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:2312.01965 · Physical Review A

Optimal finite-dimensional probe states for quantum phase estimation

Jin-Feng Qin, Yuqian Xu, Jing Liu

Published Nov 14, 2025• 3 citations
arXivHTMLPDF

Abstract

Phase estimation is a major mission in quantum metrology, especially in quantum interferometry. A full phase estimation scheme usually includes the optimal probe state and measurement. For the finite-dimensional states in Fock basis, the NOON state ceases to be optimal when the average particle number is fixed yet not equal to the Fock dimension (Fock number of the highest occupied Fock state of one mode), and what is the true optimal finite-dimensional probe state in this case is still undiscovered. Hereby we present several theorems to answer this question and provide a complete optimal scheme to realize the ultimate precision limit in practice. These optimal finite-dimensional probe states reveal an important fact that the Fock dimension could be treated as a metrological resource, and the given scheme is particularly useful in scenarios where weak light or limited particle number is demanded.

DOI
10.1103/953x-1rkm

Related papers

Optimal noisy quantum phase estimation with finite-dimensional states

Jin-Feng Qin, Jing Liu

Phase estimation in quantum interferometry is a major scenario where the quantum advantage is significantly revealed. Recently, the optimal finite-dimensional probe states (OFPSs) for phase estimation in two-mode quantum interferometry have been provided with the absence of noise [J.-F. Qin et al., Phys. Rev. A 112, 052428 (2025)]. However, the noise is inevitable in practice and the previously obtained OFPSs may cease to be optimal anymore. Hence, the forms of the true OFPSs in the existence of various noises are still open questions. Hereby, the noise of particle loss is studied and the true OFPSs under this noise have been investigated with the numerical algorithm named constrained optimization by linear approximation. Furthermore, a two-step measurement strategy is proposed to realize the ultimate precision limit in practice. The validity of this strategy is confirmed by the numerical simulation of practical experiments.

quant-pharXiv:2604.07828•Physical Review Research•Apr 10, 2026•0 citations

Optimal probe states for phase estimation with a fixed mean particle number

Jin-Feng Qin, Bo Liu

Abstract Quantum phase estimation reveals the power of quantum resources to beat the standard quantum limit and has been widely used in many fields. To improve the precision of phase estimation, we discuss the optimal probe states for phase estimation with a fixed mean particle number. By searching for the maximum quantum Fisher information, we optimize the probe states, which are superior to the path-entangled Fock states. Comparing the mean particle number ( n ¯ ) with the dimension of the probe states in Fock space (N + 1), when n ¯ ≤ N , our optimal probe states can provide a better performance than the n ¯ 00 n ¯ states. When n ¯ > N , our optimal probe states can also remain optimal if the dimension of the probe states is large enough.

Communications in Theoretical Physics•Jan 21, 2025•0 citations

Quantum parameter estimation with optimal control

Jing Liu, Haidong Yuan

A pivotal task in quantum metrology, and quantum parameter estimation in general, is to design schemes that achieve the highest precision with the given resources. Standard models of quantum metrology usually assume that the dynamics is fixed and that the highest precision is achieved by preparing the optimal probe states and performing optimal measurements. However, in many practical experimental settings, additional controls are usually available to alter the dynamics. Here we propose to use optimal control methods for further improvement of the precision limit of quantum parameter estimation. We show that, by exploring the additional degree of freedom offered by the controls, a higher-precision limit can be achieved. In particular, we show that the precision limit under the controlled schemes can go beyond the constraints put by the coherent time, which is in contrast with the standard scheme where the precision limit is always bounded by the coherent time.

arXiv:1604.04856•Physical Review A•Jul 14, 2017•133 citations

Optimal Gaussian measurements for phase estimation in single-mode Gaussian metrology

