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Post-Classical Probability Theory

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Book cover Quantum Theory: Informational Foundations and Foils

Part of the book series: Fundamental Theories of Physics ((FTPH,volume 181))

Abstract

This chapter offers a brief introduction to what is often called the convex-operational approach to the foundations of quantum mechanics, and reviews selected results, mostly by ourselves and collaborators, obtained using that approach. Broadly speaking, the goal of research in this vein is to locate quantum mechanics (henceforth: QM) within a very much more general, but conceptually very straightforward, generalization of classical probability theory. The hope is that, by regarding QM from the outside, so to say, we shall be able to understand it more clearly. And, in fact, this proves to be the case.

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Notes

  1. 1.

    The formalism easily accommodates contextual probability assignments, however: simply define \(\widetilde{X}\) to be the disjoint union of the test in \(\varvec{\mathcal {M}}\)—say, to be concrete, \(\widetilde{X} = \{ (x,E) | x \in E \in \varvec{\mathcal {M}}\}\). In effect, each outcome of \(\widetilde{X}\) consists of an outcome of X, plus a record of which test was used to secure it. For each test \(E \in \varvec{\mathcal {M}}\), let \(\widetilde{E} = \{ (x,E) | x \in E\}\), and let \(\widetilde{\varvec{\mathcal {M}}} = \{ \widetilde{E} | E \in \varvec{\mathcal {M}}\}\). Probability weights on \((\widetilde{X},\widetilde{\varvec{\mathcal {M}}})\) are exactly what one means by contextual probability weights on \((X,\varvec{\mathcal {M}})\). There is a natural surjection \(\widetilde{X} \rightarrow X\) that simply forgets these records; probability weights on \((X,\varvec{\mathcal {M}})\) pull back along this surjection to give us weights on \((\widetilde{X},\widetilde{\varvec{\mathcal {M}}})\).

  2. 2.

    A more detailed discussion of test spaces with topological structure can be found in [72].

  3. 3.

    That is, given any pair of distinct outcomes, there exists a state assigning them different probabilities.

  4. 4.

    An isomorphism of test spaces is a bijection from outcomes to outcomes, preserving tests in both directions.

  5. 5.

    A weaker notion would require only that \(\alpha _i(x_i) > 0 = \alpha _i(x_j)\) for each i, j, so that with some nonzero probability we learn which state is actual. Notice, too, that the condition of joint sharp distinguishability is a priori much stronger than pairwise sharp distinguishability.

  6. 6.

    Many authors define ordered linear spaces without requiring that the positive cone be either closed or generating. For our purposes, the present definition is more useful.

  7. 7.

    In general, one must add the requirement that \({\varvec{E}}\)’s ordering be archimedean, meaning that if \(x, y \in {\varvec{E}}\) with \(0 \le nx \le y\) for all \(n \in {\mathbb N}\), then \(x = 0\). However, in our finite-dimensional setting, any closed cone induces an archimedean ordering.

  8. 8.

    One of many uses for the test space structure is to privilege certain classes of observables on an order-unit space having special order-theoretic properties—for example, the set of observables the outcomes of which lie on extremal rays of \({\varvec{E}}_+\) forms a test space, or those whose outcomes are atomic effects, i.e., those that lie on extremal rays of \({\varvec{E}}_+\) and are extreme points of [0, u].

  9. 9.

    This definition differs from that of [17], most obviously in that objects are associated with effect spaces, rather than state spaces, but also in taking the test space X(A) to be part of the structure of \(A \in \mathcal{C}\).

  10. 10.

    More radically, one might consider models of systems interacting in such a way that the making of a particular measurement, or the preparation of a particular state, on one component, precludes the making of certain measurements, or the preparation of certain states, on the other component. Mathematically, such situations are certainly possible.

  11. 11.

    Barrett [19] calls this the global state hypothesis; the term locally tomographic seems to have become more standard.

  12. 12.

    The converse is not quite true: an order-isomorphism \({\varvec{E}}(A) \simeq {\mathbf V}(A)\) defines a non-signaling state on \(A \otimes _{\text {max}}B\) (see  definition 14 below), but need not correspond to a state of AB.

  13. 13.

    Consider, for instance, the case of \((\text{ Farmer } \otimes \text{ Hen } ) \otimes \text{ Fox } \ \ \text{ vs. } \ \ \text{ Farmer } \otimes (\text{ Hen } \otimes \text{ Fox })\).

  14. 14.

    Duals, where they exist, are canonically isomorphic. Hence, for most purposes, the choice of one rather than another object as “the” dual is irrelevant. The existence of a degenerate compact structure is, however, a real constraint [15, 62].

  15. 15.

    One might raise the aesthetic objection that it is awkward to make special reference to the interior state. But it is difficult to see how this is any worse aesthetically than making special reference to, say, pure states.

  16. 16.

    This last has sometimes been criticized for being “too mathematical”—that is, insufficiently operational and at the same time, too technically involved. It’s worth pointing out, however, that much of it becomes significantly simpler when specialized to the finite-dimensional case.

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Authors and Affiliations

  1. Department of Physics and Astronomy, University of New Mexico, Albuquerque, Mexico

    Howard Barnum

  2. Stellenbosch Institute for Advanced Study (STIAS), Wallenberg Research Centre at Stellenbosch University, Marais Street, Stellenbosch, 7600, South Africa

    Howard Barnum

  3. Department of Mathematics, Susquehanna University, Selinsgrove, PA, USA

    Alexander Wilce

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Editors and Affiliations

  1. Department of Computer Science, The University of Hong Kong, Hong Kong, China

    Giulio Chiribella

  2. Perimeter Inst Theoretical Physics , Waterloo, Ontario, Canada

    Robert W. Spekkens

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Barnum, H., Wilce, A. (2016). Post-Classical Probability Theory. In: Chiribella, G., Spekkens, R. (eds) Quantum Theory: Informational Foundations and Foils. Fundamental Theories of Physics, vol 181. Springer, Dordrecht. https://doi.org/10.1007/978-94-017-7303-4_11

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