Conjugates, Filters and Quantum Mechanics

Abstract

The Jordan structure of finite-dimensional quantum theory is derived, in a conspicuously easy way, from a few simple postulates concerning abstract probabilistic models (each defined by a set of basic measurements and a convex set of states). The key assumption is that each system A can be paired with an isomorphic $\text{conjugate}$ system, $\overline{A}$, by means of a non-signaling bipartite state ${\eta }_A$ perfectly and uniformly correlating each basic measurement on A with its counterpart on $\overline{A}$. In the case of a quantum-mechanical system associated with a complex Hilbert space $\mathcal{H}$, the conjugate system is that associated with the conjugate Hilbert space $\overline{\mathcal{H}}$, and ${\eta }_A$ corresponds to the standard maximally entangled EPR state on $\mathcal{H}\otimes \overline{\mathcal{H}}$. A second ingredient is the notion of a $\text{reversible filter}$, that is, a probabilistically reversible process that independently attenuates the sensitivity of detectors associated with a measurement. In addition to offering more flexibility than most existing reconstructions of finite-dimensional quantum theory, the approach taken here has the advantage of not relying on any form of the "no restriction" hypothesis. That is, it is not assumed that arbitrary effects are physically measurable, nor that arbitrary families of physically measurable effects summing to the unit effect, represent physically accessible observables. An appendix shows how a version of Hardy's "subspace axiom" can replace several assumptions native to this paper, although at the cost of disallowing superselection rules.

33 pp. Minor corrections throughout; some revision of Appendix A

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• Composites and Categories of Euclidean Jordan Algebras
• Some Nearly Quantum Theories
• Operational restrictions in general probabilistic theories
• Testing quantum theory by generalizing noncontextuality
• Information-theoretic foundations of thermodynamics in general probabilistic theories
• Derivation of quantum theory with superselection rules
• A Royal Road to Quantum Theory (or Thereabouts), Extended Abstract
• Process tomography in general physical theories
• The operational framework for quantum theories is both epistemologically and ontologically neutral
• Non-locality in theories without the no-restriction hypothesis
• Local tomography and the role of the complex numbers in quantum mechanics
• Symmetry-induced failures of tomographic locality: Constructing foil theories by twirling
• Conical Designs and Categorical Jordan Algebraic Post-Quantum Theories
• Characterization of the probabilistic models that can be embedded in quantum theory
• Quantum transition probability in convex sets and self-dual cones
• Can QBism exist without Q? Morphophoric measurements in generalised probabilistic theories
• Spacetime symmetries and the qubit Bloch ball: a physical derivation of finite dimensional quantum theory and the number of spatial dimensions