We approach the physical implications of the noncommutative nature of complementary observable algebras (COAs) from an information theoretic perspective. In particular, we derive a general entropic certainty principle stating that the sum of two relative entropies, naturally related to the COA, is equal to the so-called algebraic index of the associated inclusion. Uncertainty relations then arise by monotonicity of the relative entropies that participate in the underlying entropic certainty. Examples and applications are described in quantum field theories with global symmetries, where the COAs are formed by the charge-anticharge local operators (intertwiners) and the unitary representations of the symmetry group (twists), and in theories with local symmetries, where the COAs are formed by Wilson and t'Hooft loops. In general, the entropic certainty principle naturally captures the physics of order and disorder parameters, a feature that makes it a generic handle for the information theoretic characterization of quantum phases.

中文翻译：

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通过熵确定性原理的量子互补

我们从信息理论的角度研究互补可观代数（COA）的非交换性质的物理含义。特别是，我们推导出了一般的熵确定性原理指出与COA自然相关的两个相对熵的总和等于相关包含物的所谓代数指数。然后，不确定性关系由参与基本熵确定性的相对熵的单调性引起。实例和应用在具有全局对称性的量子场理论中有所描述，其中COA由荷电局部电荷算子（缠结分子）和对称群的group表示（扭曲）形成，在具有局部对称性的理论中，COA由Wilson和t'Hooft循环组成。通常，熵确定性原理自然地捕获了有序和无序参数的物理性质，这一特性使其成为量子相信息理论表征的通用方法。