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/0612080 · Physical Review A

Loss-induced limits to phase measurement precision with maximally entangled states

Mark A. Rubin, Sumanth Kaushik

Published May 7, 2007• 100 citations
arXivHTMLPDF

Abstract

The presence of loss limits the precision of an approach to phase measurement using maximally entangled states, also referred to as NOON states. A calculation using a simple beam-splitter model of loss shows that, for all nonzero values $L$ of the loss, phase measurement precision degrades with increasing number $N$ of entangled photons for $N$ sufficiently large. For $L$ above a critical value of approximately 0.785, phase measurement precision degrades with increasing $N$ for all values of $N$. For $L$ near zero, phase measurement precision improves with increasing $N$ down to a limiting precision of approximately $1.018L\phantom{\rule{0.3em}{0ex}}\text{radians}$, attained at $N$ approximately equal to $2.218∕L$, and degrades as $N$ increases beyond this value. Phase measurement precision with multiple measurements and a fixed total number of photons ${N}_{T}$ is also examined. For $L$ above a critical value of approximately 0.586, the ratio of phase measurement precision attainable with NOON states to that attainable by conventional methods using unentangled coherent states degrades with increasing $N$, the number of entangled photons employed in a single measurement, for all values of $N$. For $L$ near zero this ratio is optimized by using approximately $N=1.279∕L$ entangled photons in each measurement, yielding a precision of approximately $1.340\sqrt{L∕{N}_{T}}\phantom{\rule{0.3em}{0ex}}\text{radians}$.

DOI
10.1103/physreva.75.053805

Related papers

Loss-Induced Limits to NOON-State Phase Measurement Precision

Mark A. Rubin, Sumanth Kaushik

The presence of loss limits the precision of an approach to phase measurement using NOON states. A calculation using a simple beam-splitter model of loss shows that, for all nonzero values L of the loss, phase measurement precision degrades with increasing number N of entangled photons for N sufficiently large. For L above a critical value of approximately 0.785, phase measurement precision degrades with increasing N for all values of N. For L near zero, phase measurement precision improves with increasing N down to a limiting precision of approximately 1.018L radians, attained at N approximately equal to 2.218/L, and degrades as N increases beyond this value. Phase measurement precision with multiple measurements and a fixed total number of photons NT is optimized, for L near zero, by using approximately 1.279/L entangled photons in each measurement, yielding a precision of approximately 1.340 √ L/NT radians. 1 NOON States and the Heisenberg Limit The use of NOON states has been proposed as a means of performing phase measurements with a precision δφmin at the Heisenberg limit. In this limit, δφmin scales as δφmin ∼ 1/N, (1) This work was sponsored by the Air Force under Air Force Contract FA8721-05-C-0002. Opinions, interpretations, conclusions, and recommendations are those of the authors and are not necessarily endorsed by the U.S. Government. 1 with increasing photon number N, rather than at the standard quantum limit This fact can be seen as follows: NOON states are entangled states of the form

•Jan 1, 2006•0 citations

Precise Phase Measurements using an Entangled Coherent State

Paul Knott, Jacob Dunningham

Quantum entanglement offers the possibility of making measurements beyond the classical limit, however some issues still need to be overcome before it can be applied in realistic lossy systems. Recent work has used the quantum Fisher information (QFI) to show that entangled coherent states (ECSs) may be useful for this purpose as they combine sub-classical phase precision capabilities with robustness (Joo et al., 2011). However, to date no effective scheme for measuring a phase in lossy systems using an ECS has been devised. Here we present a scheme that does just this. We show how one could measure a phase to a precision significantly better than that attainable by both unentangled "classical" states and highly-entangled NOON states over a wide range of different losses. This brings quantum metrology closer to being a realistic and practical technology.

arXiv:1401.3969•Jan 16, 2014•0 citations

Optimal multiple-phase estimation with multi-mode NOON states against photon loss

Min Namkung, Donghyun Kim, Seongjin Hong +3 more

Abstract Multi-mode NOON states can quantum-enhance multiple-phase estimation in the absence of photon loss. However, a multi-mode NOON state is known to be vulnerable to photon loss, and its quantum-enhancement can be dissipated by lossy environment. In this work, we demonstrate that a quantum advantage in estimate precision can still be achieved in the presence of photon loss. This is accomplished by optimizing the weights of the multi-mode NOON states according to photon loss rates in the multiple modes, including the reference mode which defines the other phases. For practical relevance, we also show that photon-number counting via a multi-mode beam-splitter achieves the useful, albeit sub-optimal, quantum advantage. We expect this work to provide valuable guidance for developing quantum-enhanced multiple-phase estimation techniques in lossy environments.

