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/0506267 · Physical Review A

Optimal estimation of group transformations using entanglement

Giulio Chiribella, Giacomo Mauro D’Ariano, Massimiliano Federico Sacchi

Published Oct 31, 2005• 103 citations
arXivHTMLPDF

Abstract

We derive the optimal input states and the optimal quantum measurements for estimating the unitary action of a given symmetry group, showing how the optimal performance is obtained with a suitable use of entanglement. Optimality is defined in a Bayesian sense, as minimization of the average value of a given cost function. We introduce a class of cost functions that generalizes the Holevo class for phase estimation, and show that for states of the optimal form all functions in such a class lead to the same optimal measurement. As a first application of the main result is the complete proof of the optimal efficiency in the transmission of a Cartesian reference frame. As a second application, we derive the optimal estimation of a completely unknown two-qubit maximally entangled state, provided that $N$ copies of the state are available. In the limit of large $N$, the fidelity of the optimal estimation is shown to be $1\ensuremath{-}3∕(4N)$.

DOI
10.1103/physreva.72.042338

Related papers

Estimation of Quantum States and Operations

Ballester Sánchez, Manuel Alberto

The subject of this thesis, quantum estimation theory, is inspired by recent developments, both practical and theoretical, in the new field of quantum information and computation. The main task of quantum estimation theory, the subject of this thesis, is that given many identical copies of an unknown quantum system one wants to guess or estimate what the system is. This is done by performing an measurement and analyzing the data obtained from it. A quantum state is represented by a density matrix. In some cases one does not need to know the whole density matrix, sometimes a partial characterization is enough. Chapter 2 of this thesis is about optimally estimating the spectrum, i.e. the eigenvalues, of a density matrix. This is useful, for example, for estimating the entanglement of a bipartite pure state. The results of chapter 2 tell us that one can optimally estimate bipartite entanglement with LOCC measurements on one of the two parties. Unitary operations transform density matrices to density matrices. They are the gates of quantum information processing. Suppose one has one such unitary operation but one does not know what it does. In order to characterize it one needs to know how it transforms any input state. In order to know how a particular state is transformed, one can prepare that state, feed it to the unitary many times, and measure the output. One state is, of course, not enough. It turns out that the unitary can be completely characterized with a finite set of states, this is known as quantum process tomography. Unfortunately, in many cases it is difficult to have available in the lab all the input states needed to perform quantum process tomography. It is here where entanglement comes to the rescue. One pair of maximally entangled qubits, where one of them is used as input for the unitary gate and the other is left untouched, is sufficient. All the information about the unknown gate is contained in the joint output state. This is the problem treated in chapter 3. Now the output state needs to be measured. Again, one could measure the two qubits separately (LOCC) or jointly (collective measurement). Is there anything to gain by performing joint measurements? The answer is yes, optimal collective measurements are at least 3 times more accurate than any LOCC measurement. Chapter 4 treats the problem of estimating commuting unitary gates. There it is shown that this extra information makes it unnecessary to use an entangled input state and to use collective measurements at the output. In chapters 3 and 4 it is assumed that only one copy of the unitary gate is available. If, however, more than one copy (say N ) are available at the same time one could use an N-fold entangled state as input. If no entanglement is used, classical statistics tells us that the error of the estimation is proportional to 1/√N. In the literature, it has been shown that by using an entangled state in the input one can achieve an error proportional to 1/N. However, this has been shown only for qubit gates. In chapter 5 important steps are taken in the direction of showing the same thing for a general unitary gate, but this remains an open problem.

•Jan 1, 2001•3 citations

Optimal Probe State for Phase Estimation Under Covariant Measurement

Qipeng Qian, Christos N. Gagatsos

We study the optimization of input states for phase estimation under covariant measurements. Building on Holevo's framework, which provides the optimal covariant measurement for a fixed input state, we further optimize over the input state itself. For a general even $2π$-periodic cost function with non-negative Fourier coefficients, we derive a necessary and sufficient condition for the optimal input state: Its Fock coefficients are determined, up to arbitrary phases, by the eigenvector corresponding to the largest eigenvalue of a Toeplitz matrix defined by the cost function. This characterization yields an explicit expression for the attainable lower bound of the average cost under optimal covariant measurements and shows that this bound asymptotically approaches zero in the infinite-energy limit. For the specific cost function $W(θ,\tildeθ)=4\sin^2[(θ-\tildeθ)/2]$, we obtain the optimal input state and the corresponding minimum average cost in closed form, demonstrating Heisenberg scaling with respect to the mean photon number.

quant-pharXiv:2606.18169•Jun 17, 2026•0 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

Group theoretic structures in the estimation of an unknown unitary transformation

Giulio Chiribella

This paper presents a series of general results about the optimal estimation of physical transformations in a given symmetry group. In particular, it is shown how the different symmetries of the problem determine different properties of the optimal estimation strategy. The paper also contains a discussion about the role of entanglement between the representation and multiplicity spaces and about the optimality of square-root measurements.

