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Physical Review Letters

Operational Tomography: Fitting of Data Patterns

Jaroslav Řeháček, Dmitri S. Mogilevtsev, Z. Hradil

Published Jun 29, 2010• 62 citations
Publisher

Abstract

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.

DOI
10.1103/physrevlett.105.010402

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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.
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Tomographic reconstruction of quantum states plays a fundamental role in benchmarking quantum systems and accessing information encoded in quantum-mechanical systems. Among the informationally complete sets of quantum measurements, the tight ones provide a linear reconstruction formula and minimize the propagation of statistical errors. However, implementing tight measurements in the lab is challenging due to the high number of required measurement projections, involving a series of experimental setup preparations. In this work, we introduce the notion of cyclic tight measurements, which allow us to perform full quantum state tomography while considering only repeated application of a single unitary-based quantum device during the measurement stage. This type of measurement significantly simplifies the complexity of the experimental setup required to retrieve the quantum state of a physical system. Additionally, we design a feasible setup preparation procedure that produces well-approximated cyclic tight measurements in every finite dimension.
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.
The quantum state of a light beam can be represented as an infinite dimensional density matrix or equivalently as a density on the plane called the Wigner function. We describe quantum tomography as an inverse statistical problem in which the state is the unknown parameter and the data is given by results of measurements performed on identical quantum systems. We present consistency results for Pattern Function Projection Estimators as well as for Sieve Maximum Likelihood Estimators for both the density matrix of the quantum state and its Wigner function. Finally we illustrate via simulated data the performance of the estimators. An EM algorithm is proposed for practical implementation. There remain many open problems, e.g. rates of convergence, adaptation, studying other estimators, etc., and a main purpose of the paper is to bring these to the attention of the statistical community.