General Probabilistic Theories with a Gleason-type Theorem

Victoria J Wright1,2 and Stefan Weigert3

1International Centre for Theory of Quantum Technologies, University of Gdańsk, 80-308 Gdańsk, Poland
2ICFO-Institut de Ciencies Fotoniques, The Barcelona Institute of Science and Technology, 08860 Castelldefels, Spain
3Department of Mathematics, University of York, York YO10 5DD, United Kingdom

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Gleason-type theorems for quantum theory allow one to recover the quantum state space by assuming that (i) states consistently assign probabilities to measurement outcomes and that (ii) there is a unique state for every such assignment. We identify the class of general probabilistic theories which also admit Gleason-type theorems. It contains theories satisfying the no-restriction hypothesis as well as others which can simulate such an unrestricted theory arbitrarily well when allowing for post-selection on measurement outcomes. Our result also implies that the standard no-restriction hypothesis applied to effects is not equivalent to the dual no-restriction hypothesis applied to states which is found to be less restrictive.

Quantum theory successfully describes our world at a microscopic scale. Traditionally, the theory is formulated in terms of rather abstract postulates. They establish links between physical notions such as the state of an electron and suitable mathematical objects in a Hilbert space.

It is fair to say that the postulates are not intuitive, and many physicists are not entirely satisfied with this situation. In 1957, Andrew Gleason proved that it is possible to remove a significant layer of abstraction from the set of postulates. He did so by showing that the axiom introducing quantum states could be discarded in favour of an intuitive definition which is based on the postulate describing quantum measurements.

After this first step, the search for physically motivated axioms of quantum theory continued. It led to the formulation of the so-called general probabilistic theories, or GPTs. Unlike the traditional approach to quantum theory, the framework for GPTs is based on simple physical ideas, by relating basic probability theory and procedures that can be performed in a laboratory. Quantum theory is, in fact, an example of a GPT but there are also many other wild and wonderful theories in this broad class. The introduction of GPTs gives rise to a new perspective on quantum theory since we can search for additional properties or axioms singling out quantum theory among all other GPTs.

In this paper, we ask whether Gleason's modification of the axiomatic structure is special to quantum theory or whether his result would also apply to other GPTs describing another universe in which we might have lived. We find that GPTs can be divided into two classes, namely those which allow for an analogue of Gleason’s result (including quantum theory) and those which do not. Thus, our contribution brings us one step closer to understanding why quantum theory is special among GPTs.

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Cited by

[1] John B. DeBrota, Christopher A. Fuchs, Jacques L. Pienaar, and Blake C. Stacey, "Born's rule as a quantum extension of Bayesian coherence", Physical Review A 104 2, 022207 (2021).

[2] John B. DeBrota, Christopher A. Fuchs, Jacques L. Pienaar, and Blake C. Stacey, "The Born Rule as Dutch-Book Coherence (and only a little more)", arXiv:2012.14397.

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