Wang Pu, Guo Zhihua, Cao Huaixin
School of Mathematics and Statistics, Shaanxi Normal University, Xi'an 710119, China.
Entropy (Basel). 2022 May 7;24(5):659. doi: 10.3390/e24050659.
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.
量子相干性在许多量子信息任务中被视为一种重要资源，它是量子态的一种基于基的属性。在本文中，我们使用矩阵理论方法同时基于多个基来讨论量子非相干性。首先，通过定义两个正交归一基(e)和(f)的关联函数(m(e,f))，我们研究了关于(e)和(f)的非相干态集合(I(e))和(I(f))之间的关系。我们证明当且仅当由(e)和(f)生成的秩一投影测量相同时，(I(e)=I(f))。我们给出了交集(I(e)\cap I(f))包含除最大混合态之外的一个态的充要条件。特别地，如果两个基(e)和(f)是相互无偏的，那么该交集仅包含最大混合态。其次，我们引入了量子态相对于一组(k)个基(B)的强非相干性和弱相干性的概念，并提出了一种弱相干性的度量。在双量子比特系统中，我们证明当(k = 2)时存在相对于(B)的最大相干态，而当(k = 3)时情况并非如此。
