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Aspects of the Quantum-Classical Connection Based on Statistical Maps

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Based on three different types of affine mappings between the corresponding convex sets of states, three different kinds of relations between quantum mechanics and classical physics are reviewed.

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Notes

  1. The sharp effects are given by the extreme points of \( {\mathcal {E}}({\mathcal {H}})\), i.e., by the orthogonal projections. The other effects are called unsharp.—To the author’s knowledge, the terms “sharp” and “unsharp” for effects and observables were introduced by Paul Busch.

  2. The classical sharp effects are the extreme points of \( {\mathcal {E}}(\varOmega ,\varSigma )\), i.e., the characteristic functions \( \chi _B \), \( B \in \varSigma \).

  3. A set \( M \subseteq {\mathcal {E}}({\mathcal {H}})\) of effects is called coexistent if the effects of M can be measured jointly or, formally, if M is contained in the range of some observable. Two effects \( F,G \in {\mathcal {E}}({\mathcal {H}})\) are coexistent in this sense if and only if there exist effects \( F',G',H \in {\mathcal {E}}({\mathcal {H}})\) such that \( F = F' + H \), \( G = G' + H \), and \( F' + G' + H \le I \). Two sharp effects, i.e., orthogonal projections F and G, are coexistent if and only if they commute. The set M is called pairwise coexistent if any two effects of M are coexistent. Pairwise coexistence of M does not imply coexistence of M, an example is given in [14]. However, for a set of sharp effects the two concepts are equivalent. This follows, for instance, from von Neumann’s theorem that, for any set N of commuting self-adjoint operators acting in a separable Hilbert space, there exists a self-adjoint operator A such that every \( C \in N \) is of the form \( C = f(A) \) with some Borel function f.

    The definitions of coexistence and pairwise coexistence transfer literally to classical effects. Any two classical effects are coexistent, so the set \( {\mathcal {E}}(\varOmega ,\varSigma )\) of all classical effects is pairwise coexistent. In addition, any set of finitely many classical effects is coexistent [3]. Moreover, the set of all sharp classical effects is coexistent. In fact, \( A \mapsto \chi _A \), \( A \in \varSigma \), is a sharp classical observable; this observable corresponds to the random variable \( \omega \mapsto \omega \), \( \omega \in \varOmega \).

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    Werner Stulpe

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Dedicated to the memory of Paul Busch, a distinguished scholar and a great human being.

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Stulpe, W. Aspects of the Quantum-Classical Connection Based on Statistical Maps. Found Phys 49, 677–692 (2019). https://doi.org/10.1007/s10701-019-00269-9

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