量子信道的反变密度、完全距离和相对保真度
Contravariant Densities, Complete Distances and Relative Fidelities for Quantum Channels
摘要：
通过引入量子态的反变迹密度，我们将正常CP映射所描述的量子操作与其迹密度之间的一一对应恢复为厄米正算符值反变核.用这些密度表示了一类输入代数的两个量子操作之间的CB-范数距离，并将该公式推广到一类输入代数的量子操作之间的广义CB-距离.给出了一个更大的C距离作为通道密度的自然范数距离，并引入了另一个与量子信道纯化的极小极大均方优化问题相关的Helinger距离，并根据它们的逆变迹密度进行了计算.证明了至少对于第一类输入，两通道间的Helinger型完全保真度距离等价于CB距离，并将这种等价性推广到第二类广义CB距离.将量子编码作为与输入态和输出态相反的量子态的广义纠缠，给出了这些距离和量子信道的相对完全保真度的操作意义。
英文摘要：
Introducing contravariant trace-densities for quantum states, we restore one to one correspondence between quantum operations described by normal CP maps and their trace densities as Hermitian positive operator-valued contravariant kernels. The CB-norm distance between two quantum operations with type one input algebras is explicitly expressed in terms of these densities, and this formula is also extended to a generalized CB-distances between quantum operations with type two inputs. A larger C-distance is given as the natural norm-distance for the channel densities, and another, Helinger type distance, related to minimax mean square optimization problem for purification of quantum channels, is also introduced and evaluated in terms of their contravariant trace-densities. It is proved that the Helinger type complete fidelity distance between two channels is equivalent to the CB distance at least for type one inputs, and this equivalence is also extended to type two for the generalized CB distance. An operational meaning for these distances and relative complete fidelity for quantum channels is given in terms of quantum encodings as generalized entanglements of quantum states opposite to the inputs and the output states.
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