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

Communications in Theoretical Physics

Optimal State Estimation of Pure Qubits on Circles

Zhang Li-hua, Song Wei, Zhuo‐Liang Cao

Published Jun 1, 2008• 0 citations
Publisher

Abstract

We consider the problem of state estimation of qubits chosen from circles. It is shown that any qubit encoded in pairs chosen from a fixed circle parallel to the x-y equator with different phases contains the same information. We also investigate the problem of state estimation of qubits from three circles. The optimal estimation fidelity is derived.

DOI
10.1088/0253-6102/49/6/28

Related papers

Estimation of pure qubits on circles

Samuel L. Braunstein, Sibasish Ghosh A, Simone Severini B

Gisin and Popescu [PRL, 83, 432 (1999)] have shown that more information about their direction can be obtained from a pair of anti-parallel spins compared to a pair of parallel spins, where the first member of the pair (which we call the pointer member) can point equally along any direction in the Bloch sphere. They argued that this was due to the difference in dimensionality spanned by these two alphabets of states. Here we consider similar alphabets, but with the first spin restricted to a fixed small circle of the Bloch sphere. In this case, the dimensionality spanned by the anti-parallel versus parallel alphabet is now equal. However, the anti-parallel alphabet is found to still contain more information in general. We generalize this to having N parallel spins and M anti-parallel spins. When the pointer member is restricted to a small circle these alphabets again span spaces of equal dimension, yet in general, more directional information can be found for sets with smaller |N-M| for any fixed total number of spins. We find that the optimal POVMs for extracting directional information in these cases can always be expressed in terms of the Fourier basis. Our results show that dimensionality alone cannot explain the greater information content in anti-parallel combinations of spins compared to parallel combinations. In addition, we describe an LOCC protocol which extract optimal directional information when the pointer member is restricted to a small circle and a pair of parallel spins are supplied.

arXiv:quant-ph/0412101•Journal of Physics A Mathematical and Theoretical•Feb 6, 2007•5 citations

Fidelity-optimized quantum state estimation

Amir Kalev, Itay Hen

We describe an optimized, self-correcting procedure for the Bayesian inference of pure quantum states. By analyzing the history of measurement outcomes at each step, the procedure returns the most likely pure state, as well as the optimal basis for the measurement that is to follow. The latter is chosen to maximize, on average, the fidelity of the most likely state after the measurement. We also consider a practical variant of this protocol, where the available measurement bases are restricted to certain limited sets of bases. We demonstrate the success of our method by considering in detail the single-qubit and two-qubit cases, and comparing the performance of our method against other existing methods.

arXiv:1409.1952•New Journal of Physics•Sep 10, 2015•15 citations

State learning from pairs of states

Pranjal Agarwal, Nada Ali, Camilla Polvara +3 more

Suppose you receive a sequence of qubits where each qubit is guaranteed to be in one of two pure states, but you do not know what those states are. Your task is to determine the states. This can be viewed as a kind of quantum state learning -- or quantum state estimation. A problem is that, without more information, all that can be determined is the density matrix of the sequence and, in general, density matrices can be decomposed into pure states in many different ways. To solve the problem, additional information, either classical or quantum, is required. We show that if an additional copy of each qubit is supplied -- that is, one receives pairs of qubits, both in the same state, rather than single qubits -- the task can be accomplished. This is possible because the mixed two-qubit state has a unique decomposition into pure product states. For illustration, we simulate numerically the symmetric, informationally complete measurement of a sequence of qubit pairs and show that the unknown states and their respective probabilities of occurrence can be inferred from the data with high accuracy. Finally, we propose an experiment that employs a product measurement and can be realized with existing technology, and we demonstrate how the data tell us the states and their probabilities. We find that it is enough to detect a few thousand qubit pairs.

arXiv:2409.11120•Physical Review A•May 29, 2025•0 citations

Random qubit-states and how best to measure them

Stephen Mark Barnett

We consider the problem of measuring a single qubit, known to have been prepared in either a randomly selected pure state or a randomly selected real pure state. We seek the measurements that provide either the best estimate of the state prepared or maximise the accessible information. Surprisingly, any sensible measurement turns out to be optimal. We discuss the application of these ideas to multiple qubits and higher-dimensional systems.

