拟概率表示中的量子贝叶斯推断
Quantum Bayesian Inference in Quasiprobability Representations
AI总结：
本文推导了Petz恢复映射在任意拟概率表示下的表达式，并针对正规拟概率表示和SIC-POVMs表示给出了显式公式，揭示了经典与量子逻辑推断的核心差异在于对参考先验的操纵。
摘要：
贝叶斯规则在信息科学和物理科学中都是逻辑推断的关键部分。将其扩展到量子领域已成为近期多项研究的焦点。这些量子版贝叶斯规则此前均在希尔伯特空间的语言下表述。本文推导了Petz恢复映射在任意拟概率表示下的表达式，并针对正规拟概率表示（包括离散Wigner表示）和基于对称信息完备正定算子值测量（SIC-POVMs）的表示这两种规范选择给出了显式公式。通过使用（拟）随机矩阵作用于（拟）随机向量的相同数学语法，经典理论与量子理论在逻辑推断上的核心差异被发现在于对参考先验的操纵，而非信道的表示。
英文摘要：
Bayes' rule plays a crucial piece of logical inference in information and physical sciences alike. Its extension into the quantum regime has been the object of several recent works. These quantum versions of Bayes' rule have been expressed in the language of Hilbert spaces. In this paper, we derive the expression of the Petz recovery map within any quasiprobability representation, with explicit formulas for the two canonical choices of normal quasiprobability representations (which include Discrete Wigner representations) and of representations based on symmetric, informationally complete positive operator-valued measures (SIC-POVMs). By using the same mathematical syntax of (quasi-)stochastic matrices acting on (quasi-)stochastic vectors, the core difference in logical inference between classical and quantum theory is found in the manipulation of the reference prior rather than in the representation of the channel.
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