arXiv:quant-ph/0611244 · Physical Review A

Diluted maximum-likelihood algorithm for quantum tomography

Jaroslav Řeháček, Z. Hradil, Emanuel H. Knill, Alexander I. Lvovsky

Published Apr 18, 2007• 187 citations

Abstract

We propose a refined iterative likelihood-maximization algorithm for reconstructing a quantum state from a set of tomographic measurements. The algorithm is characterized by a very high convergence rate and features a simple adaptive procedure that ensures likelihood increase in every iteration and convergence to the maximum-likelihood state. We apply the algorithm to homodyne tomography of optical states and quantum tomography of entangled spin states of trapped ions and investigate its convergence properties.

DOI: 10.1103/physreva.75.042108

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arXiv:quant-ph/0611244 · Physical Review A

Diluted maximum-likelihood algorithm for quantum tomography

Jaroslav Řeháček, Z. Hradil, Emanuel H. Knill, Alexander I. Lvovsky

Published Apr 18, 2007• 187 citations

arXiv HTML PDF

Abstract

DOI: 10.1103/physreva.75.042108

Related papers

Maximum-Likelihood Quantum State Tomography by Cover's Method with Non-Asymptotic Analysis

Chien-Ming Lin, Hao–Chung Cheng, Yen-Huan Li

We propose an iterative algorithm that computes the maximum-likelihood estimate in quantum state tomography. The optimization error of the algorithm converges to zero at an $O ( ( 1 / k ) \log D )$ rate, where $k$ denotes the number of iterations and $D$ denotes the dimension of the quantum state. The per-iteration computational complexity of the algorithm is $O ( D ^ 3 + N D ^2 )$ , where $N$ denotes the number of measurement outcomes. The algorithm can be considered as a parameter-free correction of the $R ρR$ method [A. I. Lvovsky. Iterative maximum-likelihood reconstruction in quantum homodyne tomography. \textit{J. Opt. B: Quantum Semiclass. Opt.} 2004] [G. Molina-Terriza et al. Triggered qutrits for quantum communication protocols. \textit{Phys. Rev. Lett.} 2004.].We propose an iterative algorithm that computes the maximum-likelihood estimate in quantum state tomography. The optimization error of the algorithm converges to zero at an $O ( ( 1 / k ) \log D )$ rate, where $k$ denotes the number of iterations and $D$ denotes the dimension of the quantum state. The per-iteration computational complexity of the algorithm is $O ( D ^ 3 + N D ^2 )$, where $N$ denotes the number of measurement outcomes. The algorithm can be considered as a parameter-free correction of the $R ρR$ method [A. I. Lvovsky. Iterative maximum-likelihood reconstruction in quantum homodyne tomography. \textit{J. Opt. B: Quantum Semiclass. Opt.} 2004] [G. Molina-Terriza et al. Triggered qutrits for quantum communication protocols. \textit{Phys. Rev. Lett.} 2004.].

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In this paper we address convergence issues of the Diluted $R \rho R$ algorithm \cite{rehacek2007}, used to obtain the maximum likelihood estimate for the density matrix in quantum state tomography. We give a new interpretation to the diluted $R \rho R$ iterations that allows us to prove the global convergence under weaker assumptions. Thus, we propose a new algorithm which is globally convergent and suitable for practical implementation.In this paper we address convergence issues of the Diluted $R \rho R$ algorithm \cite{rehacek2007}, used to obtain the maximum likelihood estimate for the density matrix in quantum state tomography. We give a new interpretation to the diluted $R \rho R$ iterations that allows us to prove the global convergence under weaker assumptions. Thus, we propose a new algorithm which is globally convergent and suitable for practical implementation.

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Konrad Banaszek

arXiv:quant-ph/9901064Physical Review AJun 1, 199918 citations

We propose an iterative algorithm that computes the maximum-likelihood estimate in quantum state tomography. The optimization error of the algorithm converges to zero at an

O ( ( 1 / k ) \log D )

rate, where

k

denotes the number of iterations and

D

denotes the dimension of the quantum state. The per-iteration computational complexity of the algorithm is

O ( D ^ 3 + N D ^2 )

, where

N

denotes the number of measurement outcomes. The algorithm can be considered as a parameter-free correction of the

R ρR

method [A. I. Lvovsky. Iterative maximum-likelihood reconstruction in quantum homodyne tomography. \textit{J. Opt. B: Quantum Semiclass. Opt.} 2004] [G. Molina-Terriza et al. Triggered qutrits for quantum communication protocols. \textit{Phys. Rev. Lett.} 2004.].

In this paper we address convergence issues of the Diluted

R \rho R

algorithm \cite{rehacek2007}, used to obtain the maximum likelihood estimate for the density matrix in quantum state tomography. We give a new interpretation to the diluted

R \rho R

iterations that allows us to prove the global convergence under weaker assumptions. Thus, we propose a new algorithm which is globally convergent and suitable for practical implementation.