For a given density matrix $$\rho = \sum_{ijkl=0}^1 r_{ijkl} |i,j\rangle \langle k,l|$$ describing a bipartite two-qubit system, how can I prove for what values $r_{ij}$ the density matrix violates Bell's inequality?
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$\begingroup$ Is this an open research problem? I searched a bit, and it seems there are some criteria, but they do not appears constructive in nature, rather just existential. $\endgroup$– Mahir LokvancicCommented Feb 15, 2022 at 18:38
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$\begingroup$ Is it the Bell inequality (that's easy to find out by calculating the violation of the Bell inequality) or a Bell inequality? $\endgroup$– Norbert SchuchCommented Feb 15, 2022 at 19:35
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$\begingroup$ There are various measures of entanglement arxiv.org/abs/quant-ph/0504163. There are also some tests for density matrices that are stronger than Bell's inequality, such as arxiv.org/abs/quant-ph/9604005 $\endgroup$– alanfCommented Feb 16, 2022 at 14:51
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$\begingroup$ This paper answers your question: sciencedirect.com/science/article/abs/pii/… $\endgroup$– BardCommented Nov 1, 2022 at 17:52
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1 Answer
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The density matrix you give is diagonal in a product basis - that is, it is described by a local hidden variable model. It thus will not violate any Bell inequalities, regardless of the value of $r_{ij}$.
answered Feb 14, 2022 at 23:19
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$\begingroup$ I made a small mistake writing a density matrix in a diagonal product basis. I changed the density matrix how is it now? $\endgroup$– PhicalcCommented Feb 15, 2022 at 10:03
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$\begingroup$ Please do not edit the question in a way which completely alters its meaning. Post a new question. Otherwise I look like I did not read your question properly. $\endgroup$ Commented Feb 15, 2022 at 19:11