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Gleason-Type Derivations of the Quantum Probability Rule for Generalized Measurements

Carlton M. Caves·Christopher A. Fuchs·Kiran Manne·Joseph M. Renes
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Abstract

We prove a Gleason-type theorem for the quantum probability rule using frame functions defined on positive-operator-valued measures (POVMs), as opposed to the restricted class of orthogonal projection-valued measures used in the original theorem. The advantage of this method is that it works for two-dimensional quantum systems (qubits) and even for vector spaces over rational fields--settings where the standard theorem fails. Furthermore, unlike the method necessary for proving the original result, the present one is rather elementary. In the case of a qubit, we investigate similar results for frame functions defined upon various restricted classes of POVMs. For the so-called trine measurements, the standard quantum probability rule is again recovered.

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Carlton M. Caves, Christopher A. Fuchs, Kiran Manne, Joseph M. Renes. 2003-06-26. Gleason-Type Derivations of the Quantum Probability Rule for Generalized Measurements. https://doi.org/10.1023/b%3Afoop.0000019581.00318.a5

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