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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
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.
- oai_identifier:
- oai:arXiv.org:quant-ph/0306179
- categories:
- quant-ph
- comments:
- 10 pages RevTeX, no figures
- doi:
- 10.1023/B:FOOP.0000019581.00318.a5
- arxiv_id:
- quant-ph/0306179
- journal_ref:
- Found. Phys. 34, 193 (2004)
- created:
- 2003-06-26
Full article ▸
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