We propose an alternative framework for quantifying coherence. The framework is based on a natural property of coherence, the additivity of coherence for subspace-independent states, which is described by an operationindependent equality rather than operation-dependent inequalities and therefore applicable to various physical contexts. Our framework is compatible with all the known results on coherence measures but much more flexible and convenient for applications, and by using it many open questions can be resolved.Quantum coherence is a fundamental feature of quantum mechanics, describing the capability of a quantum state to exhibit quantum interference phenomena. It is an essential ingredient in quantum information processing [1], and plays a central role in emergent fields, such as quantum metrology [2][3][4], nanoscale thermodynamics [5][6][7][8][9][10][11], and quantum biology [12][13][14][15][16]. Although the theory of quantum coherence is historically well developed in quantum optics [17][18][19], it is only in recent years that the quantification of coherence has attracted a growing interest [20][21][22][23][24] due to the development of quantum information science.By following the approach that has been established for entanglement resource [25,26], Baumgratz et al. proposed a seminal framework for quantifying coherence as a resource in Ref. [22]. The framework comprises four conditions, of which the first two are based on the notions of free states and free operations in the resource theories, while the third and fourth conditions are two constraints imposed on coherence measures. Based on this framework, a number of coherence measures, such as the relative entropy of coherence, the l 1 norm of coherence, and the coherence of formation [20,22,27,28], have been put forward. With the coherence measures, various properties of quantum coherence, such as the relations between quantum coherence and quantum correlations [29][30][31][32][33], the freezing phenomenon of coherence [34,35], and the distillation of coherence [28,36], were investigated. Hereafter, we refer to the framework proposed by Baumgratz et al. as the BCP framework for simplicity.Although the BCP framework has been widely used as an approach to coherence measures, there are arguments against the necessity of its last two conditions [29,37], and researchers have different opinions on the definition of free operations. Besides the incoherent operations defined in the BCP framework, there have been many different suggestions on the definition of free operations, such as maximally incoherent operations [20], translationally invariant operations [23], and others [38][39][40]. These arguments against the conditions and free operations imply that the frameworks for quantifying coherence are not unique. There can be other frameworks different from the BCP framework. For instance, the framework proposed by Marvian and Spekkens in Ref. [23], called the MS framework for simplicity, is based on the translationally invariant operations, and it compris...

We find that all measures of coherence are frozen for an initial state in a strictly incoherent channel if and only if the relative entropy of coherence is frozen for the state. Our finding reveals the existence of measure-independent freezing of coherence, and provides an entropy-based dynamical condition in which the coherence of an open quantum system is totally unaffected by noise.Quantum coherence is a fundamental feature of quantum mechanics, describing the capability of a quantum state to exhibit quantum interference phenomena. The coherence effect of a state is usually ascribed to the offdiagonal elements of its density matrix with respect to a particular reference basis, which is determined according to the physical problem under consideration. It is an essential ingredient in quantum information processing [1], and plays a central role in emergent fields, such as quantum metrology [2][3][4], nanoscale thermodynamics [5][6][7][8][9][10][11], and quantum biology [12][13][14][15][16].It is only recent years that the quantification of coherence has become a hot topic due to the development of quantum information science, although the theory of quantum coherence is historically well developed in quantum optics [17][18][19]. A rigorous framework to quantify the coherence of quantum states in the resource theories has been recently proposed after a series of efforts [20][21][22][23][24][25][26][27][28][29]. By following the rigorous framework comprising four postulates [20], a number of coherence measures based on various physical contexts have been put forward. The l 1 norm of coherence and the relative entropy of coherence were first suggested as two coherence measures based on distance. The coherence measures based on entanglement [30], the coherence measures based on operation [31,32], and the coherence measures based on convex-roof construction [33,34] were subsequently proposed. With coherence measures, various properties of quantum coherence, such as the relations between quantum coherence and other quantum resources [30,35,36], the quantum coherence in infinite-dimensional systems [37,38], the complementarity relations of quantum coherence [39], and the measure of macroscopic coherence [40], have been discussed. Quantum coherence is a useful physical resource, but coherence of a quantum state is often destroyed by noise. A challenge in exploiting the resource is to protect coherence from the decoherence caused by noise, as the loss of coherence may weaken the abilities of a state to perform quantum information processing tasks. Today, after having been equipped with the knowledge of coherence measures, it becomes possible to analyze under which dynamical conditions the coherence of an open system is frozen in a noisy channel. Studies on this topic have been started in Ref. [41], where the authors found that the coherence measures based on bona fide distances are frozen for some initial states of a quantum system with even number of qubits undergoing local identical bit flip channels. This finding illu...

