论文标题
量子Aitchison几何形状
Quantum Aitchison geometry
论文作者
论文摘要
将贝叶斯统计学推断的可能性函数乘以正数不会差异，因此，在归一化之后，在许多情况下，可能性函数可以视为概率分布。这个想法导致Aitchison在1986年的概率单纯性上定义了矢量空间结构。Pawlowsky-glahn和Egozcue在2001年在该空间上提供了与统计相关的标量产品，并具有希尔伯特空间结构的概率。在本文中，我们介绍了该几何形状的非交通性对应物。我们在量子机械有限尺寸空间上引入了真正的希尔伯特空间结构。我们表明，量子设置中的标量产品尊重张量产品结构，并且可以用模块化操作员和汉密尔顿操作员表示。使用对数比转换的量子类似物，事实证明，所有新引入的操作都以吉布斯状态的语言自然出现。我们在状态空间中展示了正直基础，并研究了详细的Qubits空间的引入的几何形状。
Multiplying a likelihood function with a positive number makes no difference in Bayesian statistical inference, therefore after normalization the likelihood function in many cases can be considered as probability distribution. This idea led Aitchison to define a vector space structure on the probability simplex in 1986. Pawlowsky-Glahn and Egozcue gave a statistically relevant scalar product on this space in 2001, endowing the probability simplex with a Hilbert space structure. In this paper we present the noncommutative counterpart of this geometry. We introduce a real Hilbert space structure on the quantum mechanical finite dimensional state space. We show that the scalar product in quantum setting respects the tensor product structure and can be expressed in terms of modular operators and Hamilton operators. Using the quantum analogue of the log-ratio transformation it turns out that all the newly introduced operations emerge naturally in the language of Gibbs states. We show an orthonormal basis in the state space and study the introduced geometry on the space of qubits in details.
