Unifying probabilities for incompatible consistent sets of histories: general existence, uniqueness, and the persistence of quasiclassicality

Halliwell (Phys. Rev. A 96, 012123 (2017)) proposed classifying families of incompatible consistent sets of quantum histories according to whether a single non-negative 'unifying probability' exists whose marginals reproduce the consistent-histories probabilities of each set -- an extension of the single framework rule whose failure is read as a measure of the 'quantumness' of a system. The proposal was carried through on examples using case-by-case tools (Fine's theorem, CHSH and Leggett-Garg inequalities, non-negative Wigner functions for Gaussian states), but the underlying general theory was left open. The open core comprises: (i) general necessary and sufficient existence criteria and constructive methods for arbitrary collections of consistent sets, beyond Fine-type special cases; (ii) a classification of when the unifying probability is unique, including the source's conjecture that vanishing marginals force uniqueness; and (iii) explicit consistent-histories constructions deciding whether quasiclassical behaviour on an initial temporal window persists under future fine-grainings (the Dowker-Kent persistent-classicality question). Later work extends Fine's theorem to an arbitrary number of times (Halliwell-Mawby 2019) and exhibits a quasiclassical contrary-inference example with no unifying probability (Zampeli-Pavlou-Wallden 2026), sharpening but not resolving these questions.

Background

In the consistent histories (CH) approach to quantum theory, a closed system is described by histories -- time-ordered strings of projection operators -- with class operators $C_\alpha = P_{a_n}(t_n)\cdots P_{a_1}(t_1)$, and probabilities $p(\alpha)=\mathrm{Tr}(C_\alpha\rho C_\alpha^\dagger)$ are assigned only when the family satisfies a consistency (decoherence) condition: the interference terms $\mathrm{Tr}(C_\alpha\rho C_{\alpha'}^\dagger)$ vanish for distinct histories $\alpha\neq\alpha'$. A single physical situation typically admits several consistent sets that cannot be combined into one consistent set, and the standard interpretation imposes the single framework rule: logical or probabilistic reasoning may not mix propositions from incompatible sets. Kent showed that this multiplicity permits contrary inferences -- two consistent sets each asserting a probability-one proposition that contradicts the other's -- and, with Dowker, raised the persistent-classicality question: quasiclassical behaviour (histories of coarse-grained variables, such as localized positions, whose probabilities are strongly peaked around classical equations of motion) established on an initial temporal window need not persist, because one can always fine-grain to the future using different, non-classical variables and obtain a second, incompatible consistent set; nothing in the CH formalism by itself favours either set.

Halliwell (2017) proposed classifying families of incompatible consistent sets by a marginal-matching test. The CH quasi-probability $q(\alpha)=\mathrm{Re}\,\mathrm{Tr}(C_\alpha\rho)$ (the Goldstein-Page quasi-probability) always has non-negative marginals on each single consistent set; the test asks whether there exists a unifying probability: one genuine (non-negative) probability distribution on a classical-style sample space -- generally larger than the history sample space of any one set -- whose marginals reproduce the CH probabilities of every set in the family. Where such a distribution exists, logical deductions from different consistent sets can be consistently combined -- an extension of the single framework rule -- and where it does not, its absence is read as a measure of the quantumness of the system. In the standard examples the test runs through known results: Fine's theorem, which states that four pairwise marginals admit a common joint probability if and only if all CHSH inequalities (the Clauser-Horne-Shimony-Holt Bell inequalities) hold; its temporal analogue, the Leggett-Garg inequalities, which test macrorealism for a single dichotomic variable probed at several times; and, for Gaussian states, the Wigner function -- the standard phase-space quasi-probability, everywhere non-negative precisely for Gaussian states -- which supplies the unifying probability. The construction coincides with the earlier viability classification of quasi-probabilities (Halliwell-Yearsley 2013): a quasi-probability is viable if its non-negative marginals can be matched to a genuine probability. What the 2017 paper did not supply is the general theory behind these special cases: existence for arbitrary collections of marginals, uniqueness of the unifying distribution, and application to the persistent-classicality question. The paper itself states, as established features of the proposal rather than open questions, that the marginal-matching procedure is compatible with subsystem composition -- the Diósi test being the requirement, studied by Halliwell in the context of weaker classicality conditions, that a condition on histories be compatible with the composition of subsystems -- and that its stronger conditions are related to the existence of records (present-day correlations that encode past events).

Problem Statement

The problem is to complete the unifying-probability classification of multiple consistent sets of histories into a general theory, resolving the following jointly connected questions left open by Halliwell:

1. Existence and construction. For an arbitrary collection of consistent sets -- with arbitrary overlap patterns among their alternatives and arbitrary families of specified marginals -- establish necessary and sufficient conditions for the existence of a non-negative joint probability whose marginals match the consistent-histories probabilities of every set in the collection, and provide a practical constructive method when one exists. The known cases must emerge as instances: Fine's theorem (four pairwise marginals, CHSH), its Leggett-Garg analogues, and the non-negative Wigner function of Gaussian states.

2. Uniqueness. Classify when the unifying probability is unique; in particular, prove or refute the expectation recorded in the source that when some marginals vanish (probability-0/1 propositions) the unifying probability is fixed uniquely, and identify the general constraints that guarantee uniqueness.

3. Persistent classicality. Construct explicit consistent-histories models in which a system exhibits quasiclassical behaviour over an initial temporal window, and determine -- using (1) -- whether incompatible fine-grainings to the future (extending the same quasiclassical variables versus fine-graining in different, non-classical variables) admit a unifying probability; thereby determine whether quasiclassicality is stable or fragile under set choice (the question raised by Dowker and Kent).

These deliverables jointly serve one scientific objective: converting the example-based unifying-probability classification into a general account of the classical-quantum boundary within the consistent histories approach. A complete answer must address all of these general questions, not merely accumulate further special cases; the form of the answer may vary by facet -- general theorems, or explicit constructions and counterexamples for the persistent-classicality facet.

References

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