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

Fortschritte der Physik

Efficiency of Maximum-likelihood Reconstruction of Quantum States

Z. Hradil, Jaroslav Řeháček

Published Oct 1, 2001• 10 citations
Publisher

Abstract

We investigate the relative merits of techniques for recovery of density matrices of two qubits from experimental data. Our results are applied to measure the states of photons produced in down-conversion experiments.
DOI
10.1002/1521-3978(200110)49:10/11<1083::aid-prop1083>3.0.co;2-k

Related papers

On the Measurement of Qubits

Daniel F. V. James, Paul G. Kwiat, William J. Munro +1 more

We describe in detail the theory underpinning the measurement of density matrices of a pair of quantum two-level systems (``qubits''). Our particular emphasis is on qubits realized by the two polarization degrees of freedom of a pair of entangled photons generated in a down-conversion experiment; however the discussion applies in general, regardless of the actual physical realization. Two techniques are discussed, namely a tomographic reconstruction (in which the density matrix is linearly related to a set of measured quantities) and a maximum likelihood technique which requires numerical optimization (but has the advantage of producing density matrices which are always non-negative definite). In addition a detailed error analysis is presented, allowing errors in quantities derived from the density matrix, such as the entropy or entanglement of formation, to be estimated. Examples based on down-conversion experiments are used to illustrate our results.

WORLD SCIENTIFIC eBooks•Feb 1, 2005•32 citations

Measurement of qubits

Daniel F. V. James, Paul G. Kwiat, William J. Munro +1 more

We describe in detail the theory underpinning the measurement of density matrices of a pair of quantum two-level systems (``qubits''). Our particular emphasis is on qubits realized by the two polarization degrees of freedom of a pair of entangled photons generated in a down-conversion experiment; however, the discussion applies in general, regardless of the actual physical realization. Two techniques are discussed, namely, a tomographic reconstruction (in which the density matrix is linearly related to a set of measured quantities) and a maximum likelihood technique which requires numerical optimization (but has the advantage of producing density matrices that are always non-negative definite). In addition, a detailed error analysis is presented, allowing errors in quantities derived from the density matrix, such as the entropy or entanglement of formation, to be estimated. Examples based on down-conversion experiments are used to illustrate our results.

arXiv:quant-ph/0103121•Physical Review A•Oct 16, 2001•2,176 citations

On/off Detection Method for Reconstructing the Statistics of Quantum Optical States: An Overview

G. Brida, Marco Genovese, Marco Gramegna +3 more

We give a brief overview of the demonstration of a photon statistics reconstruction method based on Maximum Likelihood estimation and on/off detection. This method has been successfully applied to a large number of cases and, recently, extended to a scheme for a full reconstruction of the density matrix. Experimental results concerning the heralded single-photon state and the seeded parametric down-conversion are presented in some details.

International Conference on Quantum, Nano and Micro Technologies (ICQNM)•Feb 1, 2009•0 citations

Quantum-state estimation

Z. Hradil

An algorithm for quantum-state estimation based on the maximum-likelihood estimation is proposed. Existing techniques for state reconstruction based on the inversion of measured data are shown to be overestimated since they do not guarantee the positive definiteness of the reconstructed density matrix.

arXiv:quant-ph/9609012•Physical Review A•Mar 1, 1997•542 citations

Maximum-likelihood method in quantum estimation

M. G. A. Paris

The maximum-likelihood method for quantum estimation is reviewed and applied to the reconstruction of density matrix of spin and radiation as well as to the determination of several parameters of interest in quantum optics.

arXiv:quant-ph/0101071•AIP conference proceedings•Jan 1, 2001•12 citations

Maximum-likelihood method in quantum estimation

Paris, M G A, D'Ariano, G M, Sacchi, M F

The maximum-likelihood method for quantum estimation is reviewed and applied to the reconstruction of density matrix of spin and radiation as well as to the determination of several parameters of interest in quantum optics.

•Jan 1, 2001•16 citations
We describe in detail the theory underpinning the measurement of density matrices of a pair of quantum two-level systems (``qubits''). Our particular emphasis is on qubits realized by the two polarization degrees of freedom of a pair of entangled photons generated in a down-conversion experiment; however the discussion applies in general, regardless of the actual physical realization. Two techniques are discussed, namely a tomographic reconstruction (in which the density matrix is linearly related to a set of measured quantities) and a maximum likelihood technique which requires numerical optimization (but has the advantage of producing density matrices which are always non-negative definite). In addition a detailed error analysis is presented, allowing errors in quantities derived from the density matrix, such as the entropy or entanglement of formation, to be estimated. Examples based on down-conversion experiments are used to illustrate our results.
We describe in detail the theory underpinning the measurement of density matrices of a pair of quantum two-level systems (``qubits''). Our particular emphasis is on qubits realized by the two polarization degrees of freedom of a pair of entangled photons generated in a down-conversion experiment; however, the discussion applies in general, regardless of the actual physical realization. Two techniques are discussed, namely, a tomographic reconstruction (in which the density matrix is linearly related to a set of measured quantities) and a maximum likelihood technique which requires numerical optimization (but has the advantage of producing density matrices that are always non-negative definite). In addition, a detailed error analysis is presented, allowing errors in quantities derived from the density matrix, such as the entropy or entanglement of formation, to be estimated. Examples based on down-conversion experiments are used to illustrate our results.