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

Physical Review Letters

Quantum limits on precision measurements of phase

Samuel L. Braunstein

Published Dec 21, 1992• 148 citations
Publisher

Abstract

We investigate the quantum mechanical bound to how precisely we can determine a phase shift given only a constraint on the mean total number of photons available. By considering how quickly one can gain information from data analysis, we derive the sensitivity achievable (in principle) for measurements involving even highly non-Gaussian noise. Using these results we calculate the sensitivity of several recent proposals for precision phase measurement, and show that no proposal to date beats the sensitivity believed achievable by squeezed state interferometry.

DOI
10.1103/physrevlett.69.3598

Related papers

Some limits to precision phase measurement

Samuel L. Braunstein

States near the vacuum having 〈n^〉\ensuremath{\ll}1 are used for the precision determination of a phase shift. When this phase shift is sampled by enough of these near-vacuum states, its size may be extracted via data analysis. We calculate the achievable sensitivity of such schemes when we are limited in our resources to a mean total number of quanta ${\mathit{N}}_{\mathrm{tot}}$. We study in detail some examples of these near-vacuum schemes and show that in the absence of loss or noise their sensitivity approaches that of squeezed-state interferometry.

Physical Review A•Jan 1, 1994•16 citations

Heisenberg limit in phase measurements: the threshold detection approach

Dariya Salykina, V. S. Liamin, Valentin L. Gorshenin +2 more

The ultimate precision of phase estimation is limited by the Heisenberg scaling $Δϕ_0 = K/N$, where $K\sim1$ is a numerical prefactor and $N$ is the mean number of photons interacting with the phase shifting object(s). However, achieving this fundamental limit often comes at the cost of an extremely narrow high-sensitive range, rendering schemes impractical. We analyze the precision limits of phase measurements in single- and two-arm optical interferometers with input Gaussian states. We consider two detection methods: conventional homodyne measurement and non-Gaussian threshold detection that saturates the quantum Cramér-Rao bound. We characterize the performance by two complementary metrics: the peak sensitivity $Δϕ_0$ and the width $δϕ$ of the high-sensitivity range. We demonstrate that Heisenberg scaling is attainable in all configurations considered. However, we reveal that $δϕ$ strongly depends on $K$. We derive an approximate analytic expression that describes this trade-off. We show also that the two-arm interferometer with antisymmetrically squeezed inputs exhibits exceptional performance, simultaneously achieving Heisenberg-limited sensitivity and a broad high-sensitivity range $δϕ=π/2$.

arXiv:2505.06714•May 10, 2025•0 citations

Quantum limit of precision for phase estimation in squeezing-enhanced interferometry with a single-mode readout

Dmitri B. Horoshko, Fedor Jelezko

We consider an optical interferometer with coherent light in one input and a squeezed vacuum in another. Such an interferometer is known to beat the standard quantum limit of sensitivity to the difference of phase shifts in its arms. We find the ultimate limit of precision for such an interferometer by calculating the quantum Fisher information of the mixed quantum state in one of the interferometer's outputs about the difference phase. We show that, in the vicinity of the black fringe, this information is asymptotically close to the quantum Fisher information about this phase for the two-mode readout. We conclude that the single-mode readout is optimal for phase estimation in squeezing-enhanced interferometry and allows for the Heisenberg scaling of precision. We also show that the optimal local measurement in the vicinity of the black fringe consists of amplifying the output field in a phase-sensitive way and measuring its photon number.

quant-pharXiv:2603.07556•Mar 10, 2026•0 citations

Complementarity and Fundamental Limit in Precision Phase Measurement

Z. Y. Ou

By analyzing a single-photon interference experiment together with a quantum nondemolition measurement scheme for the which-path information, we prove that given the total mean number $〈n〉$ of available photons, the fundamental limit in precision measurement of a phase shift is the Heisenberg limit, i.e., $1/〈n〉$. The analysis is based on the complementarity principle and is independent of the scheme for the measurement of the phase shift. We also show that to achieve the Heisenberg limit states with photon number fluctuations of the order of or larger than the mean photon number have to be exploited.

Physical Review Letters•Sep 16, 1996•101 citations

Metrological Sensitivity beyond Gaussian Limits with Cubic Phase States

Jiajie Guo, Shuheng Liu, Boxuan Jing +2 more

Cubic phase states provide the essential non-Gaussian resource for continuous-variable quantum computing. We show that they also offer significant potential for quantum metrology, surpassing the phase-sensing sensitivity of all Gaussian states at equal average photon number. Optimal sensitivity requires only moderate initial squeezing, and the non-Gaussian advantage remains robust against loss and detection noise. We identify optimal measurement strategies and show that several experimentally relevant preparation schemes surpass Gaussian limits, in some cases reaching the sensitivity of cubic phase states. Our results establish cubic phase states as a promising resource for quantum-enhanced precision measurements beyond Gaussian limits.

