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.
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.
Quantum state tomography (QST) is a crucial technique for characterizing the state of a quantum system. However, the inherent limitations of measurement processes, including detector efficiency and photon counting statistics, introduce noise that affects the accuracy of the reconstructed quantum state. This work presents a rigorous information-theoretic framework for analyzing and quantifying the accuracy achievable in QST. We leverage concepts from information theory, specifically mutual information and channel capacity, to precisely determine the trade-offs between measurement resolution and the complexity of state reconstruction. The central argument is that the achievable accuracy is fundamentally limited by the information that can be reliably extracted from the measurement data, rather than relying solely on empirical estimates of error. This approach provides a theoretical benchmark for evaluating QST protocols and identifying optimal measurement strategies. The key contributions lie in establishing a quantifiable relationship between measurement parameters, noise levels, and the fidelity of the reconstructed quantum state. This offers a more robust and predictive method for assessing the quality of QST than traditional approaches.
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 Reliable verification of multipartite entanglement is a prerequisite for quantum information processing. However, the interplay between statistical fluctuations and systematic errors in Quantum State Tomography (QST) often obscures genuine quantum correlations. In this work, we perform a numerical benchmarking of QST performance for three-qubit GHZ and W states under a comprehensive noise model that incorporates Poissonian counting statistics, random unitary perturbations, white noise, and source instability. Beyond standard fidelity analysis, we utilize entanglement negativity and residual entanglement to investigate how these states respond to different noise sources. Our results highlight the response characteristics of entanglement measures and fidelity to experimental imperfections. We provide a quantitative analysis of the robustness profiles for both states, noting the entanglement decay and the evolution of residual entanglement under local unitary noise. Furthermore, we establish that employing an overcomplete tomographic set provides a slight advantage in mitigating systematic errors compared to standard informationally complete schemes. These findings provide a useful reference for analyzing the potential impacts of measurement artifacts and genuine decoherence in experimental setups.
The development of large-scale platforms for quantum information requires new methods for verification and validation of quantum behavior. Quantum tomography (QT) is the standard tool for diagnosing quantum states, process, and readout devices by providing complete information. However, QT is limited since it is expensive to not only implement experimentally, but also requires heavy classical post-processing of data. In this dissertation, we introduce new methods for QT that are more efficient to implement and robust to noise and errors, thereby making QT a more practical tool for current quantum information experiments. The crucial detail that makes these new, efficient, and robust methods possible is prior information about the quantum system. This prior information is prompted by the goals of most experiments in quantum information, which require pure states, unitary processes, and rank-1 POVM operators. Therefore, most experiments are designed to operate near this ideal regime. We show that when this is the case, QT can be accomplished with significantly fewer resources, and produce a robust estimate in the presence of noise and errors. Moreover, the estimate is also robust if the state is not exactly pure, the process is not exactly unitary, or the POVM is not exactly rank-1. Such compelling methods are only made possible by the positivity constraint on quantum states, processes, and POVMs. This requirement is an inherent feature of quantum mechanics, but has powerful consequences to QT. Since QT is necessarily an experimental tool, we discuss a test of these new methods in an experimental setting. The physical system is an ensemble of laser-cooled cesium atoms in the laboratory of Prof. Poul Jessen. Experiments were conducted by Hector Sosa-Martinez et al. to demonstrate different QT protocols. We compare the results, and conclude that the new methods are effective.
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.
In the light of the progress in quantum technologies, the task of verifying the correct functioning of processes and obtaining accurate tomographic information about quantum states becomes increasingly important. Compressed sensing, a machinery derived from the theory of signal processing, has emerged as a feasible tool to perform robust and significantly more resource-economical quantum state tomography for intermediate-sized quantum systems. In this work, we provide a comprehensive analysis of compressed sensing tomography in the regime in which tomographically complete data is available with reliable statistics from experimental observations of a multi-mode photonic architecture. Due to the fact that the data is known with high statistical significance, we are in a position to systematically explore the quality of reconstruction depending on the number of employed measurement settings, randomly selected from the complete set of data, and on different model assumptions. We present and test a complete prescription to perform efficient compressed sensing and are able to reliably use notions of model selection and cross validation to account for experimental imperfections and finite counting statistics. Thus, we establish compressed sensing as an effective tool for quantum state tomography, specifically suited for photonic systems.