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arXiv:quant-ph/0608004

Limitations on entropic Bell inequalities

Ian T. Durham

Published Aug 1, 2006• 1 citation
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Abstract

The derivation of Bell inequalities in terms of quantum statistical (thermodynamic) entropies is considered. Inequalities of the Wigner form are derived but shown to be extremely limiting in their applicability due to the nature of the density matrices involved. This also helps to identify a limitation in the Cerf-Adami inequalities.
DOI
10.48550/arxiv.quant-ph/0608004

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Bell's inequalities, in the form given by Cerf and Adami, are derived from the combination of the second law of thermodynamics and the Markov postulate. Violations of these inequalities are discussed in terms of the mixing characteristics of the density operators. A quantum mechanical bound for a system of three qubits, two of which are entangled, is found and is shown to lead to strong subadditivity in one case. Finally, in interpreting the results, the role the uncertainty relations play in the definition of entropy and the second law is discussed, demonstrating a link to Bell's inequalities. Additionally the second law is shown to be dependent on the type of measurement in quantum systems. Specifically, a more general statement of the second law appropriate to both quantum and classical systems is suggested.
We show that the Cerf-Adami inequalities do not necessarily depend on conditional entropies nor any reference to Markov chains. While the latter are not explicit in the original form, they are often implied in certain derivations. We also show that these inequalities are intimately related to at least one interpretation of the second law of thermodynamics. The combination of these results provides added insight into why some quantum systems violate the Cerf-Adami inequalities thereby improving our understanding of the quantum-classical boundary. As a result we suggest that the second law may serve as some type of boundary condition on classical knowledge.
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