论文标题
基于矩阵几何平均值的量子状态的保真度度量
A fidelity measure for quantum states based on the matrix geometric mean
论文作者
论文摘要
Uhlmann的保真度功能是量子理论中使用最广泛的相似性度量之一。该功能的一个定义是，它是与两个量子状态的量子到古典测量程序相关的最低经典保真度。在2010年，Matsumoto引入了另一个Fidelity函数，这是Uhlmann的双重函数，因为它是与两个量子状态的经典到量子制剂相关的最大经典忠诚度。 Matsumoto的保真度也可以使用矩阵几何平均值的良好概念定义来定义。在这项工作中，我们通过半决赛编程的镜头检查了Matsumoto的忠诚度，以提供简单的证据，表明它具有许多可取的特性，以进行相似性度量，包括量子通道下的单调性，关节凹度和单一不变性。最后，我们根据积极确定矩阵的riemannian空间提供了这种忠诚度的几何解释，并展示了这张图片如何有助于理解其某些特殊特性。
Uhlmann's fidelity function is one of the most widely used similarity measures in quantum theory. One definition of this function is that it is the minimum classical fidelity associated with a quantum-to-classical measurement procedure of two quantum states. In 2010, Matsumoto introduced another fidelity function which is dual to Uhlmann's in the sense that it is the maximimum classical fidelity associated with a classical-to-quantum preparation procedure for two quantum states. Matsumoto's fidelity can also be defined using the well-established notion definition of the matrix geometric mean. In this work, we examine Matsumoto's fidelity through the lens of semidefinite programming to give simple proofs that it possesses many desirable properties for a similarity measure, including monotonicity under quantum channels, joint concavity, and unitary invariance. Finally, we provide a geometric interpretation of this fidelity in terms of the Riemannian space of positive definite matrices, and show how this picture can be useful in understanding some of its peculiar properties.