arXiv:2401.09734•New Journal of Physics•Jul 1, 2024•13 citations

Robust quantum metrology

Paul Knott

In optical interferometry path-entangled states such as NOON states have shown to give quantum-enhanced precision measurements, but these states are notoriously fragile to particle losses, which typically collapse the quantum state and destroy the phase information. A class of inherently robust states that show the potential for great improvements over the alternatives are the entangled coherent states (ECSs). We show that these states allow substantial improvements over unentangled `classical' states and highly-entangled NOON states for a wide range of loss values. We then describe a measurement scheme that can be used to measure these states with a precision close to the theoretical bound given by the quantum Fisher information. \nWe then look at the quantum mechanisms that lead to precise measurements. In optical interferometry multi-mode entanglement is often assumed to be the driving force behind quantum enhanced measurements. Recent work has shown this assumption to be false, and here we show that when photon losses occur multi-mode entanglement is actually detrimental to obtaining quantum enhanced measurements. We specifically apply this idea to a superposition of coherent states, demonstrating that these states show a robustness to loss that allows them to significantly outperform their competitors in realistic systems. A practically viable measurement scheme is then presented that allows measurements close to the theoretical bound, even with loss. \nIn this thesis we also consider superpositions of spin coherent states and their application to quantum metrology. Compared to optical states, spin systems have a distinctly different process of decoherence known as non-Markovian dephasing, which has shown to give greatly improved robustness to loss. We see that spin cat states give an enhanced scaling over the shot noise limit, even with dephasing, whilst being realisable with current technology.

•Mar 1, 2015•2 citations

Attaining subclassical metrology in lossy systems with entangled coherent states

Paul A. Knott, William J. Munro, Jacob Dunningham

Quantum mechanics allows entanglement enhanced measurements to be performed, but loss remains an obstacle in constructing realistic quantum metrology schemes. However, recent work has revealed that entangled coherent states (ECSs) have the potential to perform robust subclassical measurements [J. Joo et al., Phys. Rev. Lett. 107, 083601 (2011)]. Up to now no read-out scheme has been devised that exploits this robust nature of ECSs, but we present here an experimentally accessible method of achieving precision close to the theoretical bound, even with loss. We show substantial improvements over unentangled classical states and highly entangled NOON states for a wide range of loss values, elevating quantum metrology to a realizable technology in the near future.

arXiv:1401.4006•Physical Review A•May 9, 2014•26 citations

Quantum metrology for nonlinear phase shifts with entangled coherent states

Jaewoo Joo, Ki-min Park B, Hyunseok Jeong +3 more

We investigate the phase enhancement of quantum states subject to nonlinear phase shifts. The optimal phase estimation of even entangled coherent states (ECSs) is shown to be better than that of NOON states with the same average particle number $\ensuremath{\langle}n\ensuremath{\rangle}$ and nonlinearity exponent $k$. We investigate the creation of an approximate entangled coherent state (AECS) from a photon-subtracted squeezed vacuum with current optical technology methods and show that a pure AECS is even better than an even ECS for large $\ensuremath{\langle}n\ensuremath{\rangle}$. Finally, we examine the simple, but physically relevant, cases of loss in the nonlinear interferometer for a fixed average photon number $\ensuremath{\langle}n\ensuremath{\rangle}$.

