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/0205026 · Physical Review Letters

Optimal Scheme for Estimating a Pure Qubit State via Local Measurements

E. Bagán, M. Baig, R. Muñoz-Tapia

Published Dec 19, 2002• 41 citations
arXivHTMLPDF

Abstract

We present the optimal scheme for estimating a pure qubit state by means of local measurements on N identical copies. We give explicit examples for low N. For large N, we show that the fidelity saturates the collective measurement bound up to order 1/N. When the signal state lays on a meridian of the Bloch sphere, we show that this can be achieved without classical communication.

DOI
10.1103/physrevlett.89.277904

Related papers

Optimal reconstruction of a pure qubit state with local measurements

E. Bagán, M. Baig, Alex Monràs +1 more

We analyse the reconstruction of an unknown pure qubit state. We derive the optimal guess that can be inferred from any set of measurements on N identical copies of the system with the fidelity as a figure of merit. We study in detail the estimation process with individual von Neumann measurements and demonstrate that they are very competitive as compared to (complicated) collective measurements. We compute the expressions of the fidelity for large $N$ and show that individual measurement schemes can perform optimally in the asymptotic regime.

arXiv:quant-ph/0312167•Dec 19, 2003•0 citations

Comprehensive analysis of quantum pure-state estimation for two-level systems

E. Bagán, Alex Monràs, R. Muñoz-Tapia

Given $N$ identical copies of the state of a quantum two-level system, we analyze its optimal estimation. We consider two situations: general pure states and (pure) states restricted to lie on the equator of the Bloch sphere. We perform a complete and comprehensive analysis of the optimal schemes based on local measurements, and give results (optimal measurements, maximum fidelity, etc.) for arbitrary $N$, not necessarily large, within the Bayesian framework. We also make a comparative analysis of the asymptotic limit of these results with those derived from a (pointwise) Cram\'er-Rao type of approach. We give explicit schemes based on local measurements and no classical communication that saturate the fidelity bounds of the most general collective schemes.

arXiv:quant-ph/0412027•Physical Review A•Jun 15, 2005•45 citations

Collective versus local measurements in a qubit mixed-state estimation

E. Bagán, M. Baig, R. Muñoz-Tapia +1 more

We discuss the problem of estimating a qubit mixed state. We give the optimal estimation that can be inferred from any given set of measurements. For collective measurements and for a large number N of copies, we show that the error in the estimation varies as $1/N.$ For local measurements, we focus on the simpler case of states lying on the equatorial plane of the Bloch sphere. We show that the error using plain tomography varies as ${1/N}^{1/4},$ while our approach leads to an error proportional to ${1/N}^{3/4}.$

arXiv:quant-ph/0307199•Physical Review A•Jan 23, 2004•45 citations

Optimal full estimation of qubit mixed states

E. Bagán, Manuel A. Ballester, Richard David Gill +2 more

We obtain the optimal scheme for estimating unknown qubit mixed states when an arbitrary number $N$ of identically prepared copies is available. We discuss the case of states in the whole Bloch sphere as well as the restricted situation where these states are known to lie on the equatorial plane. For the former case we obtain that the optimal measurement does not depend on the prior probability distribution provided it is isotropic. Although the equatorial-plane case does not have this property for arbitrary $N$, we give a prior-independent scheme which becomes optimal in the asymptotic limit of large $N$. We compute the maximum mean fidelity in this asymptotic regime for the two cases. We show that within the pointwise estimation approach these limits can be obtained in a rather easy and rapid way. This derivation is based on heuristic arguments that are made rigorous by using van Trees inequalities. The interrelation between the estimation of the purity and the direction of the state is also discussed. In the general case we show that they correspond to independent estimations whereas for the equatorial-plane states this is only true asymptotically.

