Composites and Categories of Euclidean Jordan Algebras

Abstract

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.

62 pages; 3 figures. Slight further revisions for greater clarity

References in corpus (4)

• A derivation of quantum theory from physical requirements
• Higher-order interference and single-system postulates characterizing quantum theory
• Some Nearly Quantum Theories
• Symmetry and Self-Duality in Categories of Probabilistic Models

Cited by in corpus (25)

• Operational Resource Theory of Imaginarity
• A no-go theorem on the nature of the gravitational field beyond quantum theory
• Resource Theory of Imaginarity: New Distributed Scenarios
• Reconstructing quantum theory from diagrammatic postulates
• Accessible fragments of generalized probabilistic theories, cone equivalence, and applications to witnessing nonclassicality
• Nonlocal Advantages of Quantum Imaginarity
• Post-quantum steering is a stronger-than-quantum resource for information processing
• An effect-theoretic reconstruction of quantum theory
• Testing quantum theory by generalizing noncontextuality
• Probabilistic Theories and Reconstructions of Quantum Theory (Les Houches 2019 lecture notes)
• Universal structure of objective states in all fundamental causal theories
• Pseudo standard entanglement structure cannot be distinguished from standard entanglement structure
• How dynamics constrains probabilities in general probabilistic theories
• Process tomography in general physical theories
• A computer scientist's reconstruction of quantum theory
• Partial independence suffices to rule out Real Quantum Theory experimentally
• Variations on the Choi-Jamiolkowski isomorphism
• Quantum-imaginarity-based quantum speed limit
• Symmetry-induced failures of tomographic locality: Constructing foil theories by twirling
• Quantum transition probability in convex sets and self-dual cones
• Which entropy for general physical theories?
• Information-theoretic derivation of energy, speed bounds, and quantum theory
• The operational foundations of PT-symmetric and quasi-Hermitian quantum theory
• Locally Tomographic Shadows (Extended Abstract)
• Can QBism exist without Q? Morphophoric measurements in generalised probabilistic theories