Sequential Product Spaces are Jordan Algebras

Abstract

We show that finite-dimensional order unit spaces equipped with a continuous sequential product as defined by Gudder and Greechie are homogeneous and self-dual. As a consequence of the Koecher-Vinberg theorem these spaces therefore correspond to Euclidean Jordan algebras. We remark on the significance of this result in the context of reconstructions of quantum theory. In particular, we show that sequential product spaces must be C*-algebras when their vector space tensor product is also a sequential product space (in the parlance of operational theories, when the space `allows a local composite'). We also show that sequential product spaces in infinite dimension correspond to JB-algebras when a few additional conditions are satisfied. Finally, we remark on how changing the axioms of the sequential product might lead to a new characterisation of homogeneous cones.

arXiv commentOriginal paper title was "Sequential Measurement Characterises Quantum Theory". It has been changed to reflect a change in focus of the paper. v3: Updated to journal version

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• General probabilistic theories: An introduction
• An effect-theoretic reconstruction of quantum theory
• Commutativity in Jordan Operator Algebras
• Derivation of quantum theory with superselection rules
• On the properties of spectral effect algebras
• A characterisation of ordered abstract probabilities
• A computer scientist's reconstruction of quantum theory
• The three types of normal sequential effect algebras
• Local tomography and the role of the complex numbers in quantum mechanics
• Pure Maps between Euclidean Jordan Algebras
• Quantum transition probability in convex sets and self-dual cones
• Intermediate determinism in general probabilistic theories