QUANTUM MATHEMATICAL STATISTICS

Quantum Mathematical Statistics (QMS)
studies the ways how a more complete knowledge of a quantum state can be inferred from the results of measurements of quantum occurrences (quantum measurement). Just as the classical mathematical statistics gives the decision rules to finds the probabilities of commuting events as a result of statistical hypothesis testing or parameter estimation. Unlike the Quantum Probability Theory QMS had not been developed at all until the end of 60's when Helstrom showed how the problems of optimal testing of two quantum statistical hypotheses and optimal estimation of a single unknown parameter of a quantum state can be solved mathematically. Holevo studied the mathematical aspects of the optimization problem for quantum measurements and inferences in the beginning of 70's. My own contribution was the the study of the optimal estimation problem for several parameters of non-commuting quantum states [1, 2], the solution of the quantum Bayesian problem for the linear-Gaussian case by generalized Heisenberg inequality [3, 4], the solution of the optimal multiple hypothesis testing problem for several non-orthogonal quantum pure states [5-7], right Cramer-Rao bound, uncertainty relation for shift parameter estimation, the solution of the efficiency problem using symmetrical and right quantum Cramer-Rao inequality [8], and the recursive solution of the sequential problem for quantum optimal estimation of Markov chains [9]. The extention and application of these methods to the noncommutative problems of wave pattern recognition is discribed in the book [10].

My 10 Relevant Publications:
  1. V. P. Belavkin: Optimal Estimation of Noncommutating Quantum Gaussian Variables under the Sequential Inaccurate Measurement. Radio Eng Electron Physics 17(12) 527--2532 (1972).
  2. V. P. Belavkin: Linear Estimation of Non-commuting Observables by their Indirect Measurement. Radio Eng Electron Physics 17(12) 2533--2540 (1972).
  3. V. P. Belavkin and B. Grishanin: Study of the Optimal Estimation Problem in Quantum Channels by a Generalised Heisenberg Inequality Method. Problems of Information Transmission 9(3) 44--52 (1973).
  4. V. P. Belavkin: Optimal Observation of Boson Signals in Quantum Gaussian Channels. Problems of Control and Information Theory 4 241--257 (1975).
  5. V. P. Belavkin: Optimal Discrimination of Non-orthogonal Quantum Signals. Radio Eng Electron Physics 20(6) 1177--1185 (1975).
  6. V. P. Belavkin: Optimal Multiple Quantum Statistical Hypothesis Testing. Stochastics 1 315--345 (1975).
  7. V. P. Belavkin: Hypothesis Testing for Quantum Optical Fields. Radio Eng Electron Physics 21(6) 95--104 (1976).
  8. V. P. Belavkin: Generalized Heisenberg Uncertainty Relations and Efficient Measurements in Quantum Systems. Theoret Math Physics 3 316--329 (1976). quant-ph/0412030, PDF.
  9. V. P. Belavkin: Optimal Quantum Filtration of Markovian Signals. Problems of Control and Information Theory 7 (5) 345--360 (1978).
  10. V. P. Belavkin and V. Maslov: Design of Optimal Dynamic Analyzer: Mathematical Aspects of Wave Pattern Recognition. In Mathematical Aspects of Computer Engineering, Ed V. Maslov, 146--237, Mir, Moscow, 1987. quant-ph/0412031, PDF.