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Physical Review A

Information-theoretic metric as a tool to investigate nonclassical correlations

Paweł Kurzyński, Dagomir Kaszlikowski

Published Jan 6, 2014• 21 citations
Publisher

Abstract

In this paper we study an application of an information-theoretic distance between two measurements to investigate nonclassical correlations. We postulate the triangle principle, which states that any information-theoretic distance is valid for any sets of measurements, regardless if they can be jointly measured or not. As a consequence, the triangle inequality for this distance is obeyed for any three measurements. This simple principle is valid in any classical realistic theory, however, it may not hold in quantum theory. It leads to the derivation of certain inequalities whose violations are an indicator of nonclassicality. Some of these inequalities formally look the same as those found in the literature on local realism and noncontextuality, but we also derive completely different inequalities. We also show that our geometrical approach naturally implies monogamy of nonclassical correlations.

DOI
10.1103/physreva.89.012103

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In this paper we study an application of an information-theoretic distance between two measurements to investigate nonclassical correlations. We postulate the triangle principle, which states that any information-theoretic distance is valid for any sets of measurements, regardless if they can be jointly measured or not. As a consequence, the triangle inequality for this distance is obeyed for any three measurements. This simple principle is valid in any classical realistic theory, however, it may not hold in quantum theory. It leads to the derivation of certain inequalities whose violations are an indicator of nonclassicality. Some of these inequalities formally look the same as those found in the literature on local realism and noncontextuality, but we also derive completely different inequalities. We also show that our geometrical approach naturally implies monogamy of nonclassical correlations.
In this paper we study an application of an information distance between two measurements to the problem of non-contextuality and local realism. We postulate the triangle principle which states that any information distance is well defined on any pair of measurements, even if the two measurements cannot be jointly performed. As a consequence, the triangle inequality for this distance is obeyed for any three measurements. This simple principle is valid in any classical realistic theory, however it does not hold in quantum theory. It allows us to re-derive in an astonishingly simple way a large class of non-contextuality and local realistic inequalities via multiple applications of the triangle inequality. We also show that this principle can be applied to derive monogamy relations. The triangle principle is different than the assumption of non-contextuality and local realism, which is defined as a lack of existence of a joined probability distribution giving rise to all measurable data. Therefore, we show that one can design Bell-Kochen-Specker tests using a different principle.
It has long been recognized that certain quantum correlations are incompatible with a particular assumption about classical causal structure. Given a causal structure of unknown classicality, the presence of such correlations certifies the nonclassical nature of the causal structure in a device-independent fashion. In structures where all parties share a common resource, these nonclassical correlations are also known as nonlocal correlations. Any constraint satisfied by all correlations which are classically compatible with a given causal structure defines a causal compatibility criterion. Such criteria were recently derived for the triangle structure (E. Wolfe et al., arXiv:1609.00672) in the form of polynomial inequalities, begging the question of whether any of those inequalities admit violation by quantum correlations. Numerical investigation suggests that they do not, and we further conjecture that the set of correlations admitted by the classical triangle structure is equivalent to the set of correlations admitted by its quantum generalization whenever the three observable variables are binary. Our main contribution in this work, however, is the derivation of causal compatibility inequalities for the triangle structure which do admit quantum violation. This provides a robust-to-noise witness of quantum correlations in the triangle structure. We conclude by considering the possibility of quantum resources potentially qualitatively different from those known previously.
We consider information-theoretic inequalities of the Bell type in the presence of decoherence. It is natural that too strong coupling with the environment can prevent an observation of quantum correlations. In this regard, the use of various entropic functions may give additional capabilities to reveal desired correlations. It was already shown that the Bell and Leggett-Garg inequalities in terms conditional Tsallis entropies are more sensitive in the cases of detection inefficiencies. In this paper, we study capabilities of generalized conditional entropies of the Tsallis type in analyzing the Bell theorem in decoherence scenarios. Two forms of the conditional Tsallis qqq-entropy are known in the literature. We show that each of them can be used for defining a metric in the probability space of interest. Such metrics can be used in realizing the so-called triangle principle. The triangle principle has recently been proposed as a unifying approach to questions of local realism and non-contextuality. Applying the triangle principle leads to the two families of qqq-metric inequalities of the Bell type. Information-theoretic formulations in terms of the qqq-entropic metrics are first discussed for the CHSH scenario in dephasing environment. Then we also revisit qqq-entropic inequalities of the Leggett-Garg type. An environmental influence is modeled by the phase damping channel and by the depolarizing channel.
We introduce a multipartite extension of an information-theoretic distance introduced by Zurek [Nature (London) 341, 119 (1989)]. We use it to develop a new tool for studying quantum correlations from an information-theoretic perspective. In particular, we derive entropic tests of multipartite nonlocality for three qubits and for an arbitrary even number of qubits as well as a test of state-independent contextuality. In addition, we rederive the tripartite Mermin inequality and a state-independent noncontextuality inequality by Cabello [Phys. Rev. Lett. 101, 210401 (2008)]. This suggests that the information-theoretic distance approach to multipartite nonlocality and state-independent contextuality can provide a more general treatment of nonclassical correlations than the orthodox approach based on correlation functions.
Quantum mechanics can produce correlations that are stronger than classically allowed. This stronger–than–classical correlation is the “fuel” for quantum computing. In 1991 Schumacher forwarded a beautiful geometric approach, analogous to the well-known result of Bell, to capture non-classicality of this correlation for a singlet state. He used a well-established information distance defined on an ensemble of identically–prepared states. He calculated that for certain detector settings used to measure the entangled state, the resulting geometry violated a triangle inequality — a violation that is not possible classically. This provided a novel information–based geometric Bell inequality in terms of a “covariance distance.” Here we experimentally reproduce his construction and demonstrate a definitive violation for a Bell state of two photons based on the usual spontaneous parametric down-conversion in a paired BBO crystal. The state we produced had a visibility of Vad=0.970. We discuss generalizations to higher dimensional multipartite quantum states.