态与信道相互作用下的相干性和互补性的无限维推广
Infinite Dimensional Generalization of Coherence and Complementarity in State-Channel Interaction

量子相干性和玻尔互补性是量子力学中两个重要的主题，对其的研究也在不断深入。本文给出了量子相干性和玻尔互补性在无限维态与信道相互作用下的表示。首先基于无限维的信道刻画，证明了无限维中的对称部分，非对称部分以及信道的希尔伯特–施密特范数是有限的，其中对称部分用对称的若尔当积表示，非对称部分用斜对称李积表示。然后证明了无限维情形中的非对称部分仍然满足4个性质，可以作为维格纳–亚纳斯–丹森斜信息的一种表示，最后通过证明无限维态与信道相互作用的非对称部分仍然满足10个性质，得到了无限维态与信道相互作用下的量子相干性和玻尔互补性。
Quantum coherence and Bohr’s complementarity are two significant topics in quantum mechanics. The research on them is deep-going from its very beginning. In this paper, we give the expression of quantum correlation and Bohr’s complementarity in state-channel interaction under infinite di-mensional case. First, based on the infinite dimensional channel characterization, we proved that the Hilbert Schmidt norm of the asymmetric part, the symmetric part, the channel are finite, even in the infinite dimension. The symmetric part is represented by the symmetric Jordan product, and the asymmetric part is synthesized by the skew-symmetric Lie product. Then it is proved that the asymmetric part still satisfies four properties, thus, it can represent the Wigner Yanase Dyson skew information, in the infinite dimensional case. Finally, by proving that the asymmetric part of state-channel interaction still satisfies 10 properties, we obtain the generalization of quantum co-herence and Bohr’s complementarity in state-channel interaction, under the infinite dimensional case.