How dynamics constrains probabilities in general probabilistic theories
How dynamics constrains probabilities in general probabilistic theories
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DOI：
10.22331/q-2021-05-21-457
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发表时间：
2020-02
期刊：
影响因子：
6.4
通讯作者：
Thomas D. Galley;L. Masanes
中科院分区：
文献类型：
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作者：
Thomas D. Galley;L. Masanes
我们介绍了一个一般的框架，分析一般的概率理论，强调系统的动力学和概率结构之间的区别。动力学结构是一组纯态以及可逆动力学的作用，而概率结构决定了测量和结果概率。对于传递动力学结构的动力群和稳定子群形成一个Gelfand对，我们证明了所有的概率结构是刚性的（不能无限小变形），并与动力群的球面表示一一对应。我们应用我们的方法来分类所有的概率结构时，动力结构是复杂的格拉斯曼流形上的酉群。这是量子理论的一个推广，其中纯态不是由复向量空间的一维子空间表示，而是由大于1的固定维度的子空间表示。我们还表明，系统具有紧凑的两点齐次动力学结构（即每一对纯态具有一个给定的距离可以可逆地转换到任何其他对纯态具有相同的距离），其中包括系统对应于欧几里德约旦代数，都有刚性概率结构。
We introduce a general framework for analysing general probabilistic theories, which emphasises the distinction between the dynamical and probabilistic structures of a system. The dynamical structure is the set of pure states together with the action of the reversible dynamics, whilst the probabilistic structure determines the measurements and the outcome probabilities. For transitive dynamical structures whose dynamical group and stabiliser subgroup form a Gelfand pair we show that all probabilistic structures are rigid (cannot be infinitesimally deformed) and are in one-to-one correspondence with the spherical representations of the dynamical group. We apply our methods to classify all probabilistic structures when the dynamical structure is that of complex Grassmann manifolds acted on by the unitary group. This is a generalisation of quantum theory where the pure states, instead of being represented by one-dimensional subspaces of a complex vector space, are represented by subspaces of a fixed dimension larger than one. We also show that systems with compact two-point homogeneous dynamical structures (i.e. every pair of pure states with a given distance can be reversibly transformed to any other pair of pure states with the same distance), which include systems corresponding to Euclidean Jordan Algebras, all have rigid probabilistic structures.