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Entropy

Quantum Incoherence Based Simultaneously on k Bases

Pu Perry Wang, Zhihua Guo, Huaixin Cao

Published May 7, 2022• 3 citations
Publisher

Abstract

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.

DOI
10.3390/e24050659

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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.
In the quantitative theory of quantum coherence, the amount of coherence is defined as the distance between the given state to the closest incoherent state. The set of incoherent states is conventionally defined as any state with a diagonal density matrix. One of the objections to this formulation is that the incoherent states are intrinsically basis-dependent, which makes the amount of coherence also a basis-dependent quantity. Basis-independent measures have recently been proposed where the incoherent state is taken as the maximally mixed state. We show that this is the only possible choice of reference incoherent state, without modifying the original definition of coherence. We find a relation between the two formulations by defining a contribution to the coherence due to the basis choice. The hierarchical relationship between quantum coherence and the various quantum correlations is explored in detail. Finally, we illustrate some operational uses of the basis-independent quantum coherence in quantum information theory tasks.
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.
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.
Recently various ways to quantify coherence of a quantum state have been proposed. In this paper, several reliable quantum coherence measures based on quantum Fisher information (QFI) are presented. For pure states, this coherence measure is equivalent to that based on skew information, while for mixed states, we first define two natural coherence measures via QFI and find that the two definitions are exactly identical. By virtual of the special structure of qubit, we show that the mutual unbiased bases' (MUBs) qubit coherence is constrained only by the length of the Bloch vector. Then we provide upper bounds of the coherence measure with respect to any given basis and lower bounds in the case of a set of MUBs for both pure states and mixed states of arbitrary dimension. Finally, we investigate the connection between our coherence measures and quantum metrology, thereby providing an operational meaning for these coherence measures.
A pair of quantum observables diagonal in the same basis can be measured jointly, so some coherence is obviously needed for measurement incompatibility. It is equally clear that these quantum resources are not comparable for general observables. However, in the spirit of complementarity between path and superposition, one expects a precise measurement of the incoherent basis label to be incompatible with sufficiently coherent observables. Here we formalise this idea in a very general framework, involving pairs where one observable is incoherent (diagonal), and the coherence in the other is adjusted by damping via genuinely incoherent operations. An extreme case is a basis measurement mutually unbiased to the incoherent basis, leading to an interesting interplay between coherence and complementarity. In terms of measurement theory, we show that if coherence is insufficient to sustain incompatibility, the associated joint measurements have sequential implementations via incoherent instruments. This yields a concrete method for solving joint measurability problems, tractable even in large systems without resorting to numerical optimisation: we find analytical bounds for the coherence needed for incompatibility, in terms of a divergence measure characterising basis state distinguishability of incoherent observables. We verify the consistency of our theory by a quick proof of the known noise bound for the incompatibility of mutually unbiased bases, and then apply it to study emergent classicality in the spin-boson model of pure decoherence on an N-qubit open quantum system.
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.