Quantum Incoherence Based Simultaneously on k Bases

Quantum coherence is known as an important resource in many quantum information tasks, which is a basis-dependent property of quantum states. In this paper, we discuss quantum incoherence based simultaneously on k bases using Matrix Theory Method. First, by defining a correlation function [Formula: see text] of two orthonormal bases e and f, we investigate the relationships between sets [Formula: see text] and [Formula: see text] of incoherent states with respect to e and f. We prove that [Formula: see text] if and only if the rank-one projective measurements generated by e and f are identical. We give a necessary and sufficient condition for the intersection [Formula: see text] to include a state except the maximally mixed state. Especially, if two bases e and f are mutually unbiased, then the intersection has only the maximally mixed state. Secondly, we introduce the concepts of strong incoherence and weak coherence of a quantum state with respect to a set [Formula: see text] of k bases and propose a measure for the weak coherence. In the two-qubit system, we prove that there exists a maximally coherent state with respect to [Formula: see text] when [Formula: see text] and it is not the case for [Formula: see text].