arXiv:1203.2099•Physical Review A•Oct 18, 2012•95 citations
The presence of loss limits the precision of an approach to phase measurement using maximally entangled states, also referred to as NOON states. A calculation using a simple beam-splitter model of loss shows that, for all nonzero values LLL of the loss, phase measurement precision degrades with increasing number NNN of entangled photons for NNN sufficiently large. For LLL above a critical value of approximately 0.785, phase measurement precision degrades with increasing NNN for all values of NNN. For LLL near zero, phase measurement precision improves with increasing NNN down to a limiting precision of approximately 1.018Lradians1.018L\phantom{\rule{0.3em}{0ex}}\text{radians}1.018Lradians, attained at NNN approximately equal to 2.218∕L2.218∕L2.218∕L, and degrades as NNN increases beyond this value. Phase measurement precision with multiple measurements and a fixed total number of photons NT{N}_{T}NT​ is also examined. For LLL above a critical value of approximately 0.586, the ratio of phase measurement precision attainable with NOON states to that attainable by conventional methods using unentangled coherent states degrades with increasing NNN, the number of entangled photons employed in a single measurement, for all values of NNN. For LLL near zero this ratio is optimized by using approximately N=1.279∕LN=1.279∕LN=1.279∕L entangled photons in each measurement, yielding a precision of approximately 1.340L∕NTradians1.340\sqrt{L∕{N}_{T}}\phantom{\rule{0.3em}{0ex}}\text{radians}1.340L∕NT​​radians.
The presence of loss limits the precision of an approach to phase measurement using NOON states. A calculation using a simple beam-splitter model of loss shows that, for all nonzero values L of the loss, phase measurement precision degrades with increasing number N of entangled photons for N sufficiently large. For L above a critical value of approximately 0.785, phase measurement precision degrades with increasing N for all values of N. For L near zero, phase measurement precision improves with increasing N down to a limiting precision of approximately 1.018L radians, attained at N approximately equal to 2.218/L, and degrades as N increases beyond this value. Phase measurement precision with multiple measurements and a fixed total number of photons NT is optimized, for L near zero, by using approximately 1.279/L entangled photons in each measurement, yielding a precision of approximately 1.340 √ L/NT radians. 1 NOON States and the Heisenberg Limit The use of NOON states has been proposed as a means of performing phase measurements with a precision δφmin at the Heisenberg limit. In this limit, δφmin scales as δφmin ∼ 1/N, (1) This work was sponsored by the Air Force under Air Force Contract FA8721-05-C-0002. Opinions, interpretations, conclusions, and recommendations are those of the authors and are not necessarily endorsed by the U.S. Government. 1 with increasing photon number N, rather than at the standard quantum limit This fact can be seen as follows: NOON states are entangled states of the form
Quantum entanglement offers the possibility of making measurements beyond the classical limit, however some issues still need to be overcome before it can be applied in realistic lossy systems. Recent work has used the quantum Fisher information (QFI) to show that entangled coherent states (ECSs) may be useful for this purpose as they combine sub-classical phase precision capabilities with robustness (Joo et al., 2011). However, to date no effective scheme for measuring a phase in lossy systems using an ECS has been devised. Here we present a scheme that does just this. We show how one could measure a phase to a precision significantly better than that attainable by both unentangled "classical" states and highly-entangled NOON states over a wide range of different losses. This brings quantum metrology closer to being a realistic and practical technology.
Abstract Multi-mode NOON states can quantum-enhance multiple-phase estimation in the absence of photon loss. However, a multi-mode NOON state is known to be vulnerable to photon loss, and its quantum-enhancement can be dissipated by lossy environment. In this work, we demonstrate that a quantum advantage in estimate precision can still be achieved in the presence of photon loss. This is accomplished by optimizing the weights of the multi-mode NOON states according to photon loss rates in the multiple modes, including the reference mode which defines the other phases. For practical relevance, we also show that photon-number counting via a multi-mode beam-splitter achieves the useful, albeit sub-optimal, quantum advantage. We expect this work to provide valuable guidance for developing quantum-enhanced multiple-phase estimation techniques in lossy environments.
In optical interferometry path-entangled states such as NOON states have shown to give quantum-enhanced precision measurements, but these states are notoriously fragile to particle losses, which typically collapse the quantum state and destroy the phase information. A class of inherently robust states that show the potential for great improvements over the alternatives are the entangled coherent states (ECSs). We show that these states allow substantial improvements over unentangled `classical' states and highly-entangled NOON states for a wide range of loss values. We then describe a measurement scheme that can be used to measure these states with a precision close to the theoretical bound given by the quantum Fisher information. \nWe then look at the quantum mechanisms that lead to precise measurements. In optical interferometry multi-mode entanglement is often assumed to be the driving force behind quantum enhanced measurements. Recent work has shown this assumption to be false, and here we show that when photon losses occur multi-mode entanglement is actually detrimental to obtaining quantum enhanced measurements. We specifically apply this idea to a superposition of coherent states, demonstrating that these states show a robustness to loss that allows them to significantly outperform their competitors in realistic systems. A practically viable measurement scheme is then presented that allows measurements close to the theoretical bound, even with loss. \nIn this thesis we also consider superpositions of spin coherent states and their application to quantum metrology. Compared to optical states, spin systems have a distinctly different process of decoherence known as non-Markovian dephasing, which has shown to give greatly improved robustness to loss. We see that spin cat states give an enhanced scaling over the shot noise limit, even with dephasing, whilst being realisable with current technology.
Quantum mechanics allows entanglement enhanced measurements to be performed, but loss remains an obstacle in constructing realistic quantum metrology schemes. However, recent work has revealed that entangled coherent states (ECSs) have the potential to perform robust subclassical measurements [J. Joo et al., Phys. Rev. Lett. 107, 083601 (2011)]. Up to now no read-out scheme has been devised that exploits this robust nature of ECSs, but we present here an experimentally accessible method of achieving precision close to the theoretical bound, even with loss. We show substantial improvements over unentangled classical states and highly entangled NOON states for a wide range of loss values, elevating quantum metrology to a realizable technology in the near future.
We investigate the phase enhancement of quantum states subject to nonlinear phase shifts. The optimal phase estimation of even entangled coherent states (ECSs) is shown to be better than that of NOON states with the same average particle number $\ensuremath{\langle}n\ensuremath{\rangle}$ and nonlinearity exponent kkk. We investigate the creation of an approximate entangled coherent state (AECS) from a photon-subtracted squeezed vacuum with current optical technology methods and show that a pure AECS is even better than an even ECS for large $\ensuremath{\langle}n\ensuremath{\rangle}$. Finally, we examine the simple, but physically relevant, cases of loss in the nonlinear interferometer for a fixed average photon number $\ensuremath{\langle}n\ensuremath{\rangle}$.