is affected because he does not know whether person1 has performed an operation or not.

Take the case of |00> + |11> state. If person1 measures in 0/1 basis and obtains |0>, then person2's reduced state is now |0>. But he does not know that.

This is the beauty of entanglment. It causes change instantaneously, but since the person2 does not know about the measurements of system1, hence real information is not passed. Only when person1 conveys to person2 about his measurement operator, then can person2 conclude his resuced system state.

I hope it helps.

Niraj Kumar's answer is based on an individual-particle interpretation of the wave function or state vector, to the effect that the state vector describes a single particle

(or particle pair) rather than an ensemble of such objects. Due to this interpretation the counter-intuitive nonlocal processes which Kumar refers to are thought to happen. .In an ensemble interpretation of the state vector this counter-intuitivity disappears because the preparation of the state of (an ensemble of) one particle can be understood as a selection of a subensemble based on observation of the measurement results obtained on the other particle in a so-called coincidence experiment.

I think by now most physicists are convinced that quantum mechanics describes ensembles rather than individual systems, although many of them will keep using the individual-particle talk they have learned in the old days. They see a wave packet flying around when they talk about an electron. It should be realized, however, that wave packets are in your quantum mechanics book; they fly around only if you throw your book through the room.

To Fernando,

I agree with you about a key point : when one considers interacting systems, we need to consider tensorial products. So the only true variable (which contains all the information about the interacting systems) is a tensor. And there is no canonical (basis independant) way to resume a tensor in the tensorial product of two vectors. So one cannot define the state of the interacting systems only by the knowledge of each system (because by doing this one drops the information related to the interactions). And it is wrong to consider systems which sometimes interact and sometimes do not interact. One has to be consistent. The nature of the interactions is not involved and does not matter, so there is on locality issue.

Dear Jean Claude,

Your reasoning around x:[0,T]t->x(t) is wonderful. Unusually clear for discussions on the nature of Quantum Mechanical formalism. Lack of unambiguous QM definitions of observable and state continues to be a source of enormous amount of ambiguous and confusing pseudo theories (quantum entanglement for example). Thank you.

Claude,

Entanglement can be very complicated: non-maximally entangled two particle state or entanglement of 3 particle state are headaching. But if you just want to understand maximally entangled two particle state I would suggest a very intuitive and simple exercise. But to do that you should be able to use some programming language.

Simulate Stern-Gerlach experiment in two ways.

1. Classical. Generate one random angle and "send" one electron with spin along the angle to Alice and second electron with opposite angle to Bob. Let them make measurements in a series of pairs of bases according to CHSH test. Observe the value of S that you get after accumulating many measurements for each setting of detection bases (there will be 4 or 8 bases settings)..

2. Quantum. Imagine entangled pair of electrons. Let Alice measure her electron in her basis. Imagine that electron has random spin just before it is measured. Whatever Alice gets as a result, go to Bob and let him measure spin of his electron provided that just before measurement his electron has orientation opposite to what Alice has measured. The fact that Bob's spin is not random as before but correlated to Alice's measurement is what we call entanglement. Calculate S based on many such measurements.

In both cases the initial orientation of a first spin to be measured is RANDOM, however S will be different because in case of entanglement the second spin is 100% correlated to the first one, WITHOUT NAY FORCE or communication to the first one. That will demonstrate to you what is the essence of entanglement.

@Fernando Parisio

Entanglement is not a mathematical effect. It is information effect: A special wave function that describes a TWO PARTICLE state (something without analog in Newtoinian physics) is prepared such that it contains only 1 bit of information. it is then, in course of experiment, stretched into two opposite direction such that, because of our silly thinking in terms of isolated local events, we now believe are TWO SINGLE PARTICLES. Whatever we measure on one side must be 100% correlated to what we measure on the "other" because this is the SAME system that contains only 1 bit of information. The description of this phenomenon fails miserably in terms of distance and time: those parameters have NOTHING to do with entanglement.

Fernando,

I found your blogs very relevant. In my experience of the subject, quantum contextuality is the concept that is needed to encompass both entanglement and non-locality, and that correctly accounts for the "magicity" of quantum mechanics. In my present work, I find a new mathematical approach to generate quantum geometrical structures (finite geometries) endowed with the required contextuality. For those interested you may have a look at my paper here

http://xxx.lanl.gov/abs/1404.6986

In Ref. [11 (P. Lisonek et a)] of that paper, the concept of a conceptual graph is explained in detail.

