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arXiv:1809.03321 · Physical Review A

Partial coherence and quantum correlation with fidelity and affinity distances

Chunhe Xiong, Asutosh Kumar, Minyi Huang, Sreetama Das, Ujjwal Sen, Junde Wu

Published Mar 4, 2019• 33 citations
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Abstract

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 $(2,n)$ bipartite $X$ 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.

DOI
10.1103/physreva.99.032305

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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.
A fundamental task in any physical theory is to quantify 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 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 partial coherence of a bipartite state is equal to the minimal error probability of a mixed quantum state discrimination (QSD) 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 partial coherence for (2,n)(2,n)(2,n) bipartite X-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 quantum correlation.
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.
Since quantum resource theory provides a unified framework to characterize the valuable properties of quantum states, many physically meaningful free sets in this theory often lack convexity, which limits the applicability of conventional convex resource quantifiers. It is therefore essential to develop new resource measures suitable for nonconvex free sets. We propose and systematically study a quantum resource deficiency measure DA based on quantum affinity. We verify that DA satisfies the standard axiomatic constraints and is universally valid for both coherence and entanglement. By virtue of the convex roof construction, strong monotonicity is strictly proven for coherence and entanglement. For low-dimensional systems, analytical expressions are given for coherence deficiency. For entanglement deficiency, we further provide an alternative proof which is independent of the convex roof extension. Effective upper bounds are presented for high-dimensional systems by using mixed permutation channels and maximally coherent mixed states (MCMSs). It is further proven that the deficiency of coherence of any quantum state can be estimated by the corresponding MCMS. Moreover, we reveal a general trade-off between the resource deficiency and the resource abundance (coherence or entanglement). As applications, we show that DA quantifies the operational disadvantage in subchannel discrimination and establish an uncertainty relation between the logical error probability and the deficiency of quantum states in quantum error correction, offering quantitative constraints for the preparation of logical states and the tolerance to faults.
We review the basic idea behind resource theories, where we quantify quantum resources by specifying a restricted class of operations. This divides the state space into various sets, including states which are free (because they can be created under the class of operations), and those which are a resource (because they cannot be). One can quantify the worth of the resource by the relative entropy distance to the set of free states, and under certain conditions, this is a unique measure which quantifies the rate of state to state transitions. The framework includes entanglement, asymmetry and purity theory. It also includes thermodynamics, which is a hybrid resource theory combining purity theory and asymmetry. Another hybrid resource theory which merges purity theory and entanglement can be used to study quantumness of correlations and discord, and we present quantumness in this more general framework of resource theories.
We review the basic idea behind resource theories, where we quantify quantum resources by specifying a restricted class of operations. This divides the state space into various sets, including states which are free (because they can be created under the class of operations), and those which are a resource (because they cannot be). One can quantify the worth of the resource by the relative entropy distance to the set of free states, and under certain conditions, this is a unique measure which quantifies the rate of state to state transitions. The framework includes entanglement, asymmetry and purity theory. It also includes thermodynamics, which is a hybrid resource theory combining purity theory and asymmetry. Another hybrid resource theory which merges purity theory and entanglement can be used to study quantumness of correlations and discord, and we present quantumness in this more general framework of resource theories.
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.