Quantifying asymmetry via generalized Wigner–Yanase–Dyson skew information

Abstract

Asymmetry is a useful physical resource which quantifies the extent to which a quantum state breaks a symmetry. Based on the generalization of Wigner–Yanase–Dyson skew information of quantum states for any operator (not necessarily Hermitian), we introduce an asymmetry measure of a quantum state with respect to a compact Lie group as the average skew information of this state for each operator in its unitary representation and prove that it meets the requirements for the resource theory of asymmetry. For several significant symmetry groups, such as the group U(1), the full unitary group, the local unitary group, the group UU, the group UŪ , and the orthogonal group OO, we calculate the asymmetry of quantum states. Furthermore, we compare the asymmetry measure of quantum states with respect to a compact Lie group with the asymmetry measure of quantum states with respect to the corresponding Lie algebras introduced in (2020 Europhys. Lett. 130 30004 ) and illustrate that they are closely related, although they capture different aspects of asymmetry of quantum states.