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Russian Microelectronics

Statistical Models and Adequacy Validation for Optical Quantum State Tomography with Quadrature Measurements

Yu. I. Bogdanov, Nadezhda A. Bogdanova, L. V. Belinsky, Vladimir F. Lukichev

Published Nov 1, 2017• 3 citations
Publisher

Abstract

Mutually complementary quadrature quantum measurements are analyzed and a new method to formulate statistical models of quantum states is proposed. The method is based on the root approach to quantum measurements and includes a procedure for approximating quantum states with reduced finite dimensional models. The efficiency of the proposed approach is demonstrated using numerical experiments. This approach is aimed at achieving the highest possible precision in multiphoton quantum state tomography.

DOI
10.1134/s1063739717060038

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Mutually complementary quadrature quantum measurements are analyzed and a new method to formulate statistical models of quantum states is proposed. The method is based on the root approach to quantum measurements and includes a procedure for approximating quantum states with reduced finite dimensional models. The efficiency of the proposed approach is demonstrated using numerical experiments. This approach is aimed at achieving the highest possible precision in multiphoton quantum state tomography.
We discuss a procedure of measurement followed by the reproduction of the quantum state of a three-level optical system---a frequency---and spatially degenerate two-photon field. The method of statistical estimation of the quantum state based on solving the likelihood equation and analyzing the statistical properties of the obtained estimates is developed. Using the root approach of estimating quantum states, the initial two-photon state vector is reproduced from the measured fourth moments in the field. The developed approach applied to quantum-state reconstruction is based on the amplitudes of mutually complementary processes. The classical algorithm of statistical estimation based on the Fisher information matrix is generalized to the case of quantum systems obeying Bohr's complementarity principle. It has been experimentally proved that biphoton-qutrit states can be reconstructed with the fidelity of 0.995--0.999 and higher.
Reliable generation and measurement of triphoton states has yet to be achieved in laboratory. We give an overview of the problems in generating and measuring triphoton quantum states and analyze several protocols of quantum measurements, which allow for high precision of reconstruction when sizes of available statistical data samples are limited. The tomography procedure under investigation is based on root approach to state estimation. In particular, we use the generalized Fisher information matrix to assess the accuracy of the quantum state parameters measurement. We use tomographic protocols, based on the symmetry of the Platonic solids. We demonstrate the capability to reconstruct triphoton quantum states with precision close to the maximum achievable value allowed by quantum mechanics.
A method of quantum tomography of arbitrary spin particle states is developed on the basis of the root approach. It is shown that the set of mutually complementary distributions of angular momentum projections can be naturally described by a set of basis functions based on the Kravchuk polynomials. The set of Kravchuk basis functions leads to a multiparametric statistical distribution that generalizes the binomial distribution. In order to analyze a statistical inverse problem of quantum mechanics, we investigated the likelihood equation and the statistical properties of the obtained estimates. The conclusions of the analytical researches are approved by the results of numerical calculations.