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New Journal of Physics

Tomography for quantum diagnostics

Jaroslav Řeháček, Dmitri S. Mogilevtsev, Z. Hradil

Published Apr 14, 2008• 57 citations
Publisher

Abstract

The quantification of relevant statistical errors is an indispensable but often neglected part of any tomographic scheme used for quantum diagnostic purposes. We introduce a novel resolution measure, which provides 'error bars' for any inferred quantity of interest. This is illustrated with an example of the diagnostics of non-classical states based on the value of the reconstructed Wigner function at the origin of the phase space. We show that such diagnostics is meaningful only when a lot of prior information on the measured quantum state is available. Our resolution measure also provides an effective tool for optimization and resolution tuning of tomography schemes.

DOI
10.1088/1367-2630/10/4/043022

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The quantification of relevant statistical errors is an indispensable but often neglected part of any tomographic scheme used for quantum diagnostic purposes. We introduce a novel resolution measure, which provides 'error bars' for any inferred quantity of interest. This is illustrated with an example of the diagnostics of non-classical states based on the value of the reconstructed Wigner function at the origin of the phase space. We show that such diagnostics is meaningful only when a lot of prior information on the measured quantum state is available. Our resolution measure also provides an effective tool for optimization and resolution tuning of tomography schemes.
A prime goal of quantum tomography is to provide quantitatively rigorous characterization of quantum systems, be they states, processes or measurements, particularly for the purposes of trouble-shooting and benchmarking experiments in quantum information science. A range of techniques exist to enable the calculation of errors, such as Monte-Carlo simulations, but their quantitative value is arguably fundamentally flawed without an equally rigorous way of authenticating the quality of a reconstruction to ensure it provides a reasonable representation of the data, given the known noise sources. A key motivation for developing such a tool is to enable experimentalists to rigorously diagnose the presence of technical noise in their tomographic data. In this work, I explore the performance of the chi-squared goodness-of-fit test statistic as a measure of reconstruction quality. I show that its behaviour deviates noticeably from expectations for states lying near the boundaries of physical state space, severely undermining its usefulness as a quantitative tool precisely in the region which is of most interest in quantum information processing tasks. I suggest a simple, heuristic approach to compensate for these effects and present numerical simulations showing that this approach provides substantially improved performance.
We present a novel quantum tomographic reconstruction method based on Bayesian inference via the Kalman filter update equations. The method not only yields the maximum likelihood/optimal Bayesian reconstruction but also a covariance matrix expressing the measurement uncertainties in a complete way. From this covariance matrix the error bars on any derived quantity can be easily calculated. This is a first step towards the broader goal of devising an omnibus reconstruction method that could be adapted to any tomographic setup with little effort and that treats measurement uncertainties in a statistically well-founded way. In this first part, we restrict ourselves to the important subclass of tomography based on measurements with discrete outcomes (as opposed to continuous ones), and we also ignore any measurement imperfections (dark counts, less than unit detector efficiency, etc), which will be treated in a follow-up paper. We illustrate our general theory on real tomography experiments of quantum optical information processing elements.
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.
We show that the effective resolution of quantum inference is determined by a sampling Gram operator that defines the experimentally accessible degrees of freedom and the reconstruction bandwidth supported by the data. Acting analogously to a transfer function in classical imaging, this operator together with scaling of deviance provides an operational criterion for distinguishing genuinely resolved features from structures arising from incomplete sampling, statistical fluctuations, or reconstruction assumptions. Reconstruction in the Gram eigenbasis emerges naturally as a measurement-adapted compression of the inference problem, retaining only those modes supported by typical measurement outcomes. Within finite frame theory, the same structure appears as the frame operator governing stable reconstruction. The resulting framework connects quantum tomography, inverse reconstruction, and measurement resolution, providing a practical tool for assessing whether fine phase-space structures inferred in contemporary experiments are genuinely resolved by the measurement or merely reconstructed by the model.
Quantum state tomography is the task of inferring the state of a quantum system by appropriate measurements. Since the frequency distributions of the outcomes of any finite number of measurements will generally deviate from their asymptotic limits, the estimates computed by standard methods do not in general coincide with the true state and, therefore, have no operational significance unless their accuracy is defined in terms of error bounds. Here we show that quantum state tomography, together with an appropriate data analysis procedure, yields reliable and tight error bounds, specified in terms of confidence regions-a concept originating from classical statistics. Confidence regions are subsets of the state space in which the true state lies with high probability, independently of any prior assumption on the distribution of the possible states. Our method for computing confidence regions can be applied to arbitrary measurements including fully coherent ones; it is practical and particularly well suited for tomography on systems consisting of a small number of qubits, which are currently in the focus of interest in experimental quantum information science.