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首页> 外文会议>International conference on geometric science of information >Quantum Statistical Manifolds: The Finite-Dimensional Case
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Quantum Statistical Manifolds: The Finite-Dimensional Case

机译：量子统计流形：有限维情况

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Quantum information geometry studies families of quantum states by means of differential geometry. A new approach is followed. The emphasis is shifted from a manifold of strictly positive density matrices to a manifold M of faithful quantum states on a von Neumann algebra of bounded linear operators working on a Hilbert space. In order to avoid technicalities the theory is developed for the algebra of n-by-n matrices. A chart is introduced which is centered at a given faithful state ω_p. It maps the manifold M onto a real Banach space of self-adjoint operators belonging to the commutant algebra. The operator labeling any state uja of M also determines a tangent vector in the point ω_p along the exponential geodesic in the direction of ω_σ. A link with the theory of the modular automorphism group is worked out. Explicit expressions for the chart can be derived in terms of the modular conjugation and the relative modular operators.
机译：量子信息几何学通过微分几何学研究量子态族。遵循一种新方法。重点从在希尔伯特空间上工作的有界线性算子的冯·诺伊曼代数的严格正密度矩阵的流形转变为忠实量子态的流形M。为了避免技术性，该理论被开发用于n×n矩阵的代数。引入以给定的忠实状态ω_p为中心的图表。它将流形M映射到属于交换代数的自伴算子的真实Banach空间。标记M的任何状态uja的运算符还可以确定沿ω_σ方向沿指数测地线的点ω_p中的切向量。建立了与模块自同构群理论的联系。图表的显式表达式可以根据模块共轭和相关的模块运算符得出。

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