Changhun Oh, Changhyoup Lee, Carsten Rockstuhl +4 more

Abstract The central issue in quantum parameter estimation is to find out the optimal measurement setup that leads to the ultimate lower bound of an estimation error. We address here a question of whether a Gaussian measurement scheme can achieve the ultimate bound for phase estimation in single-mode Gaussian metrology that exploits single-mode Gaussian probe states in a Gaussian environment. We identify three types of optimal Gaussian measurement setups yielding the maximal Fisher information depending on displacement, squeezing, and thermalization of the probe state. We show that the homodyne measurement attains the ultimate bound for both displaced thermal probe states and squeezed vacuum probe states, whereas for the other single-mode Gaussian probe states, the optimized Gaussian measurement cannot be the optimal setup, although they are sometimes nearly optimal. We then demonstrate that the measurement on the basis of the product quadrature operators $$\hat X\hat P + \hat P\hat X$$ X ̂ P ̂ + P ̂ X ̂ , i.e., a non-Gaussian measurement, is required to be fully optimal.

arXiv:1805.08495•npj Quantum Information•Jan 28, 2019•81 citations

Optimal Gaussian measurements for phase estimation in single-mode Gaussian metrology

Changhun Oh, Changhyoup Lee, Carsten Rockstuhl +4 more

The central issue in quantum parameter estimation is to find out the optimal measurement setup that leads to the ultimate lower bound of an estimation error. We address here a question of whether a Gaussian measurement scheme can achieve the ultimate bound for phase estimation in single-mode Gaussian metrology that exploits single-mode Gaussian probe states in a Gaussian environment. We identify three types of optimal Gaussian measurement setups yielding the maximal Fisher information depending on displacement, squeezing, and thermalization of the probe state. We show that the homodyne measurement attains the ultimate bound for both displaced thermal probe states and squeezed vacuum probe states, whereas for the other single-mode Gaussian probe states, the optimized Gaussian measurement cannot be the optimal setup, although they are sometimes nearly optimal. We then demonstrate that the measurement on the basis of the product quadrature operators XP+PX, i.e., a non-Gaussian measurement, is required to be fully optimal.

•Jan 1, 2019•65 citations

Optimal Scheme for Quantum Metrology

Jing Liu, Zhang Mao, Hongzhen Chen +2 more

Abstract Quantum metrology can achieve far better precision than classical metrology, and is one of the most important applications of quantum technologies in the real world. To attain the highest precision promised by quantum metrology, all steps of the schemes need to be optimized, which include the state preparation, parametrization, and measurement. Here the recent progresses on the optimization of these steps, which are essential for the identification and achievement of the ultimate precision limit in quantum metrology, are reviewed. It is hoped this provides a useful reference for the researchers in quantum metrology and related fields.

