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arXiv:1201.5612 · Physical Review A

Statistical analysis of sampling methods in quantum tomography

T. Kiesel

Published May 17, 2012• 8 citations
arXivHTMLPDF

Abstract

In quantum physics, all measured observables are subject to statistical uncertainties, which arise from the quantum nature as well as the experimental technique. We consider the statistical uncertainty of the so-called sampling method, in which one estimates the expectation value of a given observable by empirical means of suitable pattern functions. We show that if the observable can be written as a function of a single directly measurable operator, the variance of the estimate from the sampling method equals to the quantum-mechanical one. In this sense, we say that the estimate is on the quantum-mechanical level of uncertainty. In contrast, if the observable depends on noncommuting operators (e.g., different quadratures), the quantum-mechanical level of uncertainty is not achieved. The impact of the results on quantum tomography is discussed, and different approaches to quantum tomographic measurements are compared. It is shown explicitly for the estimation of quasiprobabilities of a quantum state, that balanced homodyne tomography does not operate on the quantum-mechanical level of uncertainty, while the unbalanced homodyne detection does.

DOI
10.1103/physreva.85.052114

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In quantum physics, all measured observables are subject to statistical uncertainties, which arise from the quantum nature as well as the experimental technique. We consider the statistical uncertainty of the so-called sampling method, in which one estimates the expectation value of a given observable by empirical means of suitable pattern functions. We show that if the observable can be written as a function of a single directly measurable operator, the variance of the estimate from the sampling method equals to the quantum-mechanical one. In this sense, we say that the estimate is on the quantum-mechanical level of uncertainty. In contrast, if the observable depends on noncommuting operators (e.g., different quadratures), the quantum-mechanical level of uncertainty is not achieved. The impact of the results on quantum tomography is discussed, and different approaches to quantum tomographic measurements are compared. It is shown explicitly for the estimation of quasiprobabilities of a quantum state, that balanced homodyne tomography does not operate on the quantum-mechanical level of uncertainty, while the unbalanced homodyne detection does.
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.
Balanced homodyning, heterodyning, and unbalanced homodyning are three well-known sampling techniques used in quantum optics to characterize photonic sources in the continuous-variable regime. We show that for all quantum states and all observable-parameter tomography schemes, which includes reconstructions of arbitrary operator moments and phase-space quasidistributions, localized sampling with unbalanced homodyning is always tomographically more powerful (gives more accurate estimators) than delocalized sampling with heterodyning. The latter is recently known to often give more accurate parameter reconstructions than conventional marginalized sampling with balanced homodyning. This result also holds for realistic photodetectors with subunit efficiency. With examples from first- through fourth-moment tomography, we demonstrate that unbalanced homodyning can outperform balanced homodyning when heterodyning fails to do so. This new benchmark takes us one step towards optimal continuous-variable tomography with conventional photodetectors and minimal experimental components.
We expand the scope of the statistical notion of error probability, that is, how often large deviations are observed in an experiment, to make it directly applicable to quantum tomography. We verify that the error probability can decrease at most exponentially in the number of trials, we derive the explicit rate that bounds this decrease, and we show that a maximum likelihood estimator achieves this bound. We also show that the statistical notion of identifiability coincides with the tomographic notion of informational completeness. Our result implies that two quantum tomographic apparatuses that have the same risk function (e.g., variance) can have different error probability, and we give an example in one-qubit-state tomography. Thus by combining these two approaches we can evaluate, in a reconstruction-independent way, the performance of such experiments more discerningly.
Abstract The quantum state associated to an unknown experimental preparation procedure can be determined by performing quantum state tomography. If the statistical uncertainty in the data dominates over other experimental errors, then a tomographic reconstruction procedure must express this uncertainty. A rigorous way to accomplish this is via statistical confidence regions (CRs) in state space. Naturally, the size of this region decreases when increasing the number of samples, but it also depends critically on the construction method of the region. We compare recent methods for constructing CRs as well as a reference method based on a Gaussian approximation. For the comparison, we propose an operational measure with the finding, that there is a significant difference between methods, but which method is preferable can depend on the details of the state preparation scenario.
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.
In this job, we will present a theory called Quantum Tomography that is the natural extension of the theory of detection of signals in classical telecommunications to Quantum Mechanics. This theory mainly consists in the reconstruction of a quantum state of a system through a probability distribution measured directly in the laboratory, usually called Tomogram. This Thesis contains five chapters. In the first one, we will show the birth of this theory as an adaptation of homodyne and heterodyne classical detection to Quantum Mechanics. In the second, we will describe a tomographic description of Quantum Mechanics on C*-algebras by splitting the theory in two parts: a Generalized Sampling Theory and a Generalized Positive Transform. In the third, we will present the first numerical algorithm that solves the Clebsh-Gordan decomposition problem for any finite-dimensional unitary representation of any finite or compact Lie group. This algorithm receives the name of SMILY algorithm. In the fourth and fifth chapters, we will discuss tomography of systems parametrized with fields. In the fourth, we will reconstruct states of classical systems and in the fifth, we will do the same with quantum ones, but here in two different ways: the first one will be done by adapting the reconstruction formula of chapter four, and the second one by means of the axiomatic description of Wightamn-Streater of a Quantum Field Theory.