A Representation of Quantum Measurement in Order-Unit Spaces

Foundations of Physics (Impact Factor: 1.14). 01/2010; DOI: 10.1007/s10701-008-9236-y
Source: arXiv

ABSTRACT A certain generalization of the mathematical formalism of quantum mechanics beyond operator algebras is considered. The approach is based on the concept of conditional probability and the interpretation of the Lueders - von Neumann quantum measurement as a probability conditionalization rule. A major result shows that the operator algebras must be replaced by order-unit spaces with some specific properties in the generalized approach, and it is analyzed under which conditions these order-unit spaces become Jordan algebras. An application of this result provides a characterization of the projection lattices in operator algebras. Comment: 12 pages, the original publication is available at htt://

  • Source
    [Show abstract] [Hide abstract]
    ABSTRACT: In quantum mechanics, the Hilbert space operators play a triple role as observables, generators of the dynamical groups and statistical operators defining the mixed states. One might expect that this is typical of Hilbert space quantum mechanics, but it is not. The same triple role occurs for the elements of a certain ordered Banach space in a much more general theory based upon quantum logics and a conditional probability calculus. It is shown how positive groups, automorphism groups, Lie algebras and statistical operators emerge from one major postulate - the non-existence of third-order interference (third-order interference and its impossibility in quantum mechanics were discovered by R. Sorkin in 1994). This again underlines the power of the combination of the conditional probability calculus with the postulate that there is no third-order interference. In two earlier papers, its impact on contextuality and nonlocality had already been revealed.
  • Source
    [Show abstract] [Hide abstract]
    ABSTRACT: We define a simple rule to describe sequences of projective measurements for such generalized probabilistic models that can be described by an Archimedean order-unit vector space. For quantum mechanics, this definition yields the established L\"uders's rule, while in the general case it can be seen as the least disturbing or most coherent way to perform sequential measurements. As example we show that Spekkens toy model is an instance of our definition. We also demonstrate the possibility of strong post-quantum correlations and triple-slit correlations for certain non-quantum toy models.
    Journal of Physics A Mathematical and Theoretical 02/2014; 47(45). DOI:10.1088/1751-8113/47/45/455304 · 1.77 Impact Factor
  • [Show abstract] [Hide abstract]
    ABSTRACT: Considering not only the well-known two-slit experiment, but also experiments with three slits, Sorkin introduced the third-order interference term I3 and discovered that the absence of third-order interference (I3=0) is typical of quantum mechanics where only second-order interference occurs. In the present paper, the interference term I3 is ported to the quantum logics with unique conditional probabilities. In this framework, the identity I3=0 does not hold in general and its consequences are analysed. A first result reveals a close link between this identity and the existence of a product in the order-unit space generated by the quantum logic. In the general case, this product is neither commutative nor associative. By a second result, the order-unit space becomes a Jordan algebra, if each element behaves like one would expect from an observable (i.e., its square is positive and there is a polynomial functional calculus). Almost all such Jordan algebras can be represented as operator algebras on a Hilbert space, and a reconstruction of quantum mechanics up to this point is thus achieved from the absence of third-order interference and a few other principles. Besides the identity I3=0, two further interesting properties of quantum mechanics distinguishing it from more general theories are studied. These are a novel bound for quantum interference and a symmetry condition for the conditional probabilities.


1 Download
Available from