arXiv:2111.12279•Advanced Quantum Technologies•Nov 23, 2021•80 citations
Phase estimation is a major mission in quantum metrology, especially in quantum interferometry. A full phase estimation scheme usually includes the optimal probe state and measurement. For the finite-dimensional states in Fock basis, the NOON state ceases to be optimal when the average particle number is fixed yet not equal to the Fock dimension (Fock number of the highest occupied Fock state of one mode), and what is the true optimal finite-dimensional probe state in this case is still undiscovered. Hereby we present several theorems to answer this question and provide a complete optimal scheme to realize the ultimate precision limit in practice. These optimal finite-dimensional probe states reveal an important fact that the Fock dimension could be treated as a metrological resource, and the given scheme is particularly useful in scenarios where weak light or limited particle number is demanded.
Phase estimation in quantum interferometry is a major scenario where the quantum advantage is significantly revealed. Recently, the optimal finite-dimensional probe states (OFPSs) for phase estimation in two-mode quantum interferometry have been provided with the absence of noise [J.-F. Qin et al., Phys. Rev. A 112, 052428 (2025)]. However, the noise is inevitable in practice and the previously obtained OFPSs may cease to be optimal anymore. Hence, the forms of the true OFPSs in the existence of various noises are still open questions. Hereby, the noise of particle loss is studied and the true OFPSs under this noise have been investigated with the numerical algorithm named constrained optimization by linear approximation. Furthermore, a two-step measurement strategy is proposed to realize the ultimate precision limit in practice. The validity of this strategy is confirmed by the numerical simulation of practical experiments.
Abstract Quantum phase estimation reveals the power of quantum resources to beat the standard quantum limit and has been widely used in many fields. To improve the precision of phase estimation, we discuss the optimal probe states for phase estimation with a fixed mean particle number. By searching for the maximum quantum Fisher information, we optimize the probe states, which are superior to the path-entangled Fock states. Comparing the mean particle number ( n ¯ ) with the dimension of the probe states in Fock space (N + 1), when n ¯ ≤ N , our optimal probe states can provide a better performance than the n ¯ 00 n ¯ states. When n ¯ > N , our optimal probe states can also remain optimal if the dimension of the probe states is large enough.
A pivotal task in quantum metrology, and quantum parameter estimation in general, is to design schemes that achieve the highest precision with the given resources. Standard models of quantum metrology usually assume that the dynamics is fixed and that the highest precision is achieved by preparing the optimal probe states and performing optimal measurements. However, in many practical experimental settings, additional controls are usually available to alter the dynamics. Here we propose to use optimal control methods for further improvement of the precision limit of quantum parameter estimation. We show that, by exploring the additional degree of freedom offered by the controls, a higher-precision limit can be achieved. In particular, we show that the precision limit under the controlled schemes can go beyond the constraints put by the coherent time, which is in contrast with the standard scheme where the precision limit is always bounded by the coherent time.
Abstract The central issue in quantum parameter estimation is to find out the optimal measurement setup that leads to the ultimate lower bound of an estimation error. We address here a question of whether a Gaussian measurement scheme can achieve the ultimate bound for phase estimation in single-mode Gaussian metrology that exploits single-mode Gaussian probe states in a Gaussian environment. We identify three types of optimal Gaussian measurement setups yielding the maximal Fisher information depending on displacement, squeezing, and thermalization of the probe state. We show that the homodyne measurement attains the ultimate bound for both displaced thermal probe states and squeezed vacuum probe states, whereas for the other single-mode Gaussian probe states, the optimized Gaussian measurement cannot be the optimal setup, although they are sometimes nearly optimal. We then demonstrate that the measurement on the basis of the product quadrature operators X^P^+P^X^\hat X\hat P + \hat P\hat XX^P^+P^X^ X ̂ P ̂ + P ̂ X ̂ , i.e., a non-Gaussian measurement, is required to be fully optimal.
The central issue in quantum parameter estimation is to find out the optimal measurement setup that leads to the ultimate lower bound of an estimation error. We address here a question of whether a Gaussian measurement scheme can achieve the ultimate bound for phase estimation in single-mode Gaussian metrology that exploits single-mode Gaussian probe states in a Gaussian environment. We identify three types of optimal Gaussian measurement setups yielding the maximal Fisher information depending on displacement, squeezing, and thermalization of the probe state. We show that the homodyne measurement attains the ultimate bound for both displaced thermal probe states and squeezed vacuum probe states, whereas for the other single-mode Gaussian probe states, the optimized Gaussian measurement cannot be the optimal setup, although they are sometimes nearly optimal. We then demonstrate that the measurement on the basis of the product quadrature operators XP+PX, i.e., a non-Gaussian measurement, is required to be fully optimal.
Abstract Quantum metrology can achieve far better precision than classical metrology, and is one of the most important applications of quantum technologies in the real world. To attain the highest precision promised by quantum metrology, all steps of the schemes need to be optimized, which include the state preparation, parametrization, and measurement. Here the recent progresses on the optimization of these steps, which are essential for the identification and achievement of the ultimate precision limit in quantum metrology, are reviewed. It is hoped this provides a useful reference for the researchers in quantum metrology and related fields.