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Fortschritte der Physik

Quantum Tomography: General Theory and New Experiments

Giacomo Mauro D’Ariano

Published May 1, 2000• 6 citations
Publisher

Abstract

Quantum tomography is a general method for estimating the ensemble average of all operators of an arbitrary quantum system from a set of measurements of a quorum of observables. A procedure for deconvolving instrumental noise is available, which makes the tomographic method feasible in different physical contexts. Recent developments are presented and new experiments are proposed, which are now made possible by the tomographic technique.

DOI
10.1002/(sici)1521-3978(200005)48:5/7<579::aid-prop579>3.0.co;2-p

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Quantum tomography is a general method for estimating the ensemble average of all operators of an arbitrary quantum system from a set of measurements of a quorum of observables. A procedure for deconvolving instrumental noise is available, which makes the tomographic method feasible in different physical contexts. Recent developments are presented and new experiments are proposed, which are now made possible by the tomographic technique.
Quantum tomography is the process of reconstructing the ensemble average of an arbitrary operator (observable or not, including the density matrix), which may not be directly accessible by feasible detection schemes, starting from the measurement of a complete set of observables i.e. a quorum . The measurement of a quorum thus represents a complete characterization of the quantum state. The operator expression in terms of a quorum corresponds to an expansion on an irreducible set of operators in the Liouville space. We give two general characterizations of these sets, and show that all the known quantum tomographies can be described in this framework. New operatorial resolutions are also given that may be used in novel reconstruction schemes.
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 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.
When the dynamics of a quantum system of interest is known, an informationally complete set of observables is not needed for state reconstruction via tomographic techniques: letting the system evolve before performing the measurement allows one to effectively extend the available ways to probe the system. This idea leads to dynamical quantum tomography, whose feasibility we characterize for general quantum dynamics using Krylov-based methods. Specializing in Markovian ones, we also provide deterministic tests and randomized ones to effectively assess parametric dynamics. The limits of the methods are explored by comparing unitary and open dynamics when a single observable is available, and the set of observables whose expectation can be reconstructed from the available ones characterized. The framework is illustrated with applications to a spin chain (with or without dissipation) and an electron-nuclear system.
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.