arXiv:1012.2130•Journal of Physics Conference Series•Mar 1, 2011•11 citations

Coherence as a Resource for Phase Estimation

Felix Ahnefeld, Thomas Theurer, Martin Bodo Plenio

Quantum phase estimation is a core task in quantum technologies ranging from metrology to quantum computing, where it appears as a key subroutine in various algorithms. Here, we quantitatively connect the performance of phase estimation protocols with quantum coherence. To achieve this, we construct and characterize resource theories of quantum networks that cannot generate coherence. Given multiple copies of a unitary encoding an unknown phase and access to a fixed coherent state, we estimate the phase using such networks. For a unified and general approach, we assess the quality of the estimate using a generic cost function that penalizes deviations from the true value. We determine the minimal average cost that can be achieved in this manner and explicitly derive optimal protocols. From this, we construct a family of coherence measures that directly connect a state's coherence with its value for phase estimation, demonstrating that every bit of coherence helps. This establishes coherence as a resource that quantifies the performance of phase estimation, and, thus, of any quantum technology relying on it as a subroutine.

arXiv:2505.18544•Physical Review Letters•Mar 31, 2026•2 citations

Efficient Characterization and Manipulation of States of Light

Arman Mansouri

Quantum science and technology have seen rapid progress in recent decades, finding applications in computation, information processing, communication, and sensing. Accurate characterization and manipulation of quantum states is essential to the successful implementation of these applications. Moreover, optimal quantum state estimation is closely related to fundamental questions, such as limits on how fast information can be extracted about quantum systems. In this thesis, we present several contributions to efficiently estimating and transforming quantum states. We discuss optimal quantum state estimation, both in the regime of few available quantum systems as well as the asymptotic regime. We numerically investigate how assuming a pure-state can affect the performance of tomographic methods, and where adaptive quantum state tomography is in fact advantageous. While collective measurements have been formally considered as optimal solutions to various problems in quantum information, their implementation has been largely neglected due to experimental challenges. We present a proof-of-concept experiment realizing Massar and Popescu's optimal collective measurement for quantum state estimation, utilizing two-photon interference and polarization-dependent loss to achieve non-maximally entangled state projections. We demonstrate that collective strategies can outperform the best separable measurements. Furthermore, we introduce and experimentally implement a fast and automated optical polarization compensation scheme using liquid crystal variable retarders and rotating quarter-wave plate tomography. Desired accuracy thresholds can be achieved within a few compensation steps. Finally, we use a matrix-model approach to investigate the realization of spatial unitary transformations via sequences of thin phase gratings and free-space propagation. We numerically demonstrate that the grating parameters and propagation distances can be optimized to achieve various two-beam and three-beam unitary transformations, such as Pauli gates and the discrete Fourier transform. Through experimental and numerical approaches, this thesis offers new perspectives on methods for quantum state estimation and unitary transformation.

