Summary
We give conditions on the existence of a Hilbert-space model for the transition probability matrix, describing a couple ofn-valued observables. In the casen=3 we prove necessary and sufficient conditions and compute explicitly statistical invariants pertinent to both complex- and real-Hilbert-space models.
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References
L. Accardi andA. Fedullo:Lett. Nuovo Cimento,34, 161 (1982).
S. Gudder andN. Zanghí:Nuovo Cimento, B,79, 291 (1984).
M. Roos:J. Math. Phys.,5, 1609 (1964). The author presented a sort of algorithm allowing one to construct, if possible, a unitary matrix, whenever they are given the moduli of its elements.
F. D. Murnaghan:The Unitary and Rotation Groups (Spartan Books Washington DC, 1962).
We must suppose that the denominators are not zero; however, the loss of generality is only apparent on account of next remark 5.1. The stated condition is necessary too, as pointed out by Roos in ref. [3],.
The method is quite similar to that employed, for different purposes, by Gudder and Zanghí in ref. [2], to which we refer for more details.
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Fedullo, A. On the existence of a Hilbert-space model for finite-valued observables. Nuov Cim B 107, 1413–1426 (1992). https://doi.org/10.1007/BF02722852
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DOI: https://doi.org/10.1007/BF02722852