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A Gleason-Type Theorem for Any Dimension Based on a Gambling Formulation of Quantum Mechanics

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Abstract

Based on a gambling formulation of quantum mechanics, we derive a Gleason-type theorem that holds for any dimension n of a quantum system, and in particular for \(n=2\). The theorem states that the only logically consistent probability assignments are exactly the ones that are definable as the trace of the product of a projector and a density matrix operator. In addition, we detail the reason why dispersion-free probabilities are actually not valid, or rational, probabilities for quantum mechanics, and hence should be excluded from consideration.

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Notes

  1. Abstract units of utility, indicating the satisfaction derived from an economic transaction.

  2. For an example see Benavoli et al. ([4], Example 1).

  3. By enforcing those requirements, partial and sure (Dutch book) losses become equivalent.

  4. We mean the eigenvectors of the density matrix of the quantum system.

  5. A projector \(\Pi \) is a set of n positive semi-definite matrices in \(\mathbb {C}_h^{n\times n}\) such that \(\Pi _i\Pi _k=0\), \((\Pi _i)^2=\Pi _i=(\Pi _i)^\dagger \), \(\sum _{i=1}^n \Pi _i=I\).

  6. In Benavoli et al. [4] we used another formulation of openness, namely (S3’): if \(G \in \mathcal {K}\) then either \(G \gneq 0\) or \(G -\Delta \in \mathcal {K}\) for some \(0<\Delta \in \mathbb {C}_h^{n\times n}\). (S3) and (S3’) are provably equivalent given (S1) and (S2).

  7. Here the gambles \(G\gneq 0 \) are treated separately because they are always desirable and, thus, they are not informative on Alice’s beliefs about the quantum system. Alice’s knowledge is determined by the gambles that are not \(G\gneq 0 \).

  8. This happens when an eigevalue has multiplicity greater than one.

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Correspondence to Alessio Benavoli.

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Benavoli, A., Facchini, A. & Zaffalon, M. A Gleason-Type Theorem for Any Dimension Based on a Gambling Formulation of Quantum Mechanics. Found Phys 47, 991–1002 (2017). https://doi.org/10.1007/s10701-017-0097-0

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