Josza's definition of fidelity for a pair of (mixed) quantum states isstudied in the context of two types of operator algebras. The first setting ismainly algebraic in that it involves unital C$^*$-algebras $A$ that possess afaithful trace functional $\tau$. In this context, the role of quantum states(that is, density operators) in the classical quantum-mechanical framework isassumed by positive elements $\rho\in A$ for which $\tau(\rho)=1$. The secondof our two settings is more operator theoretic: by fixing a faithful normalsemifinite trace $\tau$ on a semifinite von Neumann algebra $M$, we define andconsider the fidelity of pairs of positive operators in $M$ of unit trace. Themain results of this paper address monotonicity and preservation of fidelityunder the action of certain trace-preserving positive linear maps of $A$ or ofthe predual $M_*$. Our results also yield a new proof of a theorem of Moln\'aron the structure of quantum channels on the trace-class operators that preservefidelity.
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机译：在两种类型的算子代数的背景下，研究了约萨对一对（混合）量子态的保真度的定义。第一种设置主要是代数的，因为它涉及具有忠实的跟踪函数$ \ tau $的单位C $ ^ * $-代数$ A $。在这种情况下，量子状态（即密度算符）在经典量子力学框架中的作用由A $中的正元素$ \ rho \假定，其中$ \ tau（\ rho）= 1 $。我们的两个设置中的第二个是更多的算符理论：通过在半有限的von Neumann代数$ M $上固定忠实的标准半限定迹$ \ tau $，我们定义并考虑了正算符对在单元迹$ M $中的保真度。本文的主要结果解决了在某些保留迹线的正A $ A $或偶数$ M _ * $的作用下单调性和保真度的保留问题。我们的结果也为Moln'aron定理的一个新证明提供了证据，该定理是保持保真度的迹线类算子上的量子通道的结构。
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