标题： 量子理论的自对偶性和乔丹结构源于齐性和纯传递性 显示英文标题

标题： Self-duality and Jordan structure of quantum theory follow from homogeneity and pure transitivity

摘要： 量子态空间的许多重要的几何性质包括：未归一化态锥体内部对称群的传递性（齐次性），通过内积将该锥体与其效应的对偶锥体相联系（自对偶性），以及归一化态空间对称群在纯归一化态上的传递性（纯传递性）。 Koecher和Vinberg表明，齐次性和自对偶性可以表征Jordan代数型态空间：实数、复数和四元数量子理论，自旋因子，三维八元数量子态空间以及这些不可约空间的直和。 我们证明自对偶性可以从齐次性和纯传递性推导出来。 这些性质比自对偶性具有更直接的物理和信息处理意义。 例如（扩展了Barnum、Gaebeler和Wilce的结果），齐次性与量子态的操控能力密切相关。 我们对Koecher-Vinberg定理的替代方案几乎表征了相同的态空间集合：同构的Jordan代数型态空间的直和，这些可以看作是经典系统与一个不可约的Jordan代数型态空间的复合。 有各种在物理和信息上自然的附加公设已被证明能够从这些Jordan代数可能性中单独选出复数量子理论。 我们给出了基于局部层析性质的多种重建方法。 显示英文摘要

摘要： Among the many important geometric properties of quantum state space are: transitivity of the group of symmetries of the cone of unnormalized states on its interior (homogeneity), identification of this cone with its dual cone of effects via an inner product (self-duality), and transitivity of the group of symmetries of the normalized state space on the pure normalized states (pure transitivity). Koecher and Vinberg showed that homogeneity and self-duality characterize Jordan-algebraic state spaces: real, complex and quaternionic quantum theory, spin factors, 3-dimensional octonionic quantum state space and direct sums of these irreducible spaces. We show that self-duality follows from homogeneity and pure transitivity. These properties have a more direct physical and information-processing significance than self-duality. We show for instance (extending results of Barnum, Gaebeler, and Wilce) that homogeneity is closely related to the ability to steer quantum states. Our alternative to the Koecher-Vinberg theorem characterizes nearly the same set of state spaces: direct sums of isomorphic Jordan-algebraic ones, which may be viewed as composites of a classical system with an irreducible Jordan-algebraic one. There are various physically and informationally natural additional postulates that are known to single out complex quantum theory from among these Jordan-algebraic possibilities. We give various such reconstructions based on the additional property of local tomography.