Michel

I try not to confuse physics with mathematical physics or mathematics.

There are two entanglement concepts: mathematical and physical.

Mathematical: Given two vector spaces U, V, their tensor product (U tensor V) contains vectors that are not elements of the Cartesian product UxV and those vectors are called entangled. End of discussion.

Physical: here entanglement is contextual. A hydrogen atom can be modelled as a single point particle when observed as an atom in a very low energy classical regime, by a relatively simple entangled wavefunction of electron and proton coordinates when looked at in the low energy quantum chemistry regime, and an intractable highly entangled bound state described by relativistic quantum field theory when observed in the high energy regime.

In the quantum information regime you are concerned with, all questions concerning observation of entanglement components can be answered once empirical context had been specifed. By this I mean, tell me what you are going to measure and HOW (this is a critical aspect in quantum mechanics, the omission of which leads to all the conceptual messes people get into), and then the quantum formalism will give reliable predictions (we expect).

To George

Contextuality is perhaps a good concept but it adds a layer of complication to complicated issues.

The tensorial product contains all the information on the system, but in this representation one cannot, usually, define the states of each microsystem (that is the tensor cannot be expressed as the tensorial product of two vectors). But one can try to find an estimate : one replaces the certain value by a collection of tensor product of vectors, each with some probability. This way one can measure the states of each microsystem, but the state of the general system becomes a random variable.

Of course it should be possible to measure the exact value (the tensor) but this would require to measure not only the state of each microsystem, but also the value of their interactions. It is more efficient to measure the state of each system, for forget the interactions, and take a probabilist approach.

Actually all this is intuitive, and very similar to what is done in thermodynamics : entropy is linked to the distribution of the states of the microsystems, but that does not mean that the behavior of each microsystem obeys a probability law, just that this is convenient way to represent the total system.

The problem as usual comes from the fact that most physicists strive to find mysteries in QM, where common sense suffices. But this is less fashionable...

May be something useful

http://arxiv.org/abs/1310.3604

I agree with Jean-Claude and add the fact that entangled states are also eigenstates (teh Bell states) so that they become "unentagled" if represented in a "suitable" vector space. In Physics we can use different represantations for example the reciprocal space (wave vector coordinates) is more "natural" in manybody physics and is an everyday tool in semiconductor physics (band structure).

You can think of entanglement as correlations between 2 quantum systems. The common idea that a measurement on one side influences the state of the system on the other side is nonsense, there is no physical collapse of some physical quantity, but just a collapse of a probability distribution. The perfect classical analogy for quantum entanglement is the following classical scenario: Charlie prepares 2 envelopes, in one envelope he puts a red paper, in the other a blue one. He then distributes the 2 envelopes to Alice in Atlanta and to Bob in Boston. Of course when Bob opens his envelope, he sees a blue paper, and he learns that Alice's state is described by red. Is there any physical collapse? No, it is only a collapse in the probability space, namely conditioned on the result of Bob's "measurement" (i.e. opening of the envelope), his description of the system changes, i.e. he now knows for sure the result Alice will get when she will open her envelope. Things are very similar in the quantum case, however the correlations are "stronger", i.e. can not be simulated by a classical coin flip scenario (search for Bell inequalities). However the main idea is the same, entanglement is just a correlation between 2 objects, nothing more, nothing less. It has nothing to do with physical "collapses" or non-local influences. Classical states can also be non-separable, like a joint probability distribution p(i,j) = 0 except for p(0,0)=1/2 and p(1,1)=1/2. This is the mathematical description of the 2 envelopes scenario I mentioned before. So, in my opinion, entanglement is just a way of quantifying correlations in a Hilbert space, and the fact that correlations are stronger than the classical counterparts are a direct consequence of the non-commutativity of the Hilbert space structure.

Very nice comment ! Let's wait for citics...

