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Separability, Contextuality, and the Quantum Frame Problem

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Abstract

We study the relationship between assumptions of state separability and both preparation and measurement contextuality, and the relationship of both of these to the frame problem, the problem of predicting what does not change in consequence of an action. We state a quantum analog of the latter and prove its undecidability. We show how contextuality is generically induced in state preparation and measurement by basis choice, thermodynamic exchange, and the imposition of a priori causal models, and how fine-tuning assumptions appear ubiquitously in settings characterized as non-contextual.

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Notes

  1. This is sometimes called quantum or intrinsic contextuality to distinguish it from classical, causal, context dependence.

  2. The agents A and B are “observer-participants” in Wheeler’s sense; we argue in [43] that quantum theory is precisely the theory of interactions between such agents. From a philosophical perspective, our approach is thus a form of “participatory realism” in Cabello’s [53] sense. It differs from QBism [54] by rejecting the latter’s assumption that agents can interact specifically and exclusively with particular components of their environments.

  3. Contextuality in quantum mechanics is by now accepted as a state of affairs to be lived with. Curiously, noncontextuality seems to be an equally perplexing issue. Hofer-Szabó in [57] distinguishes between simultaneous and measurement noncontextuality as distinct, logically independent models. In the first case, an ontological (hidden variable) model is noncontextual if every ontic (hidden) state determines the probability of the outcomes of every measurement independently of whatever other measurements are simultaneously performed (otherwise the model is contextual). In the second case, an ontological model is noncontextual if any two measurements represented by the same self-adjoint operator, or equivalently, which have the same probability distribution of outcomes in every quantum state, also have the same probability distribution of outcomes in every ontic state (ditto). This distinction reflects upon the schism between operational and ontological models when dealing with contextuality, as exemplified by the fact that contextuality-by-default [21, 22] and causal contextuality [58,59,60], respectively, are different theories [61].

  4. Note that the diagram in Eq (2.1) depicts a generic blue-print for a variational autoencoder, an ubiquitous mechanism in Deep Learning [62].

  5. The classifiers \(\mathcal {A}\) appearing in Diag. (2.1) can be considered observables in context as discussed in [25, §3,§7]. As tokens/types, or as objects/attributes, for instance, each \(\mathcal {A}= (\textbf{E}, \textbf{X})\), where \(\textbf{E}\) is some “event” and \(\textbf{X}\) is a pair \(\textbf{X} = (\textbf{B}, \textbf{R})\), where \(\textbf{B}\) is a set of “conditions” or “influences”, and \(\textbf{R}\) is a set of “contexts” or “methods/detectors”. The set \(\textbf{X}\) can be seen as a basic ingredient for contextuality-by-default, but in contrast to [22, 23], our approach to a scale-free architecture does not necessitate assuming any specific connectivity in distributions, or any admissible probability classes of the random variables. It is the noncommutativity of the CCCD in [25, Th 7.1] that specifies (intrinsic) contextuality within a hierarchy of distributed information flow.

  6. It is worth pointing out that a distributed system of information flow as formulated in [52], of which CCCDs (when considered as systems of logical gates) are examples, already embodies context and causation, and is especially suited for modeling ontologies. These characteristics have been professed in [65, 66], in which the former uses the logical formulism of [52] to argue that causation itself may be viewed as a form of computation resulting from the regular relations in a distributed system (such as a CCCD), and the latter shows that for any pair of classifiers \(\mathcal {A},\mathcal B\) occurring in such a system, there exists some (logic) infomorphism between them such that \(\mathcal {A}\) directly causes \(\mathcal B\). In a companion theory of integrative information involving abstract logical structures known as Institutions (see e.g. [67,68,69]), the element of context is further emphasized within a category of signatures with associated sentences. As briefly recalled in the proof of Lemma 2.1, the basic mechanism of a QRF in conjunction with binary-valued classifiers provides a model for a generic topological quantum field theory [26] with no immediate need for further logical abstractions – ‘sentences’ in this case are basically qubit strings. On the other hand, classifiers (classifications) are important, generic 1-institutions from which other institutions can be generated [67, 69]. The ‘institutional’ approach to contextuality will be further addressed at a later date in relationship to the present development of ideas.

  7. As pointed out in [32], analogical reasoning is a case of informational unencapsulation in its extreme form.

  8. The FP was widely regarded as intractable, but nonetheless avoidable via circumscription and heuristics, since its first statement [30]; hence theoretical work on the problem was largely conducted with thought experiments that considered hypothetical autonomous robots embedded in open task environments, i.e. task environments having only a partial a priori model, e.g. [70, 71] where the robot suffers ‘Hamlet’s Problem’ of ‘over-thinking’ and not knowing when to stop. Such models are reviewed in [32] in the context of “Global Workspace” or shared memory architectures as models of massively parallel, distributed systems of possibly-adversarial processes. The semantically coherent, distributed systems of information flow developed in [52] provide a scale-free architectural model for any such generic Workspace (see e.g. [25, 72, 73]).

  9. Language sharing among human observers always involves a past common cause, and hence is an example of superdeterminism.

  10. This sense of ‘no disturbance’ has also been called ‘no-signaling’, where in relativistic accordance, such pairs of measurements X, Y are made in spacelike separated regions (see also e.g. [14, 103]).

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Acknowledgements

We thank Eric Dietrich and Antonino Marcianò for prior discussions on the frame problem and contextuality, respectively. We also thank a referee for careful reading of this paper, constructive criticism, and recommendations

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  1. Allen Discovery Center, Tufts University, Medford, MA, 02155, USA

    Chris Fields

  2. Department of Mathematics and Computer Science, Eastern Illinois University, 600 Lincoln Ave., Charleston, IL, 61920–3099, USA

    James F. Glazebrook

  3. Adjunct Faculty Department of Mathematics, University of Illinois at Urbana–Champaign, Urbana, IL, 61801, USA

    James F. Glazebrook

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Fields, C., Glazebrook, J.F. Separability, Contextuality, and the Quantum Frame Problem. Int J Theor Phys 62, 159 (2023). https://doi.org/10.1007/s10773-023-05406-9

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  • DOI: https://doi.org/10.1007/s10773-023-05406-9

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