Gram Matrices of Mixed-State Ensembles
Gram Matrices of Mixed-State Ensembles
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DOI：
10.1007/s10773-021-04908-8
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发表时间：
2021-07
影响因子：
1.4
通讯作者：
Yuan Sun;S. Luo;Xiangyun Lei
中科院分区：
文献类型：
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作者：
Yuan Sun;S. Luo;Xiangyun Lei
Gram矩阵是在考虑一族向量或量子纯态的成对重叠时自然产生的，并且在综合一族纯态的信息中起着重要的作用。由于量子系综通常由混合态组成，因此需要将Gram矩阵的概念扩展到混合态的情况。通过采用混合状态之间重叠的两个突出概念，即，量子亲和性和量子保真度，人们可以很容易地将纯态系综的Gram矩阵的概念扩展到混合态系综。我们研究了这两种广义Gram矩阵，并揭示了它们的基本性质。值得注意的是，虽然基于量子亲和势的Gram矩阵是非负定的（因此可以被视为虚拟系统中的量子态），但基于量子保真度的Gram矩阵可能不是非负定的。作为混合态系综的Gram矩阵的应用，我们引入了两个通过相应的Gram矩阵的相干性来表示系综量子性的量词，研究了它们的性质，并在量子比特的情况下进行了说明.
Gram matrices arise naturally in the consideration of pair-wise overlap of a family of vectors or quantum pure states, and play an important role in synthesizing information of a family of pure states. Since a quantum ensemble in general consists of mixed states, it is desirable to extend the concept of Gram matrix to the case of mixed states. By employing two prominent notions of overlap between mixed states, i.e., quantum affinity and quantum fidelity, one can readily extend the concept of Gram matrices of pure-state ensembles to that of mixed-state ensembles. We study these two extended versions of Gram matrices and reveal their fundamental properties. It is remarkable that while Gram matrix based on quantum affinity is non-negative definite (and thus can be regarded as a quantum state in a fictious system), the one based on quantum fidelity may fail to be non-negative definite. As applications of Gram matrices of mixed-state ensembles, we introduce two quantifiers of quantumness of ensembles via coherence of the corresponding Gram matrices, investigate their properties, and illustrate them in the qubit case.