Robustness of asymmetry and coherence of quantum states

Quantum states may exhibit asymmetry with respect to the action of a given group. Such an asymmetry of states can be considered 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.

Figure 1

Comparison between the ${\ell }_1$ norm of coherence ${\mathcal{C}}_{{\ell }_1}$ (horizontal axis) and the robustness of coherence ${\mathcal{C}}_{\mathcal{R}}$ (vertical axis) for $3\times {10}^4$ randomly generated $d$-dimensional states, with (a) $d=3$ and (b) $d=4$. In both panels we additionally plot some bounds to ${\mathcal{C}}_{\mathcal{R}}$ as a function of ${\mathcal{C}}_{{\ell }_1}$: the thick solid black line ${\mathcal{C}}_{\mathcal{R}}={\mathcal{C}}_{{\ell }_1}$ denotes the upper bound of Eq. (45), which is saturated by pure states; the dashed blue line ${\mathcal{C}}_{\mathcal{R}}={\mathcal{C}}_{{\ell }_1}/(d-1)$ denotes the lower bound of Eq. (45), which is saturated by the states in (49) up to ${\mathcal{C}}_{{\ell }_1}=1$; the dotted green line denotes the alternative (nontight) lower bound of Eq. (51). All the quantities plotted are dimensionless.