Quantum Incoherence Based Simultaneously on 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 bases using Matrix Theory Method. First, by defining a correlation function m(e,f) of two orthonormal bases and , we investigate the relationships between sets I(e) and I(f) of incoherent states with respect to and . We prove that I(e)=I(f) if and only if the rank-one projective measurements generated by and are identical. We give a necessary and sufficient condition for the intersection I(e)⋂I(f) to include a state except the maximally mixed state. Especially, if two bases and 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 B of 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 B when k=2 and it is not the case for k=3.