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Chinese Physics B

Quantum block coherence with respect to projective measurements

Pu Wang, Zhong-Yan Li, Hui-Xian Meng

Published May 28, 2024• 2 citations
Publisher

Abstract

Abstract Quantum coherence serves as a defining characteristic of quantum mechanics, finding extensive applications in quantum computing and quantum communication processing. This study explores quantum block coherence in the context of projective measurements, focusing on the quantification of such coherence. Firstly, we define the correlation function between the two general projective measurements P and Q, and analyze the connection between sets of block incoherent states related to two compatible projective measurements P and Q. Secondly, we discuss the measure of quantum block coherence with respect to projective measurements. Based on a given measure of quantum block coherence, we characterize the existence of maximal block coherent states through projective measurements. This research integrates the compatibility of projective measurements with the framework of quantum block coherence, contributing to the advancement of block coherence measurement theory.

DOI
10.1088/1674-1056/ad50c1

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Abstract Quantum coherence serves as a defining characteristic of quantum mechanics, finding extensive applications in quantum computing and quantum communication processing. This study explores quantum block coherence in the context of projective measurements, focusing on the quantification of such coherence. Firstly, we define the correlation function between the two general projective measurements P and Q, and analyze the connection between sets of block incoherent states related to two compatible projective measurements P and Q. Secondly, we discuss the measure of quantum block coherence with respect to projective measurements. Based on a given measure of quantum block coherence, we characterize the existence of maximal block coherent states through projective measurements. This research integrates the compatibility of projective measurements with the framework of quantum block coherence, contributing to the advancement of block coherence measurement theory.
Quantum coherence is known as an important resource in many quantum information tasks, which is a basis-dependent property of quantum states. In this paper, we discuss quantum incoherence based simultaneously on k bases using Matrix Theory Method. First, by defining a correlation function m(e,f) of two orthonormal bases e and f, we investigate the relationships between sets I(e) and I(f) of incoherent states with respect to e and f. We prove that I(e)=I(f) if and only if the rank-one projective measurements generated by e and f are identical. We give a necessary and sufficient condition for the intersection I(e)⋂I(f) to include a state except the maximally mixed state. Especially, if two bases e and f are mutually unbiased, then the intersection has only the maximally mixed state. Secondly, we introduce the concepts of strong incoherence and weak coherence of a quantum state with respect to a set B of k bases and propose a measure for the weak coherence. In the two-qubit system, we prove that there exists a maximally coherent state with respect to B when k=2 and it is not the case for k=3.
Abstract The uncertainty relation is a distinctive characteristic of quantum theory. The uncertainty is essentially rooted in quantum states. In this work we regard the uncertainty as an intrinsic property of quantum state and characterize it systematically with respect to given projective measurement. Some basic concepts about uncertainty are reformulated in this context. We prove and get the form of the uncertainty preserving operations. The quantum states with maximal uncertainty are characterized. A universal decomposition of uncertainty into classical uncertainty and quantum uncertainty is provided. Furthermore, a unified and general relation among uncertainty, coherence and coherence of assistance is established. These results are independent of any explicit uncertainty measure. At last, we propose a new uncertainty measure called the geometric uncertainty based on the fidelity and link it with the geometric coherence.
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.
Quantum coherence, as a direct manifestation of the quantum superposition principle, is a crucial resource in quantum information processing. Block coherence resource theory generalizes the traditional coherence framework by defining coherence via a set of orthogonal projectors. Within this framework, we investigates the construction and comparison of block coherence measures. First, we propose two universal methods for constructing coherence measures and introduce a two-parameter family of measures based on the ααα-zzz Rényi relative entropy and a family of measures based on the Tsallis relative operator entropy. Second, through theoretical proofs and numerical counterexamples, we compares the ordering relations and numerical magnitudes among different block coherence measures and establishes a series of universal numerical inequalities to constrain their values. Besides, we also use Cα,1C_{α,1}Cα,1​ to show the role of coherence in complex dynamic evolution of the Kominis master equation that includes recombination reactions.
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.
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.