arXiv:quant-ph/0510158•Physical Review A•Mar 2, 2006•89 citations

Optimal estimation of two-qubit pure-state entanglement

Antonio Acín, R. Tarrach, Guifré Vidal

We present optimal measuring strategies for an estimation of the entanglement of unknown two-qubit pure states and of the degree of mixing of unknown single-qubit mixed states, of which N identical copies are available. The most general measuring strategies are considered in both situations, to conclude in the first case that a local, although collective, measurement suffices to estimate entanglement, a nonlocal property, optimally.

arXiv:quant-ph/9911008•Physical Review A•May 16, 2000•32 citations

Optimal partial estimation of quantum states from several copies

Ladislav Mišta, Jaromı́r Fiurášek

We derive an analytical formula for the optimal trade-off between the mean estimation fidelity and the mean fidelity of the qubit state after a partial measurement on $N$ identically prepared qubits. We also conjecture an analytical expression for the optimal fidelity trade-off in case of a partial measurement on $N$ identical copies of a $d$-level system.

arXiv:quant-ph/0606112•Physical Review A•Aug 22, 2006•11 citations
We present the optimal scheme for estimating a pure qubit state by means of local measurements on N identical copies. We give explicit examples for low N. For large N, we show that the fidelity saturates the collective measurement bound up to order 1/N. When the signal state lays on a meridian of the Bloch sphere, we show that this can be achieved without classical communication.
We analyse the reconstruction of an unknown pure qubit state. We derive the optimal guess that can be inferred from any set of measurements on N identical copies of the system with the fidelity as a figure of merit. We study in detail the estimation process with individual von Neumann measurements and demonstrate that they are very competitive as compared to (complicated) collective measurements. We compute the expressions of the fidelity for large NNN and show that individual measurement schemes can perform optimally in the asymptotic regime.
Given NNN identical copies of the state of a quantum two-level system, we analyze its optimal estimation. We consider two situations: general pure states and (pure) states restricted to lie on the equator of the Bloch sphere. We perform a complete and comprehensive analysis of the optimal schemes based on local measurements, and give results (optimal measurements, maximum fidelity, etc.) for arbitrary NNN, not necessarily large, within the Bayesian framework. We also make a comparative analysis of the asymptotic limit of these results with those derived from a (pointwise) Cram\'er-Rao type of approach. We give explicit schemes based on local measurements and no classical communication that saturate the fidelity bounds of the most general collective schemes.
We discuss the problem of estimating a qubit mixed state. We give the optimal estimation that can be inferred from any given set of measurements. For collective measurements and for a large number N of copies, we show that the error in the estimation varies as 1/N.1/N.1/N. For local measurements, we focus on the simpler case of states lying on the equatorial plane of the Bloch sphere. We show that the error using plain tomography varies as 1/N1/4,{1/N}^{1/4},1/N1/4, while our approach leads to an error proportional to 1/N3/4.{1/N}^{3/4}.1/N3/4.
We obtain the optimal scheme for estimating unknown qubit mixed states when an arbitrary number NNN of identically prepared copies is available. We discuss the case of states in the whole Bloch sphere as well as the restricted situation where these states are known to lie on the equatorial plane. For the former case we obtain that the optimal measurement does not depend on the prior probability distribution provided it is isotropic. Although the equatorial-plane case does not have this property for arbitrary NNN, we give a prior-independent scheme which becomes optimal in the asymptotic limit of large NNN. We compute the maximum mean fidelity in this asymptotic regime for the two cases. We show that within the pointwise estimation approach these limits can be obtained in a rather easy and rapid way. This derivation is based on heuristic arguments that are made rigorous by using van Trees inequalities. The interrelation between the estimation of the purity and the direction of the state is also discussed. In the general case we show that they correspond to independent estimations whereas for the equatorial-plane states this is only true asymptotically.
We derive an analytical formula for the optimal trade-off between the mean estimation fidelity and the mean fidelity of the qubit state after a partial measurement on NNN identically prepared qubits. We also conjecture an analytical expression for the optimal fidelity trade-off in case of a partial measurement on NNN identical copies of a ddd-level system.