To Vlad,

Whenever you consider two systems, and consider the possibility that they interact, the most economical way to represent the full system is to take the tensorial product. So you have one variable, a tensor, to represent the wkole system, including the interactions. But there is no canonical way to define a tensor as the tensorial product of two vectors, so the price to pay is that you cannot define clearly the state of each individual system. However usually, because a tensor is the sum of separable tensors, it is possible to show such a sum, but the interpretation becomes probabilist : the state of the whole system can be seen as the superposition of states where each system has a clear state, with a probability law to discriminate between these states. Of course the randomization is purely formal : ths is just a convenient way to represent the interacting systems. In your example with enveloppes clearly there is no interactions between the envelops, and in this case the tensor is separable, and there is no probability.

@jean claude,

I disagree that the scenario I mentioned can be described with a separable probability distribution. There are joint correlations, and the classical description is not factorable, i.e., is represented by a classical "state" vector [1/2 0 0 1/2]. The statistics one obtains by "measuring" the envelopes (i.e. opening them and reading out the colour) is the same one as using the maximally entangled state (|01>+|10>)/sqrt(2), but restricting the measurement basis only to Z (sigma_Z). And the interaction was just the preparation phase by Charlie, so the 2 system "interacted" before.

I am not claiming that this description capture the whole notion of entanglement (even for pure states), however I find the picture useful for building up intuition without getting lost in complicated mathematics.

I do not deny that the system comprising the two envelops can be described by tensorial product (this is just the cross distribution of two values), but the fact that they do not interact is just proven by the fact that when you know the value for one envelop, you know the value for the other. You can plug probabilities on this if you want, but there are obviously defined by the cmbination of possible outcomes. There is no need to call for Hilbert spaces, scalar product, preparation phase,classical state vector,

Interaction comes from the fact that you consider the two systems together, that is that you wand to represent the two envelops in a bigger system. It does not involve any physical interaction, only logical considerations. Whenever there is no physical interaction the tensor is separable. And you canoot say that sometimes the systems interact, and sometimes they do not interacr : from the start you consider the two envelops, so you have to be consistent all over the process.

But I know that this is not fashionable. Excuse me, but for me the purpose of science is to give answers, not to provide enigma wrapped in riddles.

@jean claude,

I have to say that I do not understand entirely your answer. I see absolutely no difference between the classical scenario I mentioned and the one obtained by measuring the maximally entangled state |01>+|10> in the computational basis. For me, the whole entanglement-being-different in the q.m. setting comes from the fact that you can change the measurement basis, i.e. use a non-compatible one. And this fact is reflected in the 2 sqrt(2) bound in CHSH inequality (vs. 2 in the classical case).

Regarding the classical scenario, before the measurement, you have absolutely no knowledge about the state of the system. After the measurement, you are collapsing the probability distribution to a deterministic one, so you basically have total information about what would the other party obtain when he/she measures. In this respect, I see absolutely no difference between the classical and quantum settings. There are no non-local influences or "spooky-action-at-a-distance", but just a collapse of the probability distribution. This was my whole point, that entanglement per se is not too much different than a classical scenario, and the difference comes from the fact that the correlations are stronger (i.e. the statistics you obtain via altering the measurement basis is indeed different from the classical case, because of the Hilbert space structure).

Vlad,

The problem is in the use of a language full of mysteries (preparation, collapse, classical,...) to describe things which are simple. If you take two systems whose states are represented by discreet variables, the best way to represent the state of the full system is by taking the cross product of the variables. A better example is that of a population that you study with a breakdown by age and income. If you think that the two variables are related, you take a berakdown by age*income, which just the tensorial product. And if the two variables are not related then you have sparate variables.

In models where the variables are functions, usually (this can be made precise) you can represent the states of systems by vectors of Hilbert spaces, and the state of the whole system of interacting systems by tensors of the tensorial product of Hilbert spaces. This does not involve in any way how the systems interact, just the fact that you consider that they may. And this is logical : you need more information to account for the interactions. If it happens that the systems do not interact, then the consideration of the separate values of the states is sufficient, you have a separable tensor. The introduction of prabilities is formal : there is no random behavior involved.

There is no magic in this, this is not even Physics. And the introduction of all the vocabulary just muddles everything.

@Willem de Muynck,

I tend to disagree with your statement: "I think by now most physicists are convinced that quantum mechanics describes ensembles rather than individual systems." Relatively recently there were lots of experiments dealing with single quantum systems, a famous example being the intermittent fluorescence experiment, where one can not satisfactory describe the behaviour of the system using a density operator approach (two well known approaches are quantum jumps and consistent histories) . That is, if you plot the intensity of the absorbed light as a function of time, you get a step-like random distribution, whereas a density operator describes just the average behaviour. I think quantum mechanics can perfectly well describe individual systems and individual experiments, if one is careful enough when defining the probability sample spaces.

