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Physical Review A

Information and quantum nonseparability

Berthold W. Schumacher

Published Dec 1, 1991• 43 citations
Publisher

Abstract

An information-theoretic inequality analogous to the well-known result of Bell [Physics 1, 195 (1964)] is formulated using the concept of information distance. This inequality, like Bell's, is true for all local-hidden-variable theories, but not for quantum mechanics. The metric space structure of this new inequality suggests a reformulation of familiar Bell inequalities in terms of a ``covariance distance.'' Quantum nonseparability can be demonstrated through violations of these inequalities even in cases where the correlation between two systems is extremely weak. The connection between nonseparability and complementarity is also briefly discussed in this paper.

DOI
10.1103/physreva.44.7047

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An information-theoretic inequality analogous to the well-known result of Bell [Physics 1, 195 (1964)] is formulated using the concept of information distance. This inequality, like Bell's, is true for all local-hidden-variable theories, but not for quantum mechanics. The metric space structure of this new inequality suggests a reformulation of familiar Bell inequalities in terms of a ``covariance distance.'' Quantum nonseparability can be demonstrated through violations of these inequalities even in cases where the correlation between two systems is extremely weak. The connection between nonseparability and complementarity is also briefly discussed in this paper.
The correlation distance quantifies the statistical independence of two classical or quantum systems, via the distance from their joint state to the product of the marginal states. Tight lower bounds are given for the mutual information between pairs of two-valued classical variables and quantum qubits, in terms of the corresponding classical and quantum correlation distances. These bounds are stronger than the Pinsker inequality (and refinements thereof) for relative entropy. The classical lower bound may be used to quantify properties of statistical models that violate Bell inequalities. Entangled qubits can have a lower mutual information than can any two-valued classical variables having the same correlation distance. The qubit correlation distance also provides a direct entanglement criterion, related to the spin covariance matrix. Connections of results with classically-correlated quantum states are briefly discussed. 1
In 1935, Albert Einstein and two colleagues, Boris Podolsky and Nathan Rosen (EPR) developed a thought experiment to demonstrate what they felt was a lack of completeness in quantum mechanics (QM). EPR also postulated the existence of more fundamental theory where physical reality of any system would be completely described by the variables/states of that fundamental theory. This variable is commonly called hidden variable and the theory is called hidden variable theory (HVT). In 1964, John Bell proposed an empirically verifiable criterion to test for the existence of these HVTs. He derived an inequality, which must be satisfied by any theory that fulfill the conditions of locality and reality. He also showed that QM, as it violates this inequality, is incompatible with any local-realistic theory. Later it has been shown that Bell’s inequality (BI) can be derived from different set of assumptions and it also find applications in useful information theoretic protocols. In this review, we will discuss various foundational as well as information theoretic implications of BI. We will also discuss about some restricted nonlocal feature of quantum nonlocality and elaborate the role of Uncertainty principle and Complementarity principle in explaining this feature.
It is one of the most remarkable features of quantum physics that measurements on spatially separated systems cannot always be described by a locally causal theory. In such a theory, the outcomes of local measurements are determined in advance solely by some unknown (or hidden) variables and the choice of local measurements. Correlations that are allowed within the framework of a locally causal theory are termed classical. Typically, the fact that quantum mechanics does not always result in classical correlations is revealed by the violation of Bell inequalities, which are constraints that have to be satisfied by any classical correlations. It has been known for a long time that entanglement is necessary to demonstrate nonclassical correlations, and hence a Bell inequality violation. However, since some entangled quantum states are known to admit explicit locally causal models, the exact role of entanglement in Bell inequality violation has remained obscure. This thesis provides both a comprehensive review on these issues as well as a report on new discoveries made to clarify the relationship between entanglement and Bell inequality violation.
The concept of nonlocality and its role in the violation of Bell's inequalities are discussed.A weaker nonlocality is revealed in comparison with Albert Einstein's ``Spooky Action at a Distance'' and at the same time does not contradict the principles of the special theory of relativity (STR).A correlation statistical experiment with the violation of the Clauser±Horne±Shimony± Holt inequality in STR is described, in which weak nonlocality is manifested.The correlation in such an experiment propagates with infinite speed in the proper frame of reference of the measuring device.
The correlation distance quantifies the statistical independence of two classical or quantum systems, via the distance from their joint state to the product of the marginal states. Tight lower bounds are given for the mutual information between pairs of two-valued classical variables and quantum qubits, in terms of the corresponding classical and quantum correlation distances. These bounds are stronger than the Pinsker inequality (and refinements thereof) for relative entropy. The classical lower bound may be used to quantify properties of statistical models that violate Bell inequalities. Partially entangled qubits can have lower mutual information than can any two-valued classical variables having the same correlation distance. The qubit correlation distance also provides a direct entanglement criterion, related to the spin covariance matrix. Connections of results with classically-correlated quantum states are briefly discussed.