Robustness of asymmetry and coherence of quantum states

Abstract

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

Symmetrizing cost of quantum states

“…Following the previous literatures [2][3][4][5][6][7][8][9][10][11][12][13][14], we assume that (i) G has a unitary representation {U g } g∈G on H A , and that (ii) G is a compact Lie group or a finite group.…”

“…A central problem lying in these studies is how to quantify the degree of asymmetry of quantum states [3][4][5][6][7][10][11][12][13]. Mathematically, this problem reduces to a search for functions that are monotonically nonincreasing under operations restricted by symmetry [10].…”

“…[2]). For the other reason, it is because the concepts and techniques developed in quantum information theory are useful to analyze the notions of symmetry and asymmetry from an operational viewpoint [3][4][5][6][7][8][9].…”

Coherence, asymmetry, and quantum macroscopicity

“…Examples of such measures of asymmetry include information based measures [16,[18][19][20], robustness of asymmetry [19], and asymmetry-weight [20].…”

“…Recently, there have been attempts to interpret quantum properties as a resource. Resource theories based on the notion of free operations may provide useful tools to quantify nonclassical phenomena, covering a diverse range of topics that include entanglement [12,13], asymmetry [14][15][16][17][18][19][20], and quantum thermodynamics [21][22][23][24][25], all of which have been studied within this framework.…”