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Frontiers of Physics

Quantifying quantum correlation via quantum coherence

Guangyong Zhou, Lin-Jian Huang, Jun-Ya Pan, Liyun Hu, Jie‐Hui Huang

Published Jun 23, 2018• 7 citations
Publisher

Abstract

Resource theory is applied to quantify the quantum correlation of a bipartite state and a computable measure is proposed. Since this measure is based on quantum coherence, we present another possible physical meaning for quantum correlation, i.e., the minimum quantum coherence achieved under local unitary transformations. This measure satisfies the basic requirements for quantifying quantum correlation and coincides with concurrence for pure states. Since no optimization is involved in the final definition, this measure is easy to compute irrespective of the Hilbert space dimension of the bipartite state.

DOI
10.1007/s11467-018-0804-0

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Resource theory is applied to quantify the quantum correlation of a bipartite state and a computable measure is proposed. Since this measure is based on quantum coherence, we present another possible physical meaning for quantum correlation, i.e., the minimum quantum coherence achieved under local unitary transformations. This measure satisfies the basic requirements for quantifying quantum correlation and coincides with concurrence for pure states. Since no optimization is involved in the final definition, this measure is easy to compute irrespective of the Hilbert space dimension of the bipartite state.
To measure the quantum correlation of a bipartite state, a test matrix is constructed through the commutations among the blocks of its density matrix, which turns out to be a zero matrix for a classical state with zero quantum correlation, and a nonzero one for a quantum state with positive quantum correlation. The Frobenius norm of the test matrix is used to measure the quantum correlation, which satisfies the basic requirements for a good measure and coincides with Wootters concurrence for two-qubit pure states. Since no optimization is involved in the definition, this measure of quantum correlation is easy to compute and even can be calculated manually.
Consider a two-party correlation that can be generated by performing local measurements on a bipartite quantum system. A question of fundamental importance is to understand how many resources, which we quantify by the dimension of the underlying quantum system, are needed to reproduce this correlation. In this Letter, we identify an easy-to-compute lower bound on the smallest Hilbert space dimension needed to generate a given two-party quantum correlation. We show that our bound is tight on many well-known correlations and discuss how it can rule out correlations of having a finite-dimensional quantum representation. We show that our bound is multiplicative under product correlations and also that it can witness the nonconvexity of certain restricted-dimensional quantum correlations.
Quantum coherence and quantum correlations are of fundamental and practical significance for the development of quantum mechanics.They are also cornerstones of quantum computation and quantum communication theory. Searching physically meaningful and mathematically rigorous quantifiers of them are long-standing concerns of the community of quantum information science, and various faithful measures have been introduced so far. We review in this paper the measures of discordlike quantum correlations for bipartite and multipartite systems, the measures of quantum coherence for any single quantum system, and their relationship in different settings. Our aim is to provide a full review about the resource theory of quantum coherence, including its application in many-body systems, and the discordlike quantum correlations which were defined based on the various distance measures of states. We discuss the interrelations between quantum coherence and quantum correlations established in an operational way, and the fundamental characteristics of quantum coherence such as their complementarity under different basis sets, their duality with path information of an interference experiment, their distillation and dilution under different operations, and some new viewpoints of the superiority of the quantum algorithms from the perspective of quantum coherence. Additionally, we review properties of geometric quantum correlations and quantum coherence under noisy quantum channels. Finally, the main progresses for the study of quantum correlations and quantum coherence in the relativistic settings are reviewed. All these results provide an overview for the conceptual implications and basic connections of quantum coherence, quantum correlations, and their potential applications in various related subjects of physics.
Abstract Quantum coherence characterizes the non-classical feature of a single party system with respect to a local basis. Based on a recently introduced resource framework, coherence can be regarded as a resource and be systematically manipulated and quantified. Operationally, coherence quantifies the intrinsic randomness of the outcome of the projective measurement in the system’s computational basis. However, such a relation is only proven when randomness is characterized by the von Neumann entropy. In this work, we consider several recently proposed coherence measures and relate them to the general uncertainties of the projective measurement outcome conditioned on all the other systems. Our work thus provides a unified framework for redefining several coherence measures via general conditional entropies. Based on the relation, we numerically calculate the coherence measures via semi-definite programming. Furthermore, we discuss the operational meaning of the unified definition. Our result highlights the close relation between single partite coherence and bipartite quantum correlation.
Quantum correlations in a physical system are usually studied with respect to a unique and fixed decomposition of the system into subsystems, without fully exploiting the rich structure of the state space. Here, we show several examples in which the consideration of different ways to decompose a physical system enhances the quantum resources and accounts for a more flexible definition of quantumness measures. Furthermore, we give a different perspective regarding how to reassess the fact that local operations play a key role in general quantumness measures that go beyond entanglement---as discordlike ones. We propose a family of measures to quantify the maximum quantumness of a given state. For the discord-based case, we present some analytical results for $2\ifmmode\times\else\texttimes\fi{}d$-dimensional states. Applying our definition to low-dimensional bipartite states, we show that different behaviors can be reported for separable and entangled states vis-\`a-vis those corresponding to the usual measures of quantum correlations. We show that there is a close link between our proposal and the criterion to witness quantum correlations based on the rank of the correlation matrix, proposed by Daki\ifmmode \acute{c}\else \'{c}\fi{}, Vedral, and Brukner [Phys. Rev. Lett. 105, 190502 (2010)].
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.