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International Journal of Quantum Information

Channel-based coherence of quantum states

Chengyang Zhang, Pu Perry Wang, Li‐Hua Bai, Zhihua Guo, Huaixin Cao

Published Apr 28, 2022• 2 citations
Publisher

Abstract

Quantum coherence is one of the most fundamental and striking features in quantum physics. Considered the standard coherence (SC), the partial coherence (PC) and the block coherence (BC) as variance of quantum states under some quantum channels (QCs) [Formula: see text], we propose the concept of channel-based coherence of quantum states, called [Formula: see text]-coherence for short, which contains the SC, PC and BC, but does not contain the positive operator-valued measure (POVM)-based coherence. By our definition, a state [Formula: see text] is said to be [Formula: see text]-incoherent if it is a fixed point of a QC [Formula: see text], otherwise, it is said to be [Formula: see text]-coherent. First, we find the set [Formula: see text] of all [Formula: see text]-incoherent states for some given channels [Formula: see text] and prove that the set [Formula: see text] forms a nonempty compact convex set for any channel [Formula: see text]. Second, we define [Formula: see text]-incoherent operations ([Formula: see text]-IOs) and prove that the set of all [Formula: see text]-IOs is a nonempty convex set. We also establish some characterizations of a [Formula: see text]-IO in terms of its Kraus operators. Lastly, we discuss the problem of quantifying [Formula: see text]-coherence and prove some related properties.

DOI
10.1142/s0219749922500149

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Quantum coherence is one of the most fundamental and striking features in quantum physics. Considered the standard coherence (SC), the partial coherence (PC) and the block coherence (BC) as variance of quantum states under some quantum channels (QCs) [Formula: see text], we propose the concept of channel-based coherence of quantum states, called [Formula: see text]-coherence for short, which contains the SC, PC and BC, but does not contain the positive operator-valued measure (POVM)-based coherence. By our definition, a state [Formula: see text] is said to be [Formula: see text]-incoherent if it is a fixed point of a QC [Formula: see text], otherwise, it is said to be [Formula: see text]-coherent. First, we find the set [Formula: see text] of all [Formula: see text]-incoherent states for some given channels [Formula: see text] and prove that the set [Formula: see text] forms a nonempty compact convex set for any channel [Formula: see text]. Second, we define [Formula: see text]-incoherent operations ([Formula: see text]-IOs) and prove that the set of all [Formula: see text]-IOs is a nonempty convex set. We also establish some characterizations of a [Formula: see text]-IO in terms of its Kraus operators. Lastly, we discuss the problem of quantifying [Formula: see text]-coherence and prove some related properties.
Quantum coherence, like entanglement, is a fundamental resource in quantum information. In recent years, remarkable progress has been made in formulating resource theory of coherence from a broader perspective. The notions of block-coherence and POVM-based coherence have been established. Certain challenges, however, remain to be addressed. It is difficult to define incoherent operations directly, without requiring incoherent states, which proves a major obstacle in establishing the resource theory of dynamical coherence. In this paper, we overcome this limitation by introducing an alternate definition of incoherent operations induced via coherence measures and quantify dynamical coherence based on this definition. Finally, we apply our proposed definition to quantify POVM-based dynamical coherence.
Quantum coherence is a fundamental feature of quantum mechanics and an underlying requirement for most quantum information tasks. In the resource theory of coherence, incoherent states are diagonal with respect to a fixed orthonormal basis; i.e., they can be seen as arising from a von Neumann measurement. Here, we introduce and study a generalization to a resource theory of coherence defined with respect to the most general quantum measurements, i.e., to arbitrary positive-operator-valued measures (POVMs). We establish POVM-based coherence measures and POVM-incoherent operations that coincide for the case of von Neumann measurements with their counterparts in standard coherence theory. We provide a semidefinite program that allows us to characterize interconversion properties of resource states and exemplify our framework by means of the qubit trine POVM, for which we also show analytical results.
Quantum coherence with respect to orthonormal bases has been studied extensively in the past few years. Recently, Bischof, Kampermann, and Bru\ss{} [Phys. Rev. Lett. 123, 110402 (2019)] generalized it to the case of general positive operator-valued measure (POVM) measurements. Such POVM-based coherence, including the block coherence as a special case, have significant operational interpretations in quantifying the advantage of quantum states in quantum information processing. In this work we first establish an alternative framework for quantifying the block coherence and provide several block coherence measures. We then present several coherence measures with respect to POVM measurements, and prove a conjecture on the l1{l}_{1}l1​-norm related POVM coherence measure.
Recently, there has been a renewed interest in the quantification of coherence or other coherencelike concepts within the framework of quantum resource theory. However, rigorously defined or not, the notion of coherence or decoherence has already been used by the community for decades since the advent of quantum theory. Intuitively, the definitions of coherence and decoherence should be two sides of the same coin. Therefore, a natural question is raised: How can the conventional decoherence processes, such as the von Neumann--L\"uders (projective) measurement postulation or partially dephasing channels, fit into the bigger picture of the recently established theoretical framework? Here we show that the state collapse rules of the von Neumann or L\"uders-type measurements, as special cases of genuinely incoherent operations (GIOs), are consistent with the resource theories of quantum coherence. New hierarchical measures of coherence are proposed for the L\"uders-type measurement and their relationship with measurement-dependent discord is addressed. Moreover, utilizing the fixed-point theory for C∗{C}^{*}C∗ algebra, we prove that GIOs indeed represent a particular type of partially dephasing (phase-damping) channels which have a matrix representation based on the Schur product. By virtue of the Stinespring dilation theorem, the physical realizations of incoherent operations are investigated in detail and we find that GIOs in fact constitute the core of strictly incoherent operations and generally incoherent operations and the unspeakable notion of coherence induced by GIOs can be transferred to the theories of speakable coherence by the corresponding permutation or relabeling operators.
Abstract Quantum coherence and quantum correlation are two fundamental resources in quantum information and computation. Quantum correlated states in a bipartite system are the same as the states that are coherent with respect to any product basis, while a coherent state with respect to some given product basis may not be quantum correlated. In this work, we focus on the question of creating quantum correlation from a coherent state via incoherent quantum operations. We compare characterizations of incoherent operations (IOs), maximally incoherent operations, genuinely incoherent operations and coherence-breaking operations and then explicit interrelations among these operations. Moreover, we give the specific forms of states in two-qubit systems, called super-coherent ones, that can be mapped to quantum correlated ones via some genuinely incoherent operation. We find that there exist some coherent states that are not super-coherent. Furthermore, we prove that a state can be mapped to a quantum correlated state via an incoherent operation if and only if it is coherent. Lastly, we prove that any maximally entangled state can be mapped to an incoherent one via some one-side coherence-breaking operation.
The concept of block coherence encompasses the case where experimental capabilities are not so delicate to perform arbitrary refined measurements on individual atoms. We develop a framework which facilitates further investigation of this resource theory in several respects. Using this framework, we investigate the problem of state conversion by incoherent operations and show that a majorization condition is the necessary and sufficient condition for state transformation by block-incoherent operations. We also determine the form of the maximally coherent state from which all other states and all unitary gates can be constructed by incoherent operations. Thereafter, we define the concept of block-cohering and block-decohering powers of quantum channels and determine these powers for several types of channels. Finally, we explore the relation between block coherence and a previous extension of coherence, known as kkk coherence.