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Proceedings of SPIE, the International Society for Optical Engineering/Proceedings of SPIE

Generalized measurement of process and states

Miroslav Ježek

Published Jul 23, 2002• 0 citations
Publisher

Abstract

ABSTRACT The simultaneous inseparable reconstruction of quantum process and states is presented. The method is basedon constrained extremization of the likelihood functional. This statistical approach is ingenuous in quantummechanical frame and ensures physically correct results. Adopted scheme predicts the input density operatorsand the operator of quantum process all at once with help of noisy measured data only. Attention is paid to thecomparison of posed method with sequential measurement of the states and the process. Method is illustratedby numerical simulations.Keywords: quantum measurement, quantum state reconstruction, quantum process reconstruction, maximumlikelihood, completely positive maps 1. INTRODUCTION Measuring of quantum state of physical systems has attracted increased attention recently.15 Topical exper- iments often include extracting of quantum state from measured data as it's native part and many methods were developed for this purpose during recent years. Nevertheless most of them yield reconstructed objectswhich are not bona fide physical states. Only constrained statistical methods can simply integrate various ex-perimental data and guarantee the non-negativity of estimated density matrix. Physically the most importantis maximum likelihood principle61' representing genuine quantum measurement. Maximum likelihood methodensures existence of path in state space along which the likelihood of state monotonically increases.12Discussions of experimental results and understanding quantum nature behind them require however recog-nition of quantum processes, not only input or output quantum states. Examination of quantum communicationchannels and evaluation of performance of quantum gates are examples of practical applicability of quantumprocess reconstruction.1320 And again, maximum likelihood appears as the efficient and exceptional

DOI
10.1117/12.475887

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ABSTRACT The simultaneous inseparable reconstruction of quantum process and states is presented. The method is basedon constrained extremization of the likelihood functional. This statistical approach is ingenuous in quantummechanical frame and ensures physically correct results. Adopted scheme predicts the input density operatorsand the operator of quantum process all at once with help of noisy measured data only. Attention is paid to thecomparison of posed method with sequential measurement of the states and the process. Method is illustratedby numerical simulations.Keywords: quantum measurement, quantum state reconstruction, quantum process reconstruction, maximumlikelihood, completely positive maps 1. INTRODUCTION Measuring of quantum state of physical systems has attracted increased attention recently.15 Topical exper- iments often include extracting of quantum state from measured data as it's native part and many methods were developed for this purpose during recent years. Nevertheless most of them yield reconstructed objectswhich are not bona fide physical states. Only constrained statistical methods can simply integrate various ex-perimental data and guarantee the non-negativity of estimated density matrix. Physically the most importantis maximum likelihood principle61' representing genuine quantum measurement. Maximum likelihood methodensures existence of path in state space along which the likelihood of state monotonically increases.12Discussions of experimental results and understanding quantum nature behind them require however recog-nition of quantum processes, not only input or output quantum states. Examination of quantum communicationchannels and evaluation of performance of quantum gates are examples of practical applicability of quantumprocess reconstruction.1320 And again, maximum likelihood appears as the efficient and exceptional
Summary form only given. Experimental techniques for measurement of the quantum state of light have been the subject of intensive investigation for some time. Tomographic techniques have been applied to experiments such as the homodyne measurement of the Wigner function of a single mode of light and of the density matrix of the polarization degrees of freedom of a pair of entangled photons. In this technique, the density matrix (or Wigner function) which characterizes the quantum state of the system being measured is found from a linear transformation of experimental data. There are a number of drawbacks to the method, principally in that the recovered state might not, because of experimental noise, correspond to a physical state. For example, density matrices for any quantum state must be Hermitian, positive semi-definite matrices with unit trace. The tomographically measured matrices often fail to be positive semi-definite. To avoid this problem, the maximum likelihood approach to the estimation of quantum states has been developed. We evaluate two different approaches to the maximum likelihood technique, namely defining the likelihood function f as a weighted sum of squared variances from measured data, and defining it in terms of information content. These are compared with results of standard tomographical schemes.
We show that the method of maximum likelihood (MML) provides us with an efficient scheme for the reconstruction of quantum channels from incomplete measurement data. By construction this scheme always results in estimations of channels that are completely positive. Using this property we use the MML for a derivation of physical approximations of unphysical operations. In particular, we analyze the optimal approximation of the universal NOT gate as well as the physical approximation of a quantum nonlinear polarization rotation.
Abstract. The theory of quantum state reconstruction is illustrated here on sev-eral examples taken from modern experimental praxis. Maximum-likelihood estima-tion is applied to experiments on physical systems of increasing complexity, starting with a simple one-dimensional problem of quantum phase estimation, continuing with the absorption and phase neutron tomographies, further discussing quantum tomography of higher-dimensional discrete quantum systems, and closing with the homodyne tomography of an infinite dimensional system- a mode of light. All these experiments nicely demonstrate the utility of present state-of-art techniques for ma-nipulating states of a neutron and internal as well as external states of a photon. Over the past hundred years, quantum theory proved to be an extremely useful tool for describing and understanding nature on a microscopic level. Its usual applications consist in predicting the outcomes of future experimental results given a quantum state describing the physical system.
Coherent-state quantum process tomography (csQPT) is a method of completely characterizing a quantum-optical "black box" by probing it with coherent states and performing homodyne measurements on the output [M. Lobino et al, Science 322, 563 (2008)]. We present a technique for csQPT that is fully based on statistical inference, specifically, quantum expectation-maximization. The method relies on the Jamiolkowski isomorphism and iteratively reconstructs the process tensor in the Fock basis directly from the experimental data. This approach permits incorporation of a priori constraints into the reconstruction procedure, thereby guaranteeing that the resulting process tensor is physically consistent. Furthermore, our method is easier to implement and requires a narrower range of coherent states than its predecessors. We test its feasibility using simulations on several experimentally relevant processes.
Tomography of a quantum state is usually based on a positive-operator-valued measure (POVM) and on their experimental statistics. Among the available reconstructions, the maximum-likelihood (MaxLike) technique is an efficient one. We propose an extension of this technique when the measurement process cannot be simply described by an instantaneous POVM. Instead, the tomography relies on a set of quantum trajectories and their measurement records. This model includes the fact that, in practice, each measurement could be corrupted by imperfections and decoherence, and could also be associated with the record of continuous-time signals over a finite amount of time. The goal is then to retrieve the quantum state that was present at the start of this measurement process. The proposed extension relies on an explicit expression of the likelihood function via the effective matrices appearing in quantum smoothing and solutions of the adjoint quantum filter. It allows us to retrieve the initial quantum state as in standard MaxLike tomography, but where the traditional POVM operators are replaced by more general ones that depend on the measurement record of each trajectory. It also provides, aside from the MaxLike estimate of the quantum state, confidence intervals for any observable. Such confidence intervals are derived, as the MaxLike estimate, from an asymptotic expansion of multidimensional Laplace integrals appearing in Bayesian mean estimation. A validation is performed on two sets of experimental data: photon(s) trapped in a microwave cavity subject to quantum nondemolition measurements relying on Rydberg atoms, and heterodyne fluorescence measurements of a superconducting qubit.