A bistochastic matrix of size is called unistochastic if there exists a unitary such that for . The set of all unistochastic matrices of order forms a proper subset of the Birkhoff polytope, which contains all bistochastic (doubly stochastic) matrices. We compute the volume of the set with respect to the flat (Lebesgue) measure and analytically evaluate the mean entropy of an unistochastic matrix of this order. We also analyze the Jarlskog invariant , defined for any unitary matrix of order three, and derive its probability distribution for the ensemble of matrices distributed with respect to the Haar measure on and for the ensemble which generates the flat measure on the set of unistochastic matrices. For both measures the probability of finding smaller than the value observed for the Cabbibo–Kobayashi–Maskawa matrix, which describes the violation of the CP parity, is shown to be small. Similar statistical reasoning may also be applied to the Maki–Nakagawa–Sakata matrix, which plays role in describing the neutrino oscillations. Some conjectures are made concerning analogous probability measures in the space of unitary matrices in higher dimensions.
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December 2009
Research Article|
December 22 2009
Volume of the set of unistochastic matrices of order 3 and the mean Jarlskog invariant
Charles Dunkl;
Charles Dunkl
a)
1Department of Mathematics,
University of Virginia
, Charlottesville, Virginia 22904-4137, USA
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Karol Życzkowski
Karol Życzkowski
b)
2Institute of Physics,
Jagiellonian University
, 30-059 Cracow, Poland
and Center for Theoretical Physics, Polish Academy of Sciences
, 00-668 Warsaw, Poland
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a)
Electronic mail: [email protected].
b)
Electronic mail: [email protected].
J. Math. Phys. 50, 123521 (2009)
Article history
Received:
September 04 2009
Accepted:
November 13 2009
Citation
Charles Dunkl, Karol Życzkowski; Volume of the set of unistochastic matrices of order 3 and the mean Jarlskog invariant. J. Math. Phys. 1 December 2009; 50 (12): 123521. https://doi.org/10.1063/1.3272543
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