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New Journal of Physics

Data pattern tomography: reconstruction with an unknown apparatus

Dmitri S. Mogilevtsev, Alexandr Ignatenko, А. С. Малоштан, B. Stoklasa, Jaroslav Řeháček, Z. Hradil

Published Feb 27, 2013• 34 citations
Publisher

Abstract

We propose a scheme for the reconstruction of the quantum state without a priori knowledge about the measurement setup. Using the data pattern approach, we develop an iterative procedure for obtaining information about the measurement which is sufficient for an estimation of a particular signal state. The method is illustrated with the examples of reconstruction with on/off detection and quantum homodyne tomography.

DOI
10.1088/1367-2630/15/2/025038

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We propose a scheme for the reconstruction of the quantum state without a priori knowledge about the measurement setup. Using the data pattern approach, we develop an iterative procedure for obtaining information about the measurement which is sufficient for an estimation of a particular signal state. The method is illustrated with the examples of reconstruction with on/off detection and quantum homodyne tomography.
We give a detailed account of an efficient search algorithm for the data-pattern tomography proposed by J. Rehacek, D. Mogilevtsev, and Z. Hradil [Phys. Rev. Lett. 105, 010402 (2010)], where the quantum state of a system is reconstructed without a priori knowledge about the measuring setup. The method is especially suited for experiments involving complex detectors, which are difficult to calibrate and characterize. We illustrate the approach with the case study of the homodyne detection of a nonclassical photon state.
We present a simple and efficient Bayesian recursive algorithm for the data-pattern scheme for quantum state reconstruction, which is applicable to situations where measurement settings can be controllably varied efficiently. The algorithm predicts the best measurements required to accurately reconstruct the unknown signal state in terms of a fixed set of probe states. In each iterative step, this algorithm seeks the measurement setting that minimizes the variance of the data-pattern estimator, which essentially measures the reconstruction accuracy, with the help of a data-pattern bank that was acquired prior to the signal reconstruction. We show that, with this algorithm, it is possible to minimize the number of measurement settings required to obtain a reasonably accurate state estimator by using just the optimal settings and, at the same time, increasing the numerical efficiency of the data-pattern reconstruction.
Quantum tomography is a procedure to determine the quantum state of a physical system, or equivalently, to estimate the expectation value of any operator. It consists in appropriately averaging the outcomes of the measurement results of different observables, obtained on identical copies of the same system. Alternatively, it consists in maximizing an appropriate likelihood function defined on the same data. The procedure can be also used to completely characterize an unknown apparatus. Here we focus on the electromagnetic field, where the tomographic observables are obtained from homodyne detection.
We build an operational scheme for the quantum state reconstruction based on the fitting of data patterns. Each data pattern corresponds to the response of the measurement setup to a predefined reference state. The set of data patterns can be measured experimentally in the calibration stage preceding to the reconstruction. The quorum of reference states plays the role of a positive operator valued measure in terms of which the reconstruction is done. As the main advantage, the procedure is free of notorious problems with projections into non-normalizable quadrature eigenstates, infinite dimensionality, ill-posed inversion, or imperfect detection.
Abstract We address state reconstruction by photon-number-resolving detectors, and demonstrate that they may be effectively exploited to perform quantum tomography of states of light. In particular, we find that the pattern function technique, originally developed for optical homodyne tomography, may be also applied to discrete data. Our results open new perspectives for quantum-state reconstruction in the mesoscopic regime, and pave the way to the use of photon-number-resolving-based detection schemes in Quantum Information science.