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arXiv:1407.0959 · Physical Review A

Efficient algorithm for optimizing data-pattern tomography

L. Motka, B. Stoklasa, Jaroslav Řeháček, Z. Hradil, V. Karasek, Dmitri S. Mogilevtsev, Georg Harder, Christine Silberhorn, Luis Lorenzo Sánchez-Soto

Published May 23, 2014• 14 citations
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Abstract

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.

DOI
10.1103/physreva.89.054102

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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 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.
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.
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.
We propose an iterative algorithm that computes the maximum-likelihood estimate in quantum state tomography. The optimization error of the algorithm converges to zero at an O((1/k)log⁡D)O ( ( 1 / k ) \log D )O((1/k)logD) rate, where kkk denotes the number of iterations and DDD denotes the dimension of the quantum state. The per-iteration computational complexity of the algorithm is O(D3+ND2)O ( D ^ 3 + N D ^2 )O(D3+ND2), where NNN denotes the number of measurement outcomes. The algorithm can be considered as a parameter-free correction of the RρRR ρRRρR method [A. I. Lvovsky. Iterative maximum-likelihood reconstruction in quantum homodyne tomography. \textit{J. Opt. B: Quantum Semiclass. Opt.} 2004] [G. Molina-Terriza et al. Triggered qutrits for quantum communication protocols. \textit{Phys. Rev. Lett.} 2004.].
We suggest and demonstrate a tomographic method to characterise homodyne detectors at the quantum level.The positive operator measure associated with the detector is expanded in a quadrature basis and probed with a set of coherent states.The coefficients of the expansion are then retrieved using a least squares algorithm.Our model is general enough to describe different implementations of the homodyne setup, and it has proven capable of effectively describing the detector response to different tomographic sets.We validate the reconstructed operator measure on nonclassical states and exploit results to estimate the overall quantum efficiency of the detector.