@vlad the only difference I see for entanglement form your classical analogy is that here the letters can be in superposition state.  BTW I have another question which will go with ensemble/individual system debate. Can you prepare or measure a single spin or a single photon with a definite polarization?

Dear Chitrabhanu,

I suppose you can. It is very well possible to prepare single atoms, or even photons; it is routinely done by sending individual atoms through a cavity. Another question is whether quantum mechanics can completely describe such a single atom, because in general a measurement result of an arbitrary observable will be described only in a statistical sense.

I agree with Vlad. 

Regarding the questions of Zheng-shi Yu:

1) How can we know one quantum's state as soon as the other's state is tested?

The principal thing to keep in mind is that there are no two different quantum systems, as well as no two different quantum states when we talk about the entanglement. This is the soul of the entangled system - it is unseparable, i.e. different subsystems are the parts of one system which can only be disciplined by one global wave function. Thus by doing any measurement (even testing only one subsystem) you are getting the information about all the system. This is why by testing one of the two entangled particles you immediately know what is the state of the other one that you did not test.

2) Furthermore, can we control the state of a quantum? 

If the control is defined as the ability to transform the quantum state of the system from a given initial state in to a desirable final quantum state than the answer is YES. There exist several methods to to it by means of the interaction with another system. If in addition you would demand an online control of whether your system is on the right track along the transformation, in other  words, if you what to watch the system while it undergoes the transformation, than the answer is NO. This will not work, because the measurement itself will change the state of the system, it is like an extra interaction which is not the part of protocol, so you will have to start the preparation from the beginning.

In relation with entanglement, one may wonder if in the system of two entangled particles the state of one particle can be changed if we will manipulate only the second one. The trick is that we have to keep in mind that there only the global state of the two particles together if they are entangled! So, if one particle is manipulated, than the whole system in manipulated, yes. In fact, this in the main principle of quantum teleportation: one of the two entangled particles interacts with the system we need to teleport and aftert the proper measurement outcome the second particle will be in the quantum state of the teleported object. 

Very clear explanation

Thenk-you

      R. Feynman had wrote: "I think I can safely say that nobody understands quantum mechanics”. This situation remains till now. We are able to mathematically describe quantum processes, but we do not understand their physical nature.

      All this is true for the concept of entanglement. This concept still has not any clear physical definition. It isn't casual. We do not understand its physical nature. “Entanglement, according to Erwin Schrodinger the essence of quantum mechanics …” [arXiv:quant-ph/0106119]. That is all. Further, you will have been feeding by vague reasoning about the wave function, its collapse and the role of the observer. While this mathematics will leave out from your ears. (I am not sure that I correctly translate last two sentences.)

       I believe that we shall obtain the possibility to understand physical nature of entanglement only when we shall recognize at last the obvious experimental fact of inequality of forward and reversed processes in quantum physics (time reversal noninvariance). See A.T. Holster New J.Phys. 5, 130 (2003); arXiv:0706.2488v6 and my question in RG.

Perhaps in order to "understand" entanglement we have to look outside Physics. I often "feel" that there is something wrong with the more than hundred years debate opposing classical Physics and Quantum Physics.

Nowadays most Physicists encompass Information Theory and qualify it as "classical", associating it to Classical Mechanics. But if you look historically,  Information Theory has nothing to do with Classcial Physdics, it was motivated in the 30' and 40' for the optimisation of communications using digital pulse coded signals (the beginning of the digital era) on some transmission channels. The link was then made with Entropy (Statistical Physics), Shannon used this term because of its mysterious"buzz" potential.  But there is much debate and confusion leading to concepts such as "Neguentropy" which does not really clarify things. If you look at Physics there are many open questions even in the "classical" domain for example the concept of Entropy in XIXth century Thermodynamics .

With new "ideas" and ways of thinking (and "understanding") coming from Computer Science, Biology, Social Sciences... we could "rethink" it all over again and at the end perhaps "understand" entanglement.

This is just an opinion and is not inetnded to be a scientific demonstration.