Journal of Modern Optics•Jul 29, 2009•1 citation

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

Amount of information obtained by a quantum measurement

Serge Massar, Sandu Popescu

In this paper we address the problem of how much can be learned about an unknown quantum state by a measurement. To this end we consider optimal measurements for the state estimation problem, that is measurements that maximize the expectation of a fidelity function. We then enlarge the class of optimal measurements to measurements that act collectively on blocks of input states, and in addition we only require that the fidelity of the measurement be arbitrarily close to the optimal fidelity. We then consider the Shannon information of the outputs of optimal measurements, which is the amount of data produced by the measurements. We show that in the enlarged class of optimal measurements described above one can always construct an optimal measurement so that the Shannon information of its outputs equals the von Neumann entropy of the unknown states. Since this result is valid for all choices of fidelity functions and all distributions of input states, it provides a model independent answer to the question of how much can be learned about a quantum state by a measurement. Namely, this result shows that a measurement can extract at most one meaningful bit from every qubit carried by the unknown state.

arXiv:quant-ph/9907066•Physical Review A•May 12, 2000•30 citations
We consider the problem of state estimation of qubits chosen from circles. It is shown that any qubit encoded in pairs chosen from a fixed circle parallel to the x-y equator with different phases contains the same information. We also investigate the problem of state estimation of qubits from three circles. The optimal estimation fidelity is derived.
Gisin and Popescu [PRL, 83, 432 (1999)] have shown that more information about their direction can be obtained from a pair of anti-parallel spins compared to a pair of parallel spins, where the first member of the pair (which we call the pointer member) can point equally along any direction in the Bloch sphere. They argued that this was due to the difference in dimensionality spanned by these two alphabets of states. Here we consider similar alphabets, but with the first spin restricted to a fixed small circle of the Bloch sphere. In this case, the dimensionality spanned by the anti-parallel versus parallel alphabet is now equal. However, the anti-parallel alphabet is found to still contain more information in general. We generalize this to having N parallel spins and M anti-parallel spins. When the pointer member is restricted to a small circle these alphabets again span spaces of equal dimension, yet in general, more directional information can be found for sets with smaller |N-M| for any fixed total number of spins. We find that the optimal POVMs for extracting directional information in these cases can always be expressed in terms of the Fourier basis. Our results show that dimensionality alone cannot explain the greater information content in anti-parallel combinations of spins compared to parallel combinations. In addition, we describe an LOCC protocol which extract optimal directional information when the pointer member is restricted to a small circle and a pair of parallel spins are supplied.
We describe an optimized, self-correcting procedure for the Bayesian inference of pure quantum states. By analyzing the history of measurement outcomes at each step, the procedure returns the most likely pure state, as well as the optimal basis for the measurement that is to follow. The latter is chosen to maximize, on average, the fidelity of the most likely state after the measurement. We also consider a practical variant of this protocol, where the available measurement bases are restricted to certain limited sets of bases. We demonstrate the success of our method by considering in detail the single-qubit and two-qubit cases, and comparing the performance of our method against other existing methods.
Suppose you receive a sequence of qubits where each qubit is guaranteed to be in one of two pure states, but you do not know what those states are. Your task is to determine the states. This can be viewed as a kind of quantum state learning -- or quantum state estimation. A problem is that, without more information, all that can be determined is the density matrix of the sequence and, in general, density matrices can be decomposed into pure states in many different ways. To solve the problem, additional information, either classical or quantum, is required. We show that if an additional copy of each qubit is supplied -- that is, one receives pairs of qubits, both in the same state, rather than single qubits -- the task can be accomplished. This is possible because the mixed two-qubit state has a unique decomposition into pure product states. For illustration, we simulate numerically the symmetric, informationally complete measurement of a sequence of qubit pairs and show that the unknown states and their respective probabilities of occurrence can be inferred from the data with high accuracy. Finally, we propose an experiment that employs a product measurement and can be realized with existing technology, and we demonstrate how the data tell us the states and their probabilities. We find that it is enough to detect a few thousand qubit pairs.
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.
In this paper we address the problem of how much can be learned about an unknown quantum state by a measurement. To this end we consider optimal measurements for the state estimation problem, that is measurements that maximize the expectation of a fidelity function. We then enlarge the class of optimal measurements to measurements that act collectively on blocks of input states, and in addition we only require that the fidelity of the measurement be arbitrarily close to the optimal fidelity. We then consider the Shannon information of the outputs of optimal measurements, which is the amount of data produced by the measurements. We show that in the enlarged class of optimal measurements described above one can always construct an optimal measurement so that the Shannon information of its outputs equals the von Neumann entropy of the unknown states. Since this result is valid for all choices of fidelity functions and all distributions of input states, it provides a model independent answer to the question of how much can be learned about a quantum state by a measurement. Namely, this result shows that a measurement can extract at most one meaningful bit from every qubit carried by the unknown state.