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 conjugate system, A¯, by means of a non-signaling bipartite state ηA perfectly and uniformly correlating each basic measurement on A with its counterpart on A¯. In the case of a quantum-mechanical system associated with a complex Hilbert space H, the conjugate system is that associated with the conjugate Hilbert space H, and ηA corresponds to the standard maximally entangled EPR state on H⊗H¯. A second ingredient is the notion of a 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 ``subpace axiom" can replace several assumptions native to this paper, although at the cost of disallowing superselection rules.)
Highlights
Introduction and OverviewA number of recent papers, notably [11, 14, 20, 25, 28], have succeeded in deriving the mathematical apparatus of finite-dimensional quantum mechanics (: QM) from various packages of broadly operational, probabilistic, or information-theoretic assumptions
A euclidean Jordan algebra (EJA) J is an ordered vector space with positive cone J + = {a2|a ∈ E}, and it can be shown that this cone is self-dual with respect to the given inner product [18]
The Koecher-Vinberg Theorem [18] states that, any finite-dimensional homogeneous, self-dual, homogenous ordered vector space J can be equipped with the structure of a euclidean Jordan algebra
Summary
A number of recent papers, notably [11, 14, 20, 25, 28], have succeeded in deriving the mathematical apparatus of finite-dimensional quantum mechanics (: QM) from various packages of broadly operational, probabilistic, or information-theoretic assumptions. The state ηA gives rise to a selfdualizing inner product on E(A), with respect to which E(A) and V(A) are isomorphic as ordered vector spaces It follows that if V(A) is homogeneous, so is E(A), whence, by the Koecher-Vinberg Theorem, the latter carries a formally real Jordan structure. This is the doctrine that the state of a bipartite system is determined by the joint probabilities it assigns to outcomes of measurements on the two component systems This principle has a certain intuitive appeal; it is well known, and easy to see on dimensional grounds, that among finite-dimensional real, complex and quaternionic quantum mechanics, only in the complex version are composites locally tomographic. Several of the ideas developed here were earlier explored, and somewhat similar results derived, in [30] and [31], but the approach taken here is much simpler and more direct, and seems to go a good deal farther
Sign-in/Register to view highlights & summary Sign-in/Register to access full text options 
Free) 
Talk to us
Join us for a 30 min session where you can share your feedback and ask us any queries you have
Schedule a call
