Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

arXiv:quant-ph/0501102 · Physical Review A

Process reconstruction: From unphysical to physical maps via maximum likelihood

Mário Ziman, Martin Plesch, Vladimír Bužek, Peter Štelmachovič

Published Aug 16, 2005• 20 citations
arXivHTMLPDF

Abstract

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.

DOI
10.1103/physreva.72.022106

Related papers

Estimation of Potentially Unphysical Maps

Martin Plesch, Mário Ziman, Vladimír Bužek +1 more

When standard methods of process (black-box) estimation are applied straightforwardly then it may happen that some sets of experimental data result in unphysical estimations of the corresponding channels (maps) describing the process. To prevent this problem, one can use the method of maximum likelihood (MML), which provides an efficient scheme for reconstruction of quantum channels. This scheme always results in estimations of channels that are fully physical, e.g. the corresponding maps are linear, positive and completely positive. To show this property, we use the MML for a derivation of physical approximations of truly unphysical operations. In particular, we analyze physical approximations of the universal-NOT gate, the quantum nonlinear polarization rotation and the map ρ → ρ2. Given the result of MML, we examine retrospectively the quality of the experiment. Depending on the resulting value of the MML functional we can determine (physical) consistency of the input data.

Open Systems & Information Dynamics•Aug 31, 2006•1 citation

Optimal Approximation of Nonphysical Maps via Maximum Likelihood Estimation

Vladimír Bužek, Mário Ziman, Martin Plesch

Any quantum dynamics, i.e., the process that is described by a completely positive (CP) map of a quantum-mechanical system, can be probed in two different ways. Either we use a single entangled state of a bipartite system, or we use a collection of linearly independent single-particle test states (forming a basis of the vector space of all Hermitian operators). Given the fragility of entangled states, in this chapter we will focus our attention on the process reconstruction using only single-particle states.

SPIE eBooks•Mar 16, 2010•0 citations

Maximum-likelihood reconstruction of completely positive maps

Massimiliano Federico Sacchi

We present a method for the determination of the completely positive (CP) map describing a physical device based on random preparation of the input states, random measurements at the output, and the maximum-likelihood principle. In the numerical implementation the constraint of completely positivity can be imposed by exploiting the isomorphism between linear transformations from Hilbert spaces $\mathcal{H}$ to $\mathcal{K}$ and linear operators in $\mathcal{K}\ensuremath{\bigotimes}\mathcal{H}.$ The effectiveness of the method is shown on the basis of some examples of reconstruction of CP maps related to quantum communication channels for qubits.

arXiv:quant-ph/0009104•Physical Review A•Apr 13, 2001•64 citations

Maximum-likelihood estimation of Mueller matrices

Andrea Aiello, Graciana Puentes, D. Voigt +1 more

We show that the method of maximum-likelihood estimation, recently introduced in the context of quantum process tomography, can be applied to the determination of Mueller matrices characterizing the polarization properties of classical optical systems. Contrary to linear reconstruction algorithms, the proposed method yields physically acceptable Mueller matrices even in the presence of uncontrolled experimental errors. We illustrate our method with the case of an unphysical measured Mueller matrix taken from the literature.

arXiv:physics/0508190•Optics Letters•Mar 15, 2006•53 citations

Generalized measurement of process and states

Miroslav Ježek

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

Proceedings of SPIE, the International Society for Optical Engineering/Proceedings of SPIE•Jul 23, 2002•0 citations

Optimal Approximation of Nonphysical Maps via MaximumLikelihood Estimation

Vladimír Bužek, Mário Ziman, Martin Plesch

We will use the method of maximum likelihood to perform an estimation of an unknown quantum channel. In order to demonastrate the power of this approach, we will also apply it to obtain an approximation of nonlinear quantum mechanical maps.

•Jan 1, 2008•0 citations
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.
When standard methods of process (black-box) estimation are applied straightforwardly then it may happen that some sets of experimental data result in unphysical estimations of the corresponding channels (maps) describing the process. To prevent this problem, one can use the method of maximum likelihood (MML), which provides an efficient scheme for reconstruction of quantum channels. This scheme always results in estimations of channels that are fully physical, e.g. the corresponding maps are linear, positive and completely positive. To show this property, we use the MML for a derivation of physical approximations of truly unphysical operations. In particular, we analyze physical approximations of the universal-NOT gate, the quantum nonlinear polarization rotation and the map ρ → ρ2. Given the result of MML, we examine retrospectively the quality of the experiment. Depending on the resulting value of the MML functional we can determine (physical) consistency of the input data.
Any quantum dynamics, i.e., the process that is described by a completely positive (CP) map of a quantum-mechanical system, can be probed in two different ways. Either we use a single entangled state of a bipartite system, or we use a collection of linearly independent single-particle test states (forming a basis of the vector space of all Hermitian operators). Given the fragility of entangled states, in this chapter we will focus our attention on the process reconstruction using only single-particle states.
We present a method for the determination of the completely positive (CP) map describing a physical device based on random preparation of the input states, random measurements at the output, and the maximum-likelihood principle. In the numerical implementation the constraint of completely positivity can be imposed by exploiting the isomorphism between linear transformations from Hilbert spaces H\mathcal{H}H to K\mathcal{K}K and linear operators in $\mathcal{K}\ensuremath{\bigotimes}\mathcal{H}.$ The effectiveness of the method is shown on the basis of some examples of reconstruction of CP maps related to quantum communication channels for qubits.
We show that the method of maximum-likelihood estimation, recently introduced in the context of quantum process tomography, can be applied to the determination of Mueller matrices characterizing the polarization properties of classical optical systems. Contrary to linear reconstruction algorithms, the proposed method yields physically acceptable Mueller matrices even in the presence of uncontrolled experimental errors. We illustrate our method with the case of an unphysical measured Mueller matrix taken from the literature.
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