Conciliation of Bayes and Pointwise Quantum State Estimation
Abstract
We derive an asymptotic lower bound on the Bayes risk when N identical quantum systems whose state depends on a vector of unknown parameters are jointly measured in an arbitrary way and the parameters of interest estimated on the basis of the resulting data. The bound is an integrated version of a quantum CramérRao bound due to Holevo^{13} and it thereby links the fixed N exact Bayesian optimality usually pursued in the physics literature with the pointwise asymptotic optimality favoured in classical mathematical statistics. By heuristic arguments the bound can be expected to be sharp. This does turn out to be the case in various important examples, where it can be used to prove asymptotic optimality of interesting and useful measurementandestimation schemes. On the way we obtain a new family of "dual Holevo bounds" of independent interest.
The paper is dedicated to Slava Belavkin in recognition of his pioneering work on quantum CramérRao bounds, on the occasion of his 60th birthday. A more complete version will appear in Annals of Statistics.
 Publication:

Quantum Stochastics and Information. Statistics, Filtering and Control
 Pub Date:
 August 2008
 DOI:
 10.1142/9789812832962_0011
 arXiv:
 arXiv:math/0512443
 Bibcode:
 2008qsi..conf..239G
 Keywords:

 Statistical quantum information bounds;
 van Trees inequality;
 CramérRao bounds;
 Mathematics  Statistics Theory;
 Quantum Physics;
 62F12;
 62P35
 EPrint:
 Entitled "Asymptotic information bounds in quantum statistics", accepted for publication subject to expansion by inclusion of introductory material by Annals of Statistics, 2003. I missed deadline (Lucia de B. case). This version appeared in conference proceedings given below. Finally revised and extended, coauthor Madalin Guta, arXiv:1112.2078, published 2013 in another conference proceedings