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arXiv:1408.6851 · Physical Review Letters

Complementarity and Correlations

Lorenzo Maccone, Dagmar Bruß, Chiara Macchiavello

Published Apr 1, 2015• 51 citations
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Abstract

We provide an interpretation of entanglement based on classical correlations between measurement outcomes of complementary properties: States that have correlations beyond a certain threshold are entangled. The reverse is not true, however. We also show that, surprisingly, all separable nonclassical states exhibit smaller correlations for complementary observables than some strictly classical states. We use mutual information as a measure of classical correlations, but we conjecture that the first result holds also for other measures (e.g., the Pearson correlation coefficient or the sum of conditional probabilities).

DOI
10.1103/physrevlett.114.130401

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We provide an interpretation of entanglement based on classical correlations between measurement outcomes of complementary properties: States that have correlations beyond a certain threshold are entangled. The reverse is not true, however. We also show that, surprisingly, all separable nonclassical states exhibit smaller correlations for complementary observables than some strictly classical states. We use mutual information as a measure of classical correlations, but we conjecture that the first result holds also for other measures (e.g., the Pearson correlation coefficient or the sum of conditional probabilities).
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Quantum entanglement is one of the essential resources in quantum information processing. It is of importance to verify whether a quantum state is entangled. At present, a typical quantum certification focused on the classical correlations has attracted widespread attention. Here, we experimentally investigate the relation between quantum entanglement and the classical complementary correlations based on the mutual information, Pearson correlation coefficient and mutual predictability of two-qubit states. Our experimental results show the classical correlations for complementary properties have strong resolution capability to verify entanglement for two qubit pure states and Werner states. We find that the resolution capability has great performance improvement when the eigenstates of the measurement observables constitute a complete set of mutually unbiased bases. For Werner states in particular, the classical complementary correlations based on the Pearson correlation coefficient and mutual predictability can provide the ultimate bounds to certify entanglement.
A bipartite state is called classical (with regard to correlations) if it is left undisturbed by a certain local von Neumann measurement, and is called separable if it can be represented as a convex combination of product states. Due to the perfect distinguishability of orthogonal vectors, a classical state can essentially be identified with a convenient bivariate probability distribution, and moreover, it is separable, but not vice versa. The notion of separability plays a key role in quantum information theory because entanglement is defined via separability. However, the definition of separability is ad hoc and formal. In this paper, we present an intrinsic characterization of separable states via classical states from the measurement perspective: Separable states are precisely those states that are reductions of classical states in higher dimensions with the natural partitions. Consequently, entangled states are precisely those states that cannot be represented as such reductions of classical states. This observation highlights the hidden mutually exclusive and complementary relations between classicality and entanglement.
Within the framework of Information Theory, the existence of correlations between two random variables means that we can obtain information about one of them, just by measuring or observing the other random variable. In certain cases, this kind of relationship allows obtaining information about a variable even when the other is separated by a very large distance, that is, the process of obtaining information is non-local, an example (if not the only) is the quantum entanglement. These features of correlations make it interesting and important to study, classify and quantify them. The correlations are classified into classical correlations and quantum correlations, in addition they are quantified through the mutual information. Here we will present a natural way to define classical mutual information and then we will generalize it to the quantum case. Furthermore, every term in the definitions of mutual information will be interpreted using the concepts of classical and quantum entropy.
Nonclassical correlations between measurement results make entanglement the essence of quantum physics and the main resource for quantum information applications. Surprisingly, there are n-particle states which do not exhibit n-partite correlations at all but still are genuinely n-partite entangled. We introduce a general construction principle for such states, implement them in a multiphoton experiment and analyze their properties in detail. Remarkably, even without multipartite correlations, these states do violate Bell inequalities showing that there is no classical, i.e., local realistic model describing their properties.