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.

中文翻译：

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量子艾奇逊几何

将似然函数与正数相乘对贝叶斯统计推断没有影响，因此在归一化后，在许多情况下似然函数可以被视为概率分布。这个想法导致 Aitchison 在 1986 年定义了概率单纯形上的向量空间结构。Pawlowsky-Glahn 和 Egozcue 在 2001 年给出了这个空间的统计相关标量积，赋予概率单纯形一个希尔伯特空间结构。在本文中，我们提出了该几何的非交换对应物。我们在量子力学有限维状态空间上引入了一个真实的希尔伯特空间结构。我们证明了量子设置中的标量积尊重张量积结构，并且可以用模算子和汉密尔顿算子来表示。使用对数比变换的量子模拟，所有新引入的操作都自然地出现在吉布斯状态的语言中。我们在状态空间中展示了一个标准正交基，并详细研究了引入的量子比特空间几何。