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New Journal of Physics

Self-calibration for self-consistent tomography

Dmitri S. Mogilevtsev, Jaroslav Řeháček, Z. Hradil

Published Sep 3, 2012• 37 citations
Publisher

Abstract

Self-calibrating states are introduced as quantum resources for the simultaneous reconstruction of a quantum state and measurement device. Simple criteria for a state being self-calibrating are formulated. Theory applies to realistic state estimation scenarios, where the detection model is nonlinear in unknown parameters of the measuring device. Examples discussed include time multiplexed detection with on/off detectors and quantum homodyne tomography. The thermally noised single-photon states and Gaussian states are self-calibrating for these schemes. For homodyne tomography, statistical tests applied to raw measured data are sufficient for judging whether or not the signal state is self-calibrating without any a priori information.

DOI
10.1088/1367-2630/14/9/095001

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Self-calibrating states are introduced as quantum resources for the simultaneous reconstruction of a quantum state and measurement device. Simple criteria for a state being self-calibrating are formulated. Theory applies to realistic state estimation scenarios, where the detection model is nonlinear in unknown parameters of the measuring device. Examples discussed include time multiplexed detection with on/off detectors and quantum homodyne tomography. The thermally noised single-photon states and Gaussian states are self-calibrating for these schemes. For homodyne tomography, statistical tests applied to raw measured data are sufficient for judging whether or not the signal state is self-calibrating without any a priori information.
Self-calibrating quantum state tomography aims at reconstructing the unknown quantum state and certain properties of the measurement devices from the same data. Since the estimates of the state and device parameters come from the same data, one should employ a joint estimation scheme, including the construction and reporting of joint state-device error regions, to quantify uncertainty. We explain how to do this naturally within the framework of optimal error regions. As an illustrative example, we apply our procedure to the double-crosshair measurement of the BB84 scenario in quantum cryptography and so reconstruct the state and estimate the detection efficiencies simultaneously and reliably. We also discuss the practical situation of a satellite-based quantum key distribution scheme, for which self-calibration and proper treatment of the data are necessities.
Standard deterministic techniques for quantum state reconstruction, as for example optical homodyne tomography, photon chopping, unbalanced homodyning etc., are based on the deterministic inversion of measured data. Since the frequencies obtained in realistic experiments always differ from probabilities predicted by quantum theory due to fluctuations, imperfections and realistic restrictions, the algorithm of inversion cannot guarantee the positive definiteness of the reconstructed density matrix. Hence the estimation of the noises may appear as doubtful. Quantum states may be successfully reconstructed within quantum and information theories using the maximum likelihood estimation. The question of deterministic schemes: “What quantum state is determined by that measurement?” is replaced by the formulation consistent with quantum theory: “What quantum state(s) seems to be most likely for that measurement?” Nonlinear equation for reconstructed state is formulated. An exact solution may be approached by subsequent iterations. Reconstruction is formulated as a problem of proper normalization of incompatible (nonorthogonal) measurements. The results obtained by this novel method may differ significantly from the standard predictions. Data are fitted better keeping the constraint of positive definiteness of reconstructed density matrix. However, this interpretation may enlarge uncertainty in prediction of quantum state in comparison with deterministic schemes, since, in general, there is a whole family of states which fit the measured data equally well. The novel technique is nonlinear and the reinterpretation of existing reconstruction schemes represents an advanced program. These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.
I show that calibration of the single-photon detector can be performed without knowledge of the signal parameters. Only partial information about the state statistics is sufficient for that. If one knows that the state is the squeezed one or the squeezed one mixed with the incoherent radiation, one can infer both the parameters of the state and the efficiency of the detector. For that one needs only to measure on/off statistics of detector clicks for the number of known absorbers placed before the detector. Thus, I suggest a scheme that performs a tomography of the signal and the measuring apparatus simultaneously.
We show that the information collected in the course of a generic quantum tomography experiment can be used for verifying experimenters' assumptions about the state preparation and measurement. In particular, systematic errors, such as drifts and instabilities inherent in the tomography setup, can be identified without the need for any specific measurements designed to detect such problems. This is done by statistical analysis of available tomography data. As an example, we discuss the detection and characterization of unexpected changes in the signal state preparation during the tomography measurement and argue that minimal tomography schemes, such as symmetric informationally complete measurements, are more difficult to validate than overcomplete schemes. The method is illustrated with the analysis of experimental quantum homodyne tomography.
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.