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arXiv:1702.08751 · IEEE Journal of Selected Topics in Quantum Electronics

Optimal Quantum Tomography

Alessandro Bisio, Giulio Chiribella, Giacomo Mauro D’Ariano, S. Facchini, Paolo Perinotti

Published Jan 1, 2009• 35 citations
arXivHTMLPDF

Abstract

The present short review article illustrates the latest theoretical developments on quantum tomography, regarding general optimization methods for both data processing and setup. The basic theoretical tool is the informationally complete measurement. The optimization theory for the setup is based on the new theoretical approach of quantum combs.

DOI
10.1109/jstqe.2009.2029243

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The present short review article illustrates the latest theoretical developments on quantum tomography, regarding general optimization methods for both data processing and setup. The basic theoretical tool is the informationally complete measurement. The optimization theory for the setup is based on the new theoretical approach of quantum combs.
Latest developments in quantum tomography are presented. The method for measuring the state of radiation is derived in a simple group-theoretical framework that allows generalization to arbitrary quantum systems. Some recently developed topics are synthetically reviewed, including tomography of many radiation modes using only one local oscillator, generalization to TV-level systems, and new “adaptive” techniques for noise reduction. A set of newly proposed experiments is presented, based on a conditioned tomographic technique. 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.
Quantum tomography is a critically important tool to evaluate quantum hardware, making it essential to develop optimized measurement strategies that are both accurate and efficient. We compare a variety of strategies using nearly pure test states. Those that are informationally complete for all states are found to be accurate and reliable even in the presence of errors in the measurements themselves, while those designed to be complete only for pure states are far more efficient but highly sensitive to such errors. Our results highlight the unavoidable trade-offs inherent in quantum tomography.
Abstract In quantum-state tomography on sources with quantum degrees of freedom of large Hilbert spaces, inference of quantum states of light for instance, a complete characterization of the quantum states for these sources is often not feasible owing to limited resources. As such, the concepts of informationally incomplete state estimation becomes important. These concepts are ideal for applications to quantum channel/ process tomography, which typically requires a much larger number of measurement settings for a full characterization of a quantum channel. Some key aspects of both quantumstate and quantum-process tomography are arranged together in the form of a tutorial review article that is catered to students and researchers who are new to the field of quantum tomography, with focus on maximum-likelihood related techniques as instructive examples to illustrate these ideas.
Quantum process tomography, a primitive in many quantum information processing tasks, can be cast within the framework of the theory of design of experiments (DOE), a branch of classical statistics that deals with the relationship between inputs and outputs of an experimental setup. Such a link potentially gives access to the many ideas of the rich subject of classical DOE for use in quantum problems. The classical techniques from DOE, however, cannot be directly applied to the quantum process tomography due to the basic structural differences between the classical and quantum estimation problems. Here we properly formulate quantum process tomography as a DOE problem and examine several examples to illustrate the link and the methods. In particular, we discuss the common issue of nuisance parameters and point out interesting features in the quantum problem absent in the usual classical setting.
Engineering of quantum computers requires reliable characterization of qubits, quantum operations, and even the entire hardware. Quantum tomography is an indispensable framework in quantum characterization, verification, and validation (QCVV), which has been widely accepted by researchers. According to the tomographic target, quantum tomography can be categorized into quantum state tomography (QST), quantum process tomography (QPT), gate set tomography (GST), process tensor tomography (PTT), and instrument set tomography (IST). Standard quantum tomography toolkits generally consist of basic linear inverse methods and statistical maximum likelihood estimation (MLE) based methods. Furthermore, the performance of standard methods, including effectiveness and efficiency, has been further developed by exploiting Bayesian estimation, neural networks, matrix completion techniques, etc. In this review, we introduce the fundamental quantum tomography techniques, including QST, QPT, GST, PTT, and IST. We first introduce the details of basic linear inverse methods. Then, the framework of MLE methods with constraints is summarized. Finally, we briefly introduce recent further research in developing the performance of tomography. This review provides a primary getting-start in developing quantum tomography, which promotes quantum computer development.
Recent theoretical developments are reviewed concentrating on the description of optimal generalised quantum measurement processes in physical systems with finite dimensional Hilbert spaces. Based on a self contained presentation of the required underlying quantum theoretical concepts this review particularly addresses recently explored basic theoretical properties and possible applications of informationally complete symmetric generalised measurements, a concept unifying the description of numerous so far largely separately explored measurement scenarios, such as measurements based on complete systems of mutually unbiased bases or symmetric informationally complete measurements. The recent results concerning the construction and existence of these informationally complete symmetric generalised measurements may also shed new light on still open questions concerning some of their special cases, such as the existence of complete sets of mutually unbiased bases in Hilbert spaces of non-prime power dimensions.