No data available.

Please log in to see this content.

You have no subscription access to this content.

No metrics data to plot.

The attempt to load metrics for this article has failed.

The attempt to plot a graph for these metrics has failed.

The full text of this article is not currently available.

Volume of the set of unistochastic matrices of order 3 and the mean Jarlskog invariant

### Abstract

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 CPparity, 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.

© 2009 American Institute of Physics

Received 04 September 2009
Accepted 13 November 2009
Published online 22 December 2009

Acknowledgments:
It is a pleasure to thank I. Bengtsson and W. Tadej for numerous stimulating discussions and helpful correspondence. We acknowledge financial support by the special Grant No. DFG-SFB/38/2007 of Polish Ministry of Science and Higher Education and an European Research project COCOS (K.Ż.).

Article outline:

I. INTRODUCTION
II. THE BIRKHOFF POLYTOPE
III. THE SET OF UNISTOCHASTIC MATRICES
IV. THE VOLUME OF THE SET OF UNISTOCHASTIC MATRICES OF ORDER OF 3
V. MEAN ENTROPY OF A UNISTOCHASTIC MATRIX
VI. DISTRIBUTION OF THE JARLSKOG INVARIANT
VII. CONCLUDING REMARKS

/content/aip/journal/jmp/50/12/10.1063/1.3272543

http://aip.metastore.ingenta.com/content/aip/journal/jmp/50/12/10.1063/1.3272543

Article metrics loading...

/content/aip/journal/jmp/50/12/10.1063/1.3272543

2009-12-22

2016-10-24

Full text loading...

###
Most read this month

Article

content/aip/journal/jmp

Journal

5

3

Commenting has been disabled for this content