@M. Pitkanen ,

Your statement

"Tensor product for state spaces of systems   makes possible entanglement in quantum theory. In classical mechanics corresponds to Cartesian product.   For classical mechanism entanglement does not make sense"

Correctly, entanglement is made possible due to tensore product between Hilbert spaces. This, however, is a purely algebraic property, and does not depend on the classical or quantum nature of a theory.

Entanglement, in fact, can be as well defined for a classical system, as the paper by Eberly clearly explains. 

What distinghuis "classical entanglement" from "quantum entanglement" is not the algebraic structure of the underlying Hilbert space, but rather the non locality.

Since classical physic is local, classical entanglement must be local, meaning that one can only entangle degrees of freedom pertaining to the same classical system, and not between different classical systems (see e.g. New. J. Phys. 17, 043024 (2015)). In quantum mechanics, however, non locality is possible, and therefore entanglement between different quantum systems (as, for example, the case of two down-converted photons) is possible.

There is no clear frontier/boundary between quantum level and macro level.

When the frontiers between and , or between and is unclear, such paradoxes are called Sorites paradoxes.

See the attached paper.

Or, when the frontier between and , or between and is unclear, one also has a Sorites paradox.

 Tensor product for state spaces of systems   makes possible entanglement in quantum theory. In classical mechanics corresponds to Cartesian product.   For classical mechanism entanglement does not make sense.   

For classical field equations entanglement (and also uncertainty principle) make sense in mathematical sense.   For instance,  for the simplest solutions of  linear  wave equations the degrees of freedom are separable : solutions are products of functions depending on single coordinate -say x,y,z. General solution is superposition of these and one has entanglement in mathematical sense between these degrees of freedom. For Schrodinger equation this indeed corresponds to entanglement although one usually does not speak entanglement between coordinates.

I think that the genuinely quantal feature is quantum theory is  state function reduction: this is something purely non-classical.  In fact, quantum TGD is purely classical theory (even fermionic oscillator algebra is purely classical notion: gamma matrix algebra for infinite-D "world of classical worlds"). State function reduction making possible consciousness is the only genuinely quantal feature.  Bell inequalities implicitly contain also the theory of state function reduction so that classical field theory and quantum theory are definitely different things.

I read the paper. Very nice and interesting!

The entanglement shown is between the polarization degree of freedom and the temporal degree of freedom of an incoherent light beam. These can be described "classically", i.e. with maxwell's equations plus classical statististics. The authors find (experimentally) that for such light, indeed these degrees of freedom are entangled, and the CHSH inequality (the most commonly used version of a Bell inequality) is violated. What this practically means is that over the measurement time, which was much longer than the coherence time of the light, the polarization of the light was changing, and therefore, one could not describe the state of the light with a single polarization for the entire duration of the measurement.

Now, what would happen if the measurement time was shorter? If the total time it takes to measure the four CHSH correlations was shorter than the coherence time, than for every single measurement of those four correlations, the light would be in a definite polarization (though it migh be changing from one set of measurements to another), and for each single set of measurements, the CHSH inequality wouldn't have been violated.

This is different than what is usually dubbed as entanglement, where the CHSH inequality is violated also when the state of the system is identical for every measurment instance.   

Nevertheless, this paper is important, since it highlights the fact that, as M. Pitkanen said in his comment above, when the mathematics of two systems are the same, the physical behaviour will be the same, although the physical circumstances that enabled the mathematical description may be different.

Dear Victor,

IMHO, classicality is a feature of experiment, not theory. I would rather refrain from saying more before looking at Eberly's paper. May I ask for a reference?

Thanks

Best

Lev

Dear Lev, The paper is attached to the question.

It could also be found on http://arxiv.org/abs/1506.01305v2.

Thanks.

From a cursory reading, Eberly confuses states of the EM field with the Heisenberg field operator. Mathematics in the paper deals only with the latter. So, IMHO, the paper is simply irrelevant.

Cited from Eilon Poem:

Nevertheless, this paper is important, since it highlights the fact that, as M. Pitkanen said in his comment above, when the mathematics of two systems are the same, the physical behaviour will be the same, although the physical circumstances that enabled the mathematical description may be different.

It seems that the physical behavior depends on the mathematics.

@R De Vuyst 

"It seems that the physical behavior depends on the mathematics."

That what depends on mathematics is an artifact of mathematics and has nothing to do with physics. Pardon my bluntness, but there are things that must be put clearly.

@M. Pitkanen ,

Your statement

"Tensor product for state spaces of systems   makes possible entanglement in quantum theory. In classical mechanics corresponds to Cartesian product.   For classical mechanism entanglement does not make sense"

Correctly, entanglement is made possible due to tensore product between Hilbert spaces. This, however, is a purely algebraic property, and does not depend on the classical or quantum nature of a theory.

