欧氏Jordan代数的复合与范畴
Composites and Categories of Euclidean Jordan Algebras
摘要：
我们认为可能的非信号复合材料的概率模型的基础上欧几里德约旦代数（EJAs），满足一些合理的额外的约束，动机的愿望，以构建匕首紧凑的类别，这样的模型。我们表明，没有这样的复合有例外的约旦代数作为一个直接的被加数，也没有任何这样的复合存在，如果一个因素有一个例外的被加数，除非其他因素是一个直接和一维约旦代数（基本上代表一个经典系统）。此外，我们表明，任何简单的，非例外的EJAs的复合是一个直接的和他们的通用张量积，急剧限制的可能性。这些结果保证了我们把注意力集中在厄米矩阵的具体Jordan代数上，即，在复矩阵代数中具有优先嵌入的欧氏若当代数}。我们表明，这些可以组织在一个自然的方式作为一个对称monoidal类别，虽然不是紧凑的封闭。然后，我们构建了一个相关的类别InvQM嵌入欧几里德约旦代数，有更少的对象，但更多的态射，这不仅是紧封闭的，但匕首紧。这一范畴统一了有限维实数、复数和四元数混合态量子力学，除了两个复数量子系统的复合物带有一个额外的经典位。我们的复合概念既不需要层析局部性，也不需要在张量积下保持纯度。我们构建的类别包括这两个条件都失败的例子。在这种情况下，复合物的信息容量（可相互区分的状态的最大数量）大于其组分容量的乘积。
英文摘要：
We consider possible non-signaling composites of probabilistic models based on euclidean Jordan algebras (EJAs), satisfying some reasonable additional constraints motivated by the desire to construct dagger-compact categories of such models. We show that no such composite has the exceptional Jordan algebra as a direct summand, nor does any such composite exist if one factor has an exceptional summand, unless the other factor is a direct sum of one-dimensional Jordan algebras (representing essentially a classical system). Moreover, we show that any composite of simple, non-exceptional EJAs is a direct summand of their universal tensor product, sharply limiting the possibilities. These results warrant our focussing on concrete Jordan algebras of hermitian matrices, i.e., euclidean Jordan algebras with a preferred embedding in a complex matrix algebra}. We show that these can be organized in a natural way as a symmetric monoidal category, albeit one that is not compact closed. We then construct a related category InvQM of embedded euclidean Jordan algebras, having fewer objects but more morphisms, that is not only compact closed but dagger-compact. This category unifies finite-dimensional real, complex and quaternionic mixed-state quantum mechanics, except that the composite of two complex quantum systems comes with an extra classical bit. Our notion of composite requires neither tomographic locality, nor preservation of purity under tensor product. The categories we construct include examples in which both of these conditions fail. In such cases, the information capacity (the maximum number of mutually distinguishable states) of a composite is greater than the product of the capacities of its constituents.
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