Unified statistical method for reconstructing quantum states by purification
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Unified Statistical Method for Reconstructing Quantum States by Purification Yu. I. Bogdanov Institute of Physics and Technology, Russian Academy of Sciences, Moscow, 117218 Russia email: [email protected] Received November 17, 2008
Abstract—Mixedstate purification is used as a basis to formulate a general statistical method for recon structing the density matrix of an arbitrary quantum state. A universal statistical distribution is obtained for the fidelity of the reconstructed quantum state. The proposed theory is supported by results of numerical sim ulations. PACS numbers: 03.65.Wj, 42.50.Dv DOI: 10.1134/S106377610906003X
1. INTRODUCTION Quantum information technologies rely on the use of quantum states in novel data transmission and com puting protocols [1–3]. Control is achieved via quan tumstate reconstruction from quantum measurement data by statistical methods. Quantum measurement theory is based on von Neumann’s projection postu late [4] and its extensions [5]. Physical aspects of quantum measurements were discussed in [6–8]. In recent years, various research groups have per formed numerous successful measurements on quan tum states of light (e.g., see [9–15]). A review of research in this area can be found in [16]. This paper generalizes and extends the results reported in [10, 11, 17, 18]. In those studies, the max imum likelihood method was used to develop efficient algorithms and procedures for reconstructing the state vector of a quantum system. A theory of statistical fluctuations of the estimated state vector of a quantum system was developed, and a chisquare criterion was proposed for estimating the level of statistical fluctua tions. However, the previous approach is applicable only to pure quantum states. The present analysis deals with mixed states of general form and relies on densitymatrix purification. The resulting purified state vector is fully amenable to the procedures previ ously developed for reconstructing a quantum state and analyzing statistical fluctuations. The approach developed here is important from a theoretical perspective since mixedstate estimation is considered a difficult task [5]. Mixedstate purifica tion and Fisher information matrix can be used to explicitly decompose statistical fluctuations into phys ically relevant and “irrelevant” ones. Fluctuations of the latter type arise from the arbitrariness of the phases of the purestate mixture components, the nonu
niqueness of decomposition into pure states, and fluc tuations of state norm. An optimal projection operator is used to purify a quantum state of “irrelevant” fluc tuations. This is important for computational reasons because the algorithm and numerical procedure of quantumstate reconstruction become well condi tioned. When a mixed state is not optimally parame terized, “irrelevant” fluctuations give rise to uncon trollable errors and make the computation of a high dimensional problem numerically unstable. The fluc tuation control developed in this study makes it poss
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