The Uniqueness of the Chentsov Metric

by M. R. Grasselli and R. F. Streater, in Disordered and Complex Systems (London 2000), ed: P. Sollich, A. C. C. Coolen, L. Hughston and R. F. Streater, Amer. Inst. of Phys. Conference Proceedings, Vol. 553, 165-170, 2001.


We show that on the space of faithful density matrices, the only monotone metrics (those reduced by any completely positive trace-preserving map), for which the exponential and mixture affine connections are mutually dual, are constant multiples of the Bogoliubov-Kubo-Mori (BKM) metric.

Remark. This is a six-page summary of the full version. In the present work, we outline the proof, but also add some details concerning the extension the metric from the manifold of density matrices to the manifold of the weights, in such a way that the metric continues to be monotone.

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