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Physical Review A

Characterizing quantum correlation with the correlated part of partial coherence

Chunhe Xiong, Sunho Kim, Daowen Qiu

Published Sep 27, 2019• 6 citations
Publisher

Abstract

One of the fundamental features of the quantum world that distinguishes it from the classical counterpart is the existence of quantum correlations. In this paper, we study the quantum correlation from the viewpoint of partial coherence. Partial coherence in a bipartite system often contains local coherence of a subsystem. By removing the local coherence, we show that the remaining part of partial coherence can be regarded as quantum correlations. In this way, we introduce two quantum correlations: partial correlated coherence (PCC) and minimal partial correlated coherence (MPCC). Furthermore, we prove that PCC and MPCC induced by contractive pseudodistances are good candidates of quantum correlations. For contractive pseudodistances, we also show that the induced geometric partial correlated coherence (GPCC) and geometric minimal partial correlated coherence are actually the upper and lower bound of corresponding geometric quantum discord. Particularly, GPCCs based on a family of distances are proved to be bona fide measures of quantum correlations. By connecting to local quantum uncertainty, we provide an operational meaning for geometric quantum discord and GPCC induced by affinity of distance.

DOI
10.1103/physreva.100.032334

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One of the fundamental features of the quantum world that distinguishes it from the classical counterpart is the existence of quantum correlations. In this paper, we study the quantum correlation from the viewpoint of partial coherence. Partial coherence in a bipartite system often contains local coherence of a subsystem. By removing the local coherence, we show that the remaining part of partial coherence can be regarded as quantum correlations. In this way, we introduce two quantum correlations: partial correlated coherence (PCC) and minimal partial correlated coherence (MPCC). Furthermore, we prove that PCC and MPCC induced by contractive pseudodistances are good candidates of quantum correlations. For contractive pseudodistances, we also show that the induced geometric partial correlated coherence (GPCC) and geometric minimal partial correlated coherence are actually the upper and lower bound of corresponding geometric quantum discord. Particularly, GPCCs based on a family of distances are proved to be bona fide measures of quantum correlations. By connecting to local quantum uncertainty, we provide an operational meaning for geometric quantum discord and GPCC induced by affinity of distance.
A fundamental task in any physical theory is to quantify a certain physical quantity in a meaningful way. In this paper, we show that both fidelity distance and affinity distance satisfy the strong contractibility, and the corresponding resource quantifiers can be used to characterize a large class of resource theories. Under two assumptions, namely, convexity of ``free states'' and closure of free states under ``selective free operations,'' our general framework of resource theory includes quantum resource theories of entanglement, coherence, partial coherence, and superposition. In partial coherence theory, we show that fidelity of partial coherence of a bipartite state is equal to the minimal error probability of a mixed quantum state discrimination task and vice versa, which complements the main result in Xiong and Wu [J. Phys. A: Math. Theor. 51, 414005 (2018)]. We also compute the analytic expression of fidelity of partial coherence for (2,n)(2,n)(2,n) bipartite XXX states. At last, we study the correlated coherence in the framework of partial coherence theory. We show that partial coherence of a bipartite state, with respect to the eigenbasis of a subsystem, is actually a measure of discord-type quantum correlation.
Abstract We investigate and compare three distinguished geometric measures of bipartite quantum correlations that have been recently introduced in the literature: the geometric discord, the measurement-induced geometric discord, and the discord of response, each one defined according to three contractive distances on the set of quantum states, namely the trace, Bures, and Hellinger distances. We establish a set of exact algebraic relations and inequalities between the different measures. In particular, we show that the geometric discord and the discord of response based on the Hellinger distance are easy to compute analytically for all quantum states whenever the reference subsystem is a qubit. These two measures thus provide the first instance of discords that are simultaneously fully computable, reliable (since they satisfy all the basic Axioms that must be obeyed by a proper measure of quantum correlations), and operationally viable (in terms of state distinguishability). We apply the general mathematical structure to determine the closest classical-quantum state of a given state and the maximally quantum-correlated states at fixed global state purity according to the different distances, as well as a necessary condition for a channel to be quantumness breaking.
Quantum correlation is a key to our understanding of quantum physics. In particular, it is essential for the powerful applications to quantum information and quantum computation. There exist quantum correlations beyond entanglement, such as quantum discord (QD) and measurement-induced nonlocality (MiN) [Phys. Rev. Lett. \textbf{106}, 120401(2011)]. In [Phys. Rev. A \textbf{77}, 022113(2008)], a subclass of PPT states so-called strong positive partial transposition (SPPT) states was introduced and it was conjectured there that SPPT states are separable. However, it was illustrated with examples in [Phys. Rev. A \textbf{81}, 064101(2010)] that this conjecture is not true. Viewing the original SPPT as SPPT up to part B, in the present paper, we define SPPT state up to part A and B respectively and present a separable class of SPPT states, that is the super SPPT (SSPPT) states, in terms of local commutativity. In addition, classical-quantum (CQ) states and nullity of MiN are characterized via local commutativity. Based on CQ states, the geometric measure of quantum discord (GMQD) for infinite-dimensional case is proposed. Consequently, we highlight the relation among MiN, QD(GMQD), SSPPT and separability through a unified approach for both finite- and infinite-dimensional systems: zero MiN implies zero QD(GMQD), zero QD(GMQD) signals SSPPT and SSPPT guarantees separability, but the converses are not valid.
It has been proven that the quantum discord is a more general tool to capture non-classical correlation than quantum entanglement, because there is a non-zero quantum discord in several mixed states that could not be measured by quantum entanglement. But because of optimization part in formulating quantum discord, it is very difficult and nearly impossible to find quantum discord for some quantum states. So people proposed geometric quantum discord, which in a bipartite state could be describe as the distance of the states from the closest zero-discord state. To understand better geometric quantum discord, this paper is devoted to compare it with quantum discord and measurement induced non-locality that is in some sense dual to the geometric quantum discord. As studying the quantum correlation experimentally is one of the important thing in quantum information processing, so as our example for illustrating the difference of geometric quantum discord, quantum discord and measurement induced non-locality, our quantum system is a dimer (two spin-1/2 particles) in multiple quantum nuclear magnetic resonance.
Both coherence and entanglement stem from the superposition principle, capture quantumness of a physical system, and play a central role in quantum physics. In a multipartite quantum system, coherence and quantum correlations are closely connected. In particular, it has been established that quantum coherence of a bipartite state is an important resource for its conversion to entanglement [A. Streltsov et al., Phys. Rev. Lett. 115, 020403 (2015)] and to quantum discord [J. Ma et al., Phys. Rev. Lett. 116, 160407 (2016)]. We show here that there is a very close association between partial coherence introduced by Luo and Sun [S. Luo and Y. Sun, Phys. Rev. A 96, 022136 (2017)] and quantum correlations (quantified by quantum discord) in both directions. Furthermore, we propose families of coherence measures in terms of quantum correlations and quantum Fisher information.