Entanglement, in fact, can be as well defined for a classical system, as the paper by Eberly clearly explains. 

What distinghuis "classical entanglement" from "quantum entanglement" is not the algebraic structure of the underlying Hilbert space, but rather the non locality.

Since classical physic is local, classical entanglement must be local, meaning that one can only entangle degrees of freedom pertaining to the same classical system, and not between different classical systems (see e.g. New. J. Phys. 17, 043024 (2015)). In quantum mechanics, however, non locality is possible, and therefore entanglement between different quantum systems (as, for example, the case of two down-converted photons) is possible.

Dear Rita, 

Yes, the physical behaviour depends on the mathematics, but, one must look at what is the physical system that is described by certain mathematics, and in what circumstances it can be described in this way.

This may seem unnecessay at first, but it may be detrimental for real applications.

For example, if I have a system of two entangled modes, and I want to use them for communications, I could do that only if I can physically put them in different places. In the example described in the paper, this is not possible: one cannot take the polarization of light and put it in one place, and take the temporal profile of the light and put it in another place.

Furthermore, as I said in my previous comment, this mathematical description is not always valid, as these degrees of freedom may not be entangled if the measurements were faster than the coherence decay time.

So, while the mathematics of the system presented in the paper is the same as that of two entangled qubits, and have the same meaning in terms of the separability of the modes, which is the interesting point of the paper, one should always be aware of the physical context in which this mathematical description is applicable. I think the authors of the paper are aware of this, as they write:

"Repeated tests confirmed that such a [classical] field can strongly violate the CHSH Bell inequality and can attain Bell-violating levels of correlation similar to those found in tests of maximally entangled quantum systems... Thus one sees that Bell violation has less to do with quantum theory than previously thought, but everything to do with entanglement".

Eilon.

Not exactly. Single mode or particle also can show quantum /nonclassical behavor. For example, g2

In my view it does not - please see an old paper by of ours "Persistent entanglement in the classical limit" as linked below.

http://iopscience.iop.org/1367-2630/7/1/064

One can use entanglement not only to detect but also to quantify nonclassicality as defined in our recent paper

Article Nonclassicality quantified by standard entanglement potentia...

To Margo Orginotti and those others who possible notice my sloppy statement:

You are of course right. My own intention was to say exactly what you said and in the rest of comment I  tried my best to do it. Entanglement is purely algebraic property and does not distinguish between quantum and classical  in  mathematical sense.

Unfortunately there was however  this single hastily written  sentence.

"For classical mechanism entanglement does not make sense"

I should have said: "In  classical mechanics - understood as classical physics of point particles rather than fields - entanglement does not make sense mathematically but makes sense quantum mechanically".

The mathematical reason is that quantum  states as points of  configuration space are replaced by functions in it.  Quite generally, quantisation can be seen as transition from the configuration space to space of functions in it - kind of abstraction.  The idea about repeated second quantisation is a natural outcome and leads to the   notion of infinite primes as  repeated second quantisation of supersymmetric arithmetic QFT with states labelled by primes. This gives new number theoretic view about infinite numbers different from Cantorian view involving only the notion of "follower" (n->n+1)

Entanglement would only arise well within a quantum regime; a comon sense

answer is that the two things are unrelated.

States  are not separable if their joint probability distributions cannot be  expressed as a mere product of the marginal probabilities; they are entangled; they exhibit nonlocality and  violate Bell inequality.

The work of Vedral & coworkers (L. Henderson and V. Vedral: J. Phys. A: Math. Gen. 34, 6899, 2001) and others, have shown that the mutual information is a useful entity that quantifies correlations.There are other similar quantities, and the search is still on for  newer notion of correlations.

The question is whether we can use the concept of correlation in itself to  set a boundary between classical and quantum descriptions. Is correlation an artifical artifact that may give rise to spurious results?  Are there other yet to be discovered measures  that has been left out in our quest to seek a divide between the classical and quantum world? 

A simplier way to understand the dilemma just posed above is to ask whether  the set of all separable states is classical? We know that  the quantum discord is not zero for all separable states. So classicality does not mean separability. And unfortunately classical and entangled correlations do not always add up to give the total correlations in a system. Hence there is  need to investigate other forms of correlations besides entanglement. We must be mindful that these "other quantum correlations" need to satisfy 1) positivity, 2) be invariant under local unitary transformations and 3) should not increase with attachment of an ancilla.

So the answer to your questions is an emphatic "NO".

I do not see the existence of boundary between classical and quantum as a fruitful idea. In quantum field theories classical solutions are assigned with externals for which action is stationary.  In contrast to this, I see classical physics as an  *exact* aspect  of quantum physics.   The  quantum states are quantum superpositions of classical time evolutions: this conforms with holography.