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arXiv:quant-ph/0210146 · Physical Review A

Quantum inference of states and processes

Miroslav Ježek, Jaromı́r Fiurášek, Z. Hradil

Published Jul 9, 2003• 175 citations
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Abstract

The maximum-likelihood principle unifies inference of quantum states and processes from experimental noisy data. Particularly, a generic quantum process may be estimated simultaneously with unknown quantum probe states, provided that measurements on probe and transformed probe states are available. Drawbacks of various approximate treatments are also considered.

DOI
10.1103/physreva.68.012305

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The maximum-likelihood principle unifies inference of quantum states and processes from experimental noisy data. Particularly, a generic quantum process may be estimated simultaneously with unknown quantum probe states, provided that measurements on probe and transformed probe states are available. Drawbacks of various approximate treatments are also considered.
Maximum-likelihood estimation is applied to the determination of an unknown quantum measurement. The calibrated measuring apparatus carries out measurements on many different quantum states and the positive operator-valued measure governing the measurement statistics is then inferred from the collected data via the maximum-likelihood principle. In contrast to the procedures based on linear inversion, our approach always provides a physically sensible result. We illustrate the method on the case of the Stern-Gerlach apparatus.
The question of what information about a quantum state may be inferred from a sequence of measurements made on it, is addressed. The main result is that maximum-likelihood estimation gives an arguably natural optimum approach in quantum theory. It singles out a particular choice of basis, in which the state is expressed. Various incompatible and incomplete observations made on the ensemble are treated as a complete generalized measurement. Consequently, the maximum likelihood (relative entropy) is the best measure for relating experimental data with theoretical predictions of quantum theory.
ABSTRACT The simultaneous inseparable reconstruction of quantum process and states is presented. The method is basedon constrained extremization of the likelihood functional. This statistical approach is ingenuous in quantummechanical frame and ensures physically correct results. Adopted scheme predicts the input density operatorsand the operator of quantum process all at once with help of noisy measured data only. Attention is paid to thecomparison of posed method with sequential measurement of the states and the process. Method is illustratedby numerical simulations.Keywords: quantum measurement, quantum state reconstruction, quantum process reconstruction, maximumlikelihood, completely positive maps 1. INTRODUCTION Measuring of quantum state of physical systems has attracted increased attention recently.15 Topical exper- iments often include extracting of quantum state from measured data as it's native part and many methods were developed for this purpose during recent years. Nevertheless most of them yield reconstructed objectswhich are not bona fide physical states. Only constrained statistical methods can simply integrate various ex-perimental data and guarantee the non-negativity of estimated density matrix. Physically the most importantis maximum likelihood principle61' representing genuine quantum measurement. Maximum likelihood methodensures existence of path in state space along which the likelihood of state monotonically increases.12Discussions of experimental results and understanding quantum nature behind them require however recog-nition of quantum processes, not only input or output quantum states. Examination of quantum communicationchannels and evaluation of performance of quantum gates are examples of practical applicability of quantumprocess reconstruction.1320 And again, maximum likelihood appears as the efficient and exceptional
We propose an iterative algorithm for incomplete quantum process tomography with the help of quantum state estimation. The algorithm, which is based on the combined principles of maximum likelihood and maximum entropy, yields a unique estimator for an unknown quantum process when one has less than a complete set of linearly independent measurement data to specify the quantum process uniquely. We apply this iterative algorithm adaptively in various situations and so optimize the amount of resources required to estimate a quantum process with incomplete data.