Wigner function, Wigner-Yanase skew information, and parity asymmetry

Abstract

The Wigner function, a position-momentum joint distribution introduced by Wigner in 1932 as a quantum correction for thermodynamics, plays a fundamental role in the phase-space formulation of quantum mechanics. The Wigner-Yanase skew information, an information-theoretic concept introduced by Wigner and Yanase in 1963 for quantifying information contents of quantum states, remains almost unnoticed until 2000s and is now playing an increasingly important role in quantum information theory. Apart from the appearance of the name Wigner, the relations between these two important concepts seem to have attracted little attention. In this work, by exploiting symmetry and asymmetry of bosonic field states with respect to the parity, we quantify parity asymmetry in term of the Wigner-Yanase skew information, and reveal some intrinsic complementary relations between the Wigner function and the Wigner-Yanase skew information. The results can be extended to a rather general setup.