A series of geometric concepts are formulated for PT -symmetric quantum mechanics and they are further unified into one entity, i.e., an extended quantum geometric tensor (QGT). The imaginary part of the extended QGT gives a Berry curvature whereas the real part induces a metric tensor on system's parameter manifold. This results in a unified conceptual framework to understand and explore physical properties of PT -symmetric systems from a geometric perspective. To illustrate the usefulness of the extended QGT, we show how its real part, i.e., the metric tensor, can be exploited as a tool to detect quantum phase transitions as well as spontaneous PT -symmetry breaking in PT -symmetric systems.Given a family of Hamiltonians depending smoothly on a manifold of parameters, e.g., external field strengths, a problem of great importance is how to characterize geometric aspects of the eigenstates of the Hamiltonians. In standard quantum mechanics (QM) where Hamiltonians are Hermitian operators, the solution to the problem is to use the quantum geometric tensor (QGT), of which the imaginary part determines the Berry curvature [1] and the real part induces a Riemannian metric tensor [2] on the manifold. The QGT has played an indispensable role in various frontier topics of quantum computation, quantum information, and condensed-matter physics [3,4], where both its imaginary and real parts serve as versatile tools.Since the pioneering work of Bender and Boettcher [5], however, it has been realized that Hamiltonians can be non-Hermitian but still possess real spectra due to paritytime reversal (PT ) symmetry. This has led to a complex extension of standard QM called PT -symmetric QM (PT QM) [6,7], with one main conceptual advance being the introduction of a nontrivial inner-product metric to define its Hilbert space. Over the past decade, PTsymmetric systems have been experimentally realized by spatially engineering gain-loss structures [8], thus further boosting PT QM as an important research area. A current stream of development is towards extensive studies of physical properties, especially topological properties, of PT -symmetric systems [9][10][11][12][13][14][15][16][17].Unfortunately, a systematic geometric concept like the QGT is still elusive in PT QM. Without such a concept, it is difficult to extend geometric understandings, such as those of quantum phase transitions (QPTs) [18][19][20][21][22][23][24][25], to PT -symmetric systems. On the other hand, the interplay between the nontrivial inner-product metric and geometric aspects of PT QM is rarely understood to date. In particular, in the course of varying parameters of a PT -symmetric system, the inner-product metric varies as well [26]. How this feature impacts on previous geometric perspectives in standard QM (such as curvature and metric tensor) remains unknown.In this Letter, we report the finding of an extended QGT in PT QM, of which the imaginary part gives a Berry curvature whereas the real part induces a metric tensor on system's parameter manifold. Thi...

Asymmetry of quantum states is a useful resource in applications such as quantum metrology, quantum communication, and reference frame alignment. However, asymmetry of a state tends to be degraded in physical scenarios where environment-induced noise is described by covariant operations, e.g., open systems constrained by superselection rules, and such degradations weaken the abilities of the state to implement quantum information processing tasks. In this paper, we investigate under which dynamical conditions asymmetry of a state is totally unaffected by the noise described by covariant operations. We find that all asymmetry measures are frozen for a state under a covariant operation if and only if the relative entropy of asymmetry is frozen for the state. Our finding reveals the existence of universal freezing of asymmetry, and provides a necessary and sufficient condition under which asymmetry is totally unaffected by the noise.Symmetry is a central concept in quantum mechanics, describing invariant features of a quantum system with respect to the action of a group of transformations [1]. For a specific symmetry, two relevant notions are asymmetric states and covariant operations, which are the states that break the symmetry and the quantum operations that respect the symmetry, respectively. In the physical world, all elementary interactions are expected to have specific symmetries [2]. For example, the interactions that do not have preferred direction are rotationally invariant and hence have SO(3) symmetry. The presence of a symmetry in a system generally imposes restrictions on the manipulation of the system, which results in nontrivial limitations on the implementation of quantum information processing tasks. Interestingly, asymmetric states can be exploited to overcome the restrictions and allow one to implement quantum information processing tasks that would otherwise be forbidden [3]. For example, in the presence of a conservation law, it is forbidden to measure exactly an observable that does not commute with a conserved quantity, but it is still possible to measure approximatively the observable with the aid of asymmetric states [4,5]. Asymmetry of states is a useful resource for implementing quantum information processing tasks [3], and the exploitation of asymmetric states has been carried out in applications, such as quantum metrology [6][7][8][9], quantum communication [10,11], and reference frame alignment [12][13][14].By taking asymmetry as a physical resource, a resource theory of asymmetry, just like the resource theory of entanglement [15], has been recently developed. The abilities of an asymmetric state to overcome the restriction imposed by a symmetry are analogous to the abilities of an entangled state to overcome the restriction of local operations and classical communication (LOCC). Asymmetric states and covariant operations in the asymmetry theory correspond respectively to entangled states and LOCC in the entanglement theory, or resource states and free operations in a general resourc...

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