•Mar 5, 2026•0 citations
We derive the optimal input states and the optimal quantum measurements for estimating the unitary action of a given symmetry group, showing how the optimal performance is obtained with a suitable use of entanglement. Optimality is defined in a Bayesian sense, as minimization of the average value of a given cost function. We introduce a class of cost functions that generalizes the Holevo class for phase estimation, and show that for states of the optimal form all functions in such a class lead to the same optimal measurement. As a first application of the main result is the complete proof of the optimal efficiency in the transmission of a Cartesian reference frame. As a second application, we derive the optimal estimation of a completely unknown two-qubit maximally entangled state, provided that NNN copies of the state are available. In the limit of large NNN, the fidelity of the optimal estimation is shown to be 1\ensuremath−3∕(4N)1\ensuremath{-}3∕(4N)1\ensuremath−3∕(4N).
The subject of this thesis, quantum estimation theory, is inspired by recent developments, both practical and theoretical, in the new field of quantum information and computation. The main task of quantum estimation theory, the subject of this thesis, is that given many identical copies of an unknown quantum system one wants to guess or estimate what the system is. This is done by performing an measurement and analyzing the data obtained from it. A quantum state is represented by a density matrix. In some cases one does not need to know the whole density matrix, sometimes a partial characterization is enough. Chapter 2 of this thesis is about optimally estimating the spectrum, i.e. the eigenvalues, of a density matrix. This is useful, for example, for estimating the entanglement of a bipartite pure state. The results of chapter 2 tell us that one can optimally estimate bipartite entanglement with LOCC measurements on one of the two parties. Unitary operations transform density matrices to density matrices. They are the gates of quantum information processing. Suppose one has one such unitary operation but one does not know what it does. In order to characterize it one needs to know how it transforms any input state. In order to know how a particular state is transformed, one can prepare that state, feed it to the unitary many times, and measure the output. One state is, of course, not enough. It turns out that the unitary can be completely characterized with a finite set of states, this is known as quantum process tomography. Unfortunately, in many cases it is difficult to have available in the lab all the input states needed to perform quantum process tomography. It is here where entanglement comes to the rescue. One pair of maximally entangled qubits, where one of them is used as input for the unitary gate and the other is left untouched, is sufficient. All the information about the unknown gate is contained in the joint output state. This is the problem treated in chapter 3. Now the output state needs to be measured. Again, one could measure the two qubits separately (LOCC) or jointly (collective measurement). Is there anything to gain by performing joint measurements? The answer is yes, optimal collective measurements are at least 3 times more accurate than any LOCC measurement. Chapter 4 treats the problem of estimating commuting unitary gates. There it is shown that this extra information makes it unnecessary to use an entangled input state and to use collective measurements at the output. In chapters 3 and 4 it is assumed that only one copy of the unitary gate is available. If, however, more than one copy (say N ) are available at the same time one could use an N-fold entangled state as input. If no entanglement is used, classical statistics tells us that the error of the estimation is proportional to 1/√N. In the literature, it has been shown that by using an entangled state in the input one can achieve an error proportional to 1/N. However, this has been shown only for qubit gates. In chapter 5 important steps are taken in the direction of showing the same thing for a general unitary gate, but this remains an open problem.
We study the optimization of input states for phase estimation under covariant measurements. Building on Holevo's framework, which provides the optimal covariant measurement for a fixed input state, we further optimize over the input state itself. For a general even 2π2π2π-periodic cost function with non-negative Fourier coefficients, we derive a necessary and sufficient condition for the optimal input state: Its Fock coefficients are determined, up to arbitrary phases, by the eigenvector corresponding to the largest eigenvalue of a Toeplitz matrix defined by the cost function. This characterization yields an explicit expression for the attainable lower bound of the average cost under optimal covariant measurements and shows that this bound asymptotically approaches zero in the infinite-energy limit. For the specific cost function W(θ,θ~)=4sin⁡2[(θ−θ~)/2]W(θ,\tildeθ)=4\sin^2[(θ-\tildeθ)/2]W(θ,θ~)=4sin2[(θ−θ~)/2], we obtain the optimal input state and the corresponding minimum average cost in closed form, demonstrating Heisenberg scaling with respect to the mean photon number.
Quantum phase estimation is a core task in quantum technologies ranging from metrology to quantum computing, where it appears as a key subroutine in various algorithms. Here, we quantitatively connect the performance of phase estimation protocols with quantum coherence. To achieve this, we construct and characterize resource theories of quantum networks that cannot generate coherence. Given multiple copies of a unitary encoding an unknown phase and access to a fixed coherent state, we estimate the phase using such networks. For a unified and general approach, we assess the quality of the estimate using a generic cost function that penalizes deviations from the true value. We determine the minimal average cost that can be achieved in this manner and explicitly derive optimal protocols. From this, we construct a family of coherence measures that directly connect a state's coherence with its value for phase estimation, demonstrating that every bit of coherence helps. This establishes coherence as a resource that quantifies the performance of phase estimation, and, thus, of any quantum technology relying on it as a subroutine.
Quantum science and technology have seen rapid progress in recent decades, finding applications in computation, information processing, communication, and sensing. Accurate characterization and manipulation of quantum states is essential to the successful implementation of these applications. Moreover, optimal quantum state estimation is closely related to fundamental questions, such as limits on how fast information can be extracted about quantum systems. In this thesis, we present several contributions to efficiently estimating and transforming quantum states. We discuss optimal quantum state estimation, both in the regime of few available quantum systems as well as the asymptotic regime. We numerically investigate how assuming a pure-state can affect the performance of tomographic methods, and where adaptive quantum state tomography is in fact advantageous. While collective measurements have been formally considered as optimal solutions to various problems in quantum information, their implementation has been largely neglected due to experimental challenges. We present a proof-of-concept experiment realizing Massar and Popescu's optimal collective measurement for quantum state estimation, utilizing two-photon interference and polarization-dependent loss to achieve non-maximally entangled state projections. We demonstrate that collective strategies can outperform the best separable measurements. Furthermore, we introduce and experimentally implement a fast and automated optical polarization compensation scheme using liquid crystal variable retarders and rotating quarter-wave plate tomography. Desired accuracy thresholds can be achieved within a few compensation steps. Finally, we use a matrix-model approach to investigate the realization of spatial unitary transformations via sequences of thin phase gratings and free-space propagation. We numerically demonstrate that the grating parameters and propagation distances can be optimized to achieve various two-beam and three-beam unitary transformations, such as Pauli gates and the discrete Fourier transform. Through experimental and numerical approaches, this thesis offers new perspectives on methods for quantum state estimation and unitary transformation.