Universal freezing of asymmetry

Abstract

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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