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International Journal of Quantum Information

Tripartite Bell-type inequalities for quantum coherence and skew information

Liang Qiu, Fei Pan, Zhi Liu

Published Jun 1, 2017• 5 citations
Publisher

Abstract

We construct the tripartite Bell-type inequalities of product states for [Formula: see text]-norm of coherence, relative entropy of coherence and skew information. We give examples of three-qubit entangled states that violate these inequalities. Particularly, the tripartite Bell-type inequality for relative entropy of coherence is always violated by the W class pure or mixed states as well as the GHZ class pure or mixed states, being used as entanglement witness.

DOI
10.1142/s0219749917500253

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We construct the tripartite Bell-type inequalities of product states for [Formula: see text]-norm of coherence, relative entropy of coherence and skew information. We give examples of three-qubit entangled states that violate these inequalities. Particularly, the tripartite Bell-type inequality for relative entropy of coherence is always violated by the W class pure or mixed states as well as the GHZ class pure or mixed states, being used as entanglement witness.
Bell inequality is a mathematical inequality derived using the assumptions of locality and realism. Its violation guarantees the existence of quantum correlations in a quantum state. Bell inequality acts as an entanglement witness in the sense that a pure bipartite quantum state, having nonvanishing entanglement, always violates a Bell inequality. We construct Bell-type inequalities for product states in quantum coherence theory for different measures of coherence, and find that the maximally entangled states violate these inequalities. We further show that Bell-type inequalities for relative entropy of coherence is violated by all two-qubit pure entangled states, serving as an entanglement witness.
It is well known that Bell inequalities are derived under the assumptions of locality and realism. Bell inequalities impose strict constraints on the statistical correlations of measurements of multipartite systems. Violating each of them guarantees the existence of quantum correlations in a quantum state. A quantum state with non-vanishing entanglement may violate some Bell inequalities. Recent progress of the fields like quantum biology and quantum thermodynamics reveals a particular role of quantum coherence in quantum information processing. Quantum coherence is identified by the presence of off-diagonal terms in the density matrix. To quantify quantum coherence of a given state, Baumgratz et al. (Baumgratz T, Cramer M, Plenio M B 2014 Phys. Rev. Lett. 113 140401) provided several kinds of coherence measures such as l1-norm of coherence and relative entropy of coherence. In this paper, we propose to use quantum coherence to derive Bell inequalities. We construct the Bell inequalities of four-partite product states with l1-norm of coherence, relative entropy of coherence. In the Bell inequalities of four-partite correlations, measurement operators are products of local measurement operators. Each local operator is one of the two arbitrary observables. We consider the violations of the four-partite Bell inequalities by the four-partite general pure Greenberger-Horne-Zeilinger (GHZ) state, cluster states, W states with real coefficients. We also investigate the violations of the four-partite Bell inequalities by the four-partite GHZ class mixed states, cluster class mixed states, W class mixed states and Dicke class mixed states. It is shown that the four-partite Bell inequalities in terms of relative entropy of coherence are always violated by the four-partite general pure GHZ states, cluster states with the real coefficients. Hence there is non-vanishing entanglement for these states.
Bell inequality is a mathematical inequality derived using the assumptions of locality and realism. Its violation guarantees the existence of quantum correlations in a quantum state. Bell inequality acts as an entanglement witness in the sense that a pure bipartite quantum state, having nonvanishing entanglement, always violates a Bell inequality. We construct Bell-type inequalities for product states in quantum coherence theory for different measures of coherence, and find that the maximally entangled states violate these inequalities. We further show that Bell-type inequalities for relative entropy of coherence is violated by all two-qubit pure entangled states, serving as an entanglement witness.
We put forward complementary relations of entanglement, coherence, steering inequality violation, and Bell nonlocality for arbitrary three-qubit states. We show that two families of genuinely entangled three-qubit pure states with a single parameter exist, and they exhibit maximum coherence and steering inequality violation for a fixed amount of negativity, respectively. It is found that the negativity is exactly equal to the geometric mean of bipartite concurrences for the three-qubit pure states, although the negativity is always less than or equal to the latter for three-qubit mixed states. Moreover, the complementary relation between negativity and first-order coherence for tripartite entanglement states is established. Furthermore, we investigate the close relation between the negativity and the maximum steering inequality violation. In addition, the complementary relation between negativity and the maximum Bell-inequality violation for arbitrary three-qubit states is obtained. The results provide reliable evidence of fundamental connections among entanglement, coherence, steering inequality violation, and Bell nonlocality.
Abstract We study the skew information-based coherence of quantum states and derive explicit formulas for Werner states and isotropic states in a set of autotensors of mutually unbiased bases (MUBs). We also give surfaces of skew information-based coherence for Bell-diagonal states and a special class of X states in both computational basis and in MUBs. Moreover, we depict the surfaces of the skew information-based coherence for Bell-diagonal states under various types of local nondissipative quantum channels. The results show similar as well as different features compared with relative entropy of coherence and l 1 norm of coherence.