Abstract
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.
Highlights
One of the greatest challenges associated with trying to demonstrate a quantum information processing (QIP) protocol experimentally is to be able to verify and quantify its successful operation
Any experiment will be affected by technical noise sources, errors that arise from inaccuracies or instabilities in the measurement apparatus
Most errors reported in QIP experiments consider only fundamental noise sources and ignore errors which arise from technical noise
Summary
I provide a brief description of a specific simple form of state tomography loosely following the treatment of QPT described in Ref. [20]. The typical way to solve this problem is to use maximum likelihood tomography [10, 11], an optimisation procedure which is very similar to common curve-fitting data analysis techniques This requires no assumptions about the form of the measured state except that it be physical. (Note that the more standard, linear weighted least squares problem would involve fixed variances which do not depend on the optimisation parameters.) This can be recast as follows as a particular form of convex optimisation called a semidefinite programme [15, 22, 28] Such problems are the focus of a large body of work and many good numerical routines are readily and freely available to solve them.
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