Principal Investigator: Professor V. P. Belavkin

In the frame of the project we studied:

1. The quantum analogies of Cramer Rao inequality and their relation to the generalized uncertainty relations corresponding to the symmetric and right logarithmic derivatives from the point of view of monotone Riemanian metrics.

2. The optimization problem of quantum efficiency for quantum statistical estimation of one or several parameters, and the corresponding exponential canonical families of quantum states.

3. Central limit in quantum mathematical statistics for the asymptotically efficient estimation of quantum identical independent states.

4. The noncommutative quantum-quantum analogies of the covariance's and corresponding Fisher informations for quantum information channels described by unital completely positive maps.

5. The relations between quantum relative entropies and quantum Fisher informations.

Carrying out this program we achieved the following results:

1. It has been shown that symmetric logarithmic derivatives and the symmetric (Helstrom) quantum Fisher information corresponds to the unbiased estimations of a single real or several commuting parameters, while the right derivatives and right Fisher information is more appropriate for estimation of a complex, or several non-commuting parameters of quantum states.

2. A one-to-one correspondence has been found between the efficient quantum statistics and special exponential families of quantum states for both symmetric and the right logarithmic derivatives.

3. It was proved that the maximal likelihood estimation is asymptotically efficient at least in the special central limit aproximations.

4. It was found that the generalized uncertainty relations correspond to the new, antisymmetric logarithmic derivatives. This result was also generalized to the quantum-quantum information channels described by the unital CP maps.

The results obtained and methods developed open a new interesting area belonging to both quantum statistics and quantum information. This area is planned to be investigated in an follow up research project on asymptotic methods quantum statistics and information theory. It will find various applications: In quantum probability, in information and communication via quantum channels. The results of this project should be considered as only first step in the realization of a long term project related to these fields.

The results of the project are included into three new papers, two already submitted for publication.

Acknowledgement: The Principle Investigator and the Visiting Fellow acknowledge the EPSRC funding of the four months research in the UK.