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arXiv:1106.2056 · Physical Review A

Statistical estimation of the quality of quantum-tomography protocols

Yu. I. Bogdanov, G. Brida, I. D. Bukeev, Marco Genovese, Konstantin Kravtsov, S. P. Kulik, Ekaterina V. Moreva, A. A. Soloviev, Alexander P. Shurupov

Published Oct 10, 2011• 57 citations
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Abstract

We present a complete methodology for testing the performances of quantum tomography protocols. The theory is validated by several numerical examples and by the comparison with experimental results achieved with various protocols for whole families of polarization states of qubits and ququarts including pure, mixed, entangled, and separable.

DOI
10.1103/physreva.84.042108

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We present a complete methodology for testing the performances of quantum tomography protocols. The theory is validated by several numerical examples and by the comparison with experimental results achieved with various protocols for whole families of polarization states of qubits and ququarts including pure, mixed, entangled, and separable.
Quantum tomography is the most informative tool for estimating the quality of preparation and transformation of quantum states. Its development is crucially necessary for debugging of developed quantum processors. Many existing methods of quantum tomography differ in types of performed measurements and in procedures of their processing. The practical implementation of quantum tomography requires the comparison of different methods, which is complicated because of the absence of a general methodology of estimation. A universal methodology based on numerical experiments has been proposed in this work to estimate the quality of quantum state tomography methods. The developed methodology has been applied to three quantum tomography methods (root approach, compressed sensing, and adaptive tomography) efficiently operating with almost pure states, which is relevant for the current technological foundation of the experiments.
An operational method has been proposed for estimating the efficiency of quantum cryptography protocols for quantum states with discrete variables. The method is based on the estimate of the quality of the protocol by means of a universal asymptotic distribution of the characteristic of the accuracy of the reconstruction of the quantum state, which is called fidelity. For a specially designed measurement matrix, the condition number, which is minimal for the optimal protocol, is introduced. The method has been tested in a numerical simulation and real experiments with polarization qubits. It has been shown that the optimal choice of a polarization transformer makes it possible to strongly improve the quality of the reconstruction of states.