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Deriving Born’s Rule from an Inference to the Best Explanation


In previous articles we presented a simple set of axioms named “Contexts, Systems and Modalities” (CSM), where the structure of quantum mechanics appears as a result of the interplay between the quantized number of modalities accessible to a quantum system, and the continuum of contexts that are required to define these modalities. In the present article we discuss further how to obtain (or rather infer) Born’s rule within this framework. Our approach is compared with other former and recent derivations, and its strong links with Gleason’s theorem are particularly emphasized.

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  1. 1.

    The word “context” includes the actual settings of the device, e.g. measurement of \(S_z\) rather than \(S_x\): the context must be factual, not contrafactual. On the other hand all devices able to measure \(S_z\) are equivalent as a context, in a (Bohrian) sense that they all define the same conditions for predicting the future behaviour of the system.

  2. 2.

    We omit the free evolution of the system; if it is present, the result of a new measurement can still be predicted with certainty, but in another context that can be deduced from the free evolution. Mutatis mutandis, this is equivalent to full repeatability.

  3. 3.

    In [10] extravalent modalities in different contexts are considered to be the same modality, transferred from a context to another. This is however not satisfactory, since a modality belongs to a specific context and system. The notions of extracontextuality and extravalence are thus more suitable, as explained in [12].

  4. 4.

    Note that extravalent modalities appear only if \(N \ge 3\), this has an obvious geometrical interpretation in relation with Gleason’s theorem (see below).

  5. 5.

    In order to make sense of Theorem 2, it is essential to distinguish between modalities and vectors in an Hilbert space, that will correspond to extravalence classes of modalities (see below). This issue is also essential for a good understanding of Gleason’s hypotheses.

  6. 6.

    A density operator is a positive semidefinite Hermitian operator with unit trace. It describes a pure state if it is a rank one projector.

  7. 7.

    In the general case in \(\mathcal{{R}}_{\text {3}}\), the maximum (resp. minimum) value of f is \(0 \le M \le 1\) (resp. \(0 \le m \le 1\)), and one shows [15] that there exist an orthonormal basis \(\{ p, q, r \}\) such that \(f(u) = M \cos ^2(u, p) + m \cos ^2(u, q) + (1-M-m) \cos ^2(u, r)\) with \(M+m \le 1\).

  8. 8.

    A detailed review on quantum measurements (including the algebraic framework) is presented in Landsman [22]. Note however that the ontological views expressed in this article are quite different from ours.

  9. 9.

    A real-valued monotone function on [a,b] is continuous, except in an at most countable set of discontinuities (Froda–Darboux theorem).


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The authors thank Franck Laloë and Roger Balian for many useful discussions, and Nayla Farouki for continuous support.

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Correspondence to Philippe Grangier.

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Auffèves, A., Grangier, P. Deriving Born’s Rule from an Inference to the Best Explanation. Found Phys (2020). https://doi.org/10.1007/s10701-020-00326-8

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  • Quantum mechanics
  • Born’s rule
  • Gleason’s theorem