Robustness of asymmetry and coherence of quantum states

Abstract

Quantum states may exhibit asymmetry with respect to the action of a given\ngroup. Such an asymmetry of states can be considered as a resource in\napplications such as quantum metrology, and it is a concept that encompasses\nquantum coherence as a special case. We introduce explicitly and study the\nrobustness of asymmetry, a quantifier of asymmetry of states that we prove to\nhave many attractive properties, including efficient numerical computability\nvia semidefinite programming, and an operational interpretation in a channel\ndiscrimination context. We also introduce the notion of asymmetry witnesses,\nwhose measurement in a laboratory detects the presence of asymmetry. We prove\nthat properly constrained asymmetry witnesses provide lower bounds to the\nrobustness of asymmetry, which is shown to be a directly measurable quantity\nitself. We then focus our attention on coherence witnesses and the robustness\nof coherence, for which we prove a number of additional results; these include\nan analysis of its specific relevance in phase discrimination and quantum\nmetrology, an analytical calculation of its value for a relevant class of\nquantum states, and tight bounds that relate it to another previously defined\ncoherence monotone.\n