Conjugates, Filters and Quantum Mechanics

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 systemAcan be paired with an isomorphicconjugatesystem,A¯, by means of a non-signaling bipartite stateηAperfectly and uniformly correlating each basic measurement onAwith its counterpart onA¯. In the case of a quantum-mechanical system associated with a complex Hilbert spaceH, the conjugate system is that associated with the conjugate Hilbert spaceH, andηAcorresponds to the standard maximally entangled EPR state onH⊗H¯. A second ingredient is the notion of areversible 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 ``subpace axiom" can replace several assumptions native to this paper, although at the cost of disallowing superselection rules.)