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CHSH violation and entanglement of quantum states

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How is the violation of the usual CHSH inequality by a quantum state related to the entanglement of that quantum state?

Say we know that exist Hermitian and unitary operators $A_{0}$, $A_{1}$, $B_{0}$ and $B_{1}$ such that $$\mathrm{tr} ( \rho ( A_{0}\otimes B_{0} + A_{0} \otimes B_{1} + A_{1}\otimes B_{0} - A_{1} \otimes B_{1} )) = 2+ c > 2,$$ then we know that the state $\rho$ must be entangled. But what else do we know? If we know the form of the operators $A_{j}$ and $B_{j}$, then there is certainly more to be said (see e.g. http://prl.aps.org/abstract/PRL/v87/i23/e230402 ). However, what if I do not want to assume anything about the measurements performed?

Can the value of $c$ be used to give a rigourous lower bound on any of the familar entanglement measures, such as log-negativity or relative entropy of entanglement?

Clearly, one could argue in a slightly circular fashion and define an entanglement measure as the maximal possible CHSH violation over all possible measurements. But is there anything else one can say?

This post has been migrated from (A51.SE)
asked Nov 18, 2011 in Theoretical Physics by Earl (405 points) [ no revision ]
@PiotrMigdal: Perhaps you should post that as an answer.

This post has been migrated from (A51.SE)
@PiotrMigdal Thanks for your comment. I had not thought of goggling for "device-independent" and was not aware of that paper. It seems to answer my question, though I'm still going through some of the details.

This post has been migrated from (A51.SE)
Your questions is answered here: [arXiv:0907.2170](http://arxiv.org/abs/0907.2170). BTW, _device-independent_ is the key phrase to search for.

This post has been migrated from (A51.SE)

1 Answer

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In a paper C.-E. Bardyn et al., PRA 80(6): 062327 (2009), arXiv:0907.2170, they discuss constrains on the state, given how much the CHSH equality is violated ($S=2+c$), but without putting any assumptions on the operator used.

In general people consider schemes, when operators (for a Bell-type measurement) are random or one or more parties cannot be trusted. One of the key phrases is device-independent and maybe also loophole-free (as even a slight misalignment of operators may change the results dramatically).

This post has been migrated from (A51.SE)
answered Nov 21, 2011 by Piotr Migdal (1,250 points) [ no revision ]

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