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This article has been brought to my attention recently (free access) : Shifting the quantum-classical boundary: theory and experiment for statistically classical optical fields

In the abstract, they claim that "The growing recognition that entanglement is not exclusively a quantum property [...] prompts the examination of its role in marking the quantum-classical boundary." To this purpose, they used a classical (statistically speaking) ergodic stochastic optical field and proved theoretically and experimentally that the CHSH inequality could be violated, if instead of using quantum states, one uses classical internal degrees of freedom of the field.

More precisely, they consider the case of completely unpolarized light for which it is impossible to write the electrical field as a product $E(t)\:\vec{u}$ for all $t$. Considering the temporal amplitude $E(t)$ and the polarization vector $\vec{u}$ as two internal degrees of freedom, they use linear algebra to show that the situation is equivalent to two entangled quantum states and that it then should violate Bell inequality.

I have no problem admitting that even for statistically classical fields, one can find strong correlations between two degrees of freedom, thus violating the CHSH inequality. But all that means to me is that, these degrees of freedom are not locally independent, ie that one cannot associate local hidden variables to each one of them that would explain the behaviour of the field. That doesn't mean at all that they are actually entangled, because for me entanglement is purely a quantum property.

My question is then the following : are strong classical correlations violating Bell inequality really implying any form of "classical entanglement" or is this just a fancy name ? A precise characterization of entanglement would also be appreciated to see if it is (as I thought until now) intrinsically a quantum property.

Thanks for the help.

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It is important to make a distinction between the "entanglement" they are talking about, and the entanglement that occurs in quantum mechanics. The way I see it, the usual definition of entanglement is purely quantum, in virtue of explicitly referring to the Hilbert space structure of quantum mechanics. An entangled state is, by definition, a state for which there exists no basis in which it is a product state.

Also it is worth pointing out that by itself, a violation of the CHSH inequality does not allow to conclude anything about non-locality. Indeed, an easy way to violate the CHSH inequality would be to use a computer to simulate the results of a Bell experiment. In order for a violation of the CHSH inequality to mean that local hidden variables are insufficient for describing the results of an experiment, you need a reason to believe that the degrees of freedom under study cannot causally affect each other. Typically we require two parties to be space like separated, and then it make sense to assume that the measurement settings of party A should not affect the notice state at B. It is perhaps worth pointing out that for concrete experiments, there is a lot of room for debate (https://en.wikipedia.org/wiki/Loopholes_in_Bell_test_experiments)

In the experiment with classical light that is under discussion, it is clear that Maxwell's electromagnetism can be seen as a ¨local hidden variable model¨ that describes the outcomes of the measurements.

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  • $\begingroup$ Thanks. I am still confused about the word "entanglement". In my quantum optics lecture notes, it is shown that a standing wave of one photon can obviously be expressed as a factorized state in the basis of standing wave modes. However in the basis of travelling wave modes, a standing wave of one photon is expressed as a Bell state, ie a superposition of one photon travelling to the right and one photon travelling to the left . The conclusion of the notes is : the entanglement of a quantum state is not necessarily an intrinsic property. Is this a misuse of the word "entanglement" ? $\endgroup$ Dec 24, 2015 at 9:03
  • $\begingroup$ The entanglement of a quantum state is always an intrinsic property, in the sense that it is independent of the basis in which the state is expressed. When talking about quantum fields such as photons, the discussion is a bit more subtle, but there can be entanglement between the modes of a single photon. You might find this paper interesting journals.aps.org/pra/abstract/10.1103/PhysRevA.72.064306 $\endgroup$
    – philippe
    May 5, 2016 at 10:17

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