quant-pharXiv:2512.03769•Dec 4, 2025•0 citations

Quantum metrology from an information theory perspective

Sergio Boixo, Animesh Datta, Matthew J. Davis +5 more

Questions about quantum limits on measurement precision were once viewed from the perspective of how to reduce or avoid the effects of quantum noise. With the advent of quantum information science came a paradigm shift to proving rigorous bounds on measurement precision. These bounds have been interpreted as saying, first, that the best achievable sensitivity scales as 1/n, where n is the number of particles one has available for a measurement and, second, that the only way to achieve this Heisenberg-limited sensitivity is to use quantum entanglement. We review these results and show that using quadratic couplings of n particles to a parameter to be estimated, one can achieve sensitivities that scale as 1/n^2 if one uses entanglement, but even in the absence of any entanglement at any time during the measurement protocol, one can achieve a super-Heisenberg scaling of 1/n^(3/2)

AIP conference proceedings•Jan 1, 2009•3 citations
We investigate the quantum mechanical bound to how precisely we can determine a phase shift given only a constraint on the mean total number of photons available. By considering how quickly one can gain information from data analysis, we derive the sensitivity achievable (in principle) for measurements involving even highly non-Gaussian noise. Using these results we calculate the sensitivity of several recent proposals for precision phase measurement, and show that no proposal to date beats the sensitivity believed achievable by squeezed state interferometry.
States near the vacuum having 〈n^〉\ensuremath{\ll}1 are used for the precision determination of a phase shift. When this phase shift is sampled by enough of these near-vacuum states, its size may be extracted via data analysis. We calculate the achievable sensitivity of such schemes when we are limited in our resources to a mean total number of quanta Ntot{\mathit{N}}_{\mathrm{tot}}Ntot​. We study in detail some examples of these near-vacuum schemes and show that in the absence of loss or noise their sensitivity approaches that of squeezed-state interferometry.
The ultimate precision of phase estimation is limited by the Heisenberg scaling Δϕ0=K/NΔϕ_0 = K/NΔϕ0​=K/N, where K∼1K\sim1K∼1 is a numerical prefactor and NNN is the mean number of photons interacting with the phase shifting object(s). However, achieving this fundamental limit often comes at the cost of an extremely narrow high-sensitive range, rendering schemes impractical. We analyze the precision limits of phase measurements in single- and two-arm optical interferometers with input Gaussian states. We consider two detection methods: conventional homodyne measurement and non-Gaussian threshold detection that saturates the quantum Cramér-Rao bound. We characterize the performance by two complementary metrics: the peak sensitivity Δϕ0Δϕ_0Δϕ0​ and the width δϕδϕδϕ of the high-sensitivity range. We demonstrate that Heisenberg scaling is attainable in all configurations considered. However, we reveal that δϕδϕδϕ strongly depends on KKK. We derive an approximate analytic expression that describes this trade-off. We show also that the two-arm interferometer with antisymmetrically squeezed inputs exhibits exceptional performance, simultaneously achieving Heisenberg-limited sensitivity and a broad high-sensitivity range δϕ=π/2δϕ=π/2δϕ=π/2.
We consider an optical interferometer with coherent light in one input and a squeezed vacuum in another. Such an interferometer is known to beat the standard quantum limit of sensitivity to the difference of phase shifts in its arms. We find the ultimate limit of precision for such an interferometer by calculating the quantum Fisher information of the mixed quantum state in one of the interferometer's outputs about the difference phase. We show that, in the vicinity of the black fringe, this information is asymptotically close to the quantum Fisher information about this phase for the two-mode readout. We conclude that the single-mode readout is optimal for phase estimation in squeezing-enhanced interferometry and allows for the Heisenberg scaling of precision. We also show that the optimal local measurement in the vicinity of the black fringe consists of amplifying the output field in a phase-sensitive way and measuring its photon number.
By analyzing a single-photon interference experiment together with a quantum nondemolition measurement scheme for the which-path information, we prove that given the total mean number 〈n〉〈n〉〈n〉 of available photons, the fundamental limit in precision measurement of a phase shift is the Heisenberg limit, i.e., 1/〈n〉1/〈n〉1/〈n〉. The analysis is based on the complementarity principle and is independent of the scheme for the measurement of the phase shift. We also show that to achieve the Heisenberg limit states with photon number fluctuations of the order of or larger than the mean photon number have to be exploited.
Cubic phase states provide the essential non-Gaussian resource for continuous-variable quantum computing. We show that they also offer significant potential for quantum metrology, surpassing the phase-sensing sensitivity of all Gaussian states at equal average photon number. Optimal sensitivity requires only moderate initial squeezing, and the non-Gaussian advantage remains robust against loss and detection noise. We identify optimal measurement strategies and show that several experimentally relevant preparation schemes surpass Gaussian limits, in some cases reaching the sensitivity of cubic phase states. Our results establish cubic phase states as a promising resource for quantum-enhanced precision measurements beyond Gaussian limits.
Questions about quantum limits on measurement precision were once viewed from the perspective of how to reduce or avoid the effects of quantum noise. With the advent of quantum information science came a paradigm shift to proving rigorous bounds on measurement precision. These bounds have been interpreted as saying, first, that the best achievable sensitivity scales as 1/n, where n is the number of particles one has available for a measurement and, second, that the only way to achieve this Heisenberg-limited sensitivity is to use quantum entanglement. We review these results and show that using quadratic couplings of n particles to a parameter to be estimated, one can achieve sensitivities that scale as 1/n^2 if one uses entanglement, but even in the absence of any entanglement at any time during the measurement protocol, one can achieve a super-Heisenberg scaling of 1/n^(3/2)