Partial coherence with application to the monotonicity problem of coherence involving skew information

Abstract

Quantifications of coherence are intensively studied in the context of completely decoherent operations (i.e., von Neuamnn measurements, or equivalently, orthonormal bases) in recent years. Here we investigate partial coherence (i.e., coherence in the context of partially decoherent operations such as L\uders measurements). A bona fide measure of partial coherence is introduced. As an application, we address the monotonicity problem of $K$-coherence (a quantifier for coherence in terms of Wigner-Yanase skew information) [Girolami, Phys. Rev. Lett. 113, 170401 (2014)], which is introduced to realize a measure of coherence as axiomatized by Baumgratz, Cramer, and Plenio [Phys. Rev. Lett. 113, 140401 (2014)]. Since $K$-coherence fails to meet the necessary requirement of monotonicity under incoherent operations, it is desirable to remedy this monotonicity problem. We show that if we modify the original measure by taking skew information with respect to the spectral decomposition of an observable, rather than the observable itself, as a measure of coherence, then the problem disappears, and the resultant coherence measure satisfies the monotonicity. Some concrete examples are discussed and related open issues are indicated.