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Post-measurement states in generalized probabilistic theories

  1. Sep 14, 2011 #1
    Hello everybody

    I am currently trying to understand attempts to create a framework of generalized probabilistic theories in which quantum theory and classical theory appear as special cases. More precisely, I try to understand the framework which is sometimes called the framework of "convex theories", in which the set of states is considered to be a convex subset an ordered vector space. For those not familiar with this framework, have a look at e.g. the following papers:
    • H. Barnum: Ordered linear spaces and categories as frameworks for information-processing characterizations of quantum and classical theory, Arxiv preprint arXiv:0908.2354
    • J. Barret et al.: Limits on nonlocal correlations from the structure of the local state space, New Journal of Physics 13 (2011) 063024 (24pp)
    • O. Dahlsten et al.: Unifying typical entanglement and coin tossing: on randomization in probabilistic theories, Arxiv preprint arXiv:1107.6029

    There is one thing I don't understand. So far, I have not found a paper in which it is explained how performing a measurement changes the state of a system. I'm trying to establish a rule for updating the state from the pre- to the post-measurement state. Let me tell you what I have come up with so far:

    I think it is clear that one cannot give a rule for updating the state for every effect of a theory (see papers above to learn what it means to be an effect). If this were possible, then one could assign a post-measurement state to every element of a POVM in quantum mechanics. But in the case of a POVM which is not a projective measurement, this is not possible, because if one regards this POVM as being induced by a measurement on a larger system containing the system in question as a subsystem, then the post-measurement state might depend on the state of the ancilla system which is needed to extend the system in question to the larger system.
    But (still talking about the quantum case) one can assign a post-measurement state to the system if the the POVM is a projective measurement. Noting that a POVM element is a projector if and only if it is an extremal point of the set of proper effects, one might hope that one can generalize the state-update-rule to the case of a generalized probabilistic theory.

    One property which, I would say, is save to demand from a post-measurement state is the following one:
    If one measures a state [itex]\rho[/itex] and gets the result corresponding to effect [itex]f[/itex], then the post-measurement state [itex]\rho_{post}[/itex] should satisfy [itex]f(\rho_{post}) = 1[/itex].
    The problem is that in general, this condition does not specify [itex]\rho_{post}[/itex] uniquely. In the quantum case, the condition specifies [itex]\rho_{post}[/itex] uniquely if [itex]f[/itex] is induced by a one-dimensional projector, but it does not in the case where it is a 2- or higher-dimensional projector.

    Now one might say the following (I hope this claim is correct; i haven't proved it so far): In the quantum case, [itex]f(\rho_{post})=1[/itex] specifies [itex]\rho_{post}[/itex] uniquely (i.e. [itex]f[/itex] is a one-dimensional projector) if and only if [itex]f[/itex] (in addition to being an extreme point of the set of proper effects) is ray extremal (i.e. lies on an extremal ray of [itex]V^*_+[/itex]). In the classical and quantum case, restriction to the set of extremal and ray extremal points of the set of proper effects is sufficient to get unique post-measurement states by the condition [itex]f(\rho_{post})=1[/itex].
    But again, this does not suffice. In boxworld (see the n=4 polygon theory of the second paper listed above), there are extremal and ray-extremal effects for which [itex]f(\rho_{post})=1[/itex] is satisfied for a whole 1-dimensional face of the set of states.

    After all, I don't know how to establish a rule for assigning post-measurement states in generalized probabilistic theories. Can anyone help me? Any ideas or even solutions to this problem?

    Thanks in advance.
  2. jcsd
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