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

IEEE Transactions on Information Theory

Application of quantum estimation theory to analog communication over quantum channels

Stewart D. Personick

Published May 1, 1971• 121 citations
Publisher

Abstract

Recent inquiries into optical communication have raised questions as to the validity of classical detection and estimation theory for weak light fields. Helstrom [1] proposed that the axioms of quantum mechanics be incorporated into a quantum approach to optical estimation and detection. In this paper, we discuss two important results, the quantum equivalent of the minimum-mean-square-error (MMSE) estimator and the quantum Cramér-Rao bound for estimation of the parameters of an electromagnetic field. The first result, a new one, is applied to linear modulation. We show that homodyning is the optimal demodulation scheme in that case. Parallels to the classical MMSE estimator are drawn. The Cramér-Rao bound, first derived in the quantum ease by Helstrom [1], is applied to specific estimation problems. Details are left to the references, but interesting results are presented.

DOI
10.1109/tit.1971.1054643

Related papers

Multiple-parameter quantum estimation and measurement of nonselfadjoint observables

Horace P. Yuen, M. Lax

A quantum mechanical form of the Cramér-Rao inequality and a minimum-mcan-square-error quantum estimator for multiple parameters are derived, allowing all possible quantum measurements of the received field. The role of nonselfadjoint operators is emphasized in the formulation. Relations of our results to previous work on quantum estimation are discussed. For the estimation of complex mode amplitudes of coherent signals in Gaussian noise, it is shown that the optimal receiver measures the photon annihilation operator, which corresponds to optical heterodyning. This demonstrates the possible optimality of nonselfadjoint operators and clearly indicates the importance of considering more general quantum measurements in quantum signal detection.

IEEE Transactions on Information Theory•Nov 1, 1973•253 citations

Fundamental quantum limits in optical metrology from rate-distortion theory

Ranjith Nair

Abstract We derive fundamental lower bounds on the performance of optical metrology and communication systems in a Bayesian framework. The derivation uses classical rate-distortion theory in conjunction with bounds on the capacity to transmit classical information of various optical channels specified by the system design. The bounds are expressed in terms of the system parameters, the prior probability distribution of the parameter, and the average energy, i.e. number of photons E in the probe state. For phase estimation, the bounds pertain to a cyclic mean squared error criterion incorporating the cyclic nature of the phase. In the absence of optical loss, our bounds are applicable to multimode linear phase modulation schemes (including ancilla-assisted ones), and to nonlinear modulations on a single mode. The bounds display inverse-quadratic Heisenberg-limit scaling of the cyclic mean square error with respect to E . In the presence of any finite amount of loss, a lower bound on ancilla-assisted phase estimation with standard quantum limit scaling is derived, which is shown to be little different from a similar bound for coherent-state probes. For systems involving a single optical mode, we also obtain lower bounds on the mean squared error of estimating any classical parameter, and on the average error probability of any M -ary communication system under an average energy constraint. The bounds are valid for arbitrary quantum measurements, for any prior probability distribution, and do not rely on unbiasedness assumptions.

arXiv:1204.3761•Journal of Physics A Mathematical and Theoretical•Sep 18, 2018•12 citations

Two kinds of asymptotic bounds in quantum channel estimation

Masahito Hayashi

In order to treat the covariant approach and the Cramer-Rao approach in a unified viewpoint, we compare the Cramer-Rao bound and the local asymptotic mini-max bound, which coincides the limit of Bayesian bound in the group covariant case. it is shown that the local asymptotic mini-max bound is strictly larger than the Cramer-Rao bound in the phase estimation case while the both bounds coincide when the minimum mean square error decreases with the order O(1/n). We also derive a sufficient condition for that the minimum mean square error decreases with the order O(1/n).

•Mar 24, 2010•1 citation

Transmission estimation at the quantum Cramér-Rao bound with macroscopic quantum light

Timothy S. Woodworth, Carla Hermann-Avigliano, Kam Wai Clifford Chan +1 more

The field of quantum metrology seeks to apply quantum techniques and/or resources to classical sensing approaches with the goal of enhancing the precision in the estimation of a parameter beyond what can be achieved with classical resources. Theoretically, the fundamental minimum uncertainty in the estimation of a parameter for a given probing state is bounded by the quantum Cramér-Rao bound. From a practical perspective, it is necessary to find physical measurements that can saturate this fundamental limit and to show experimentally that it is possible to perform measurements with the required precision to do so. Here we perform experiments that saturate the quantum Cramér-Rao bound for transmission estimation over a wide range of transmissions when probing the system under study with a continuous wave bright two-mode squeezed state. To properly take into account the imperfections in the generation of the quantum state, we extend our previous theoretical results to incorporate the measured properties of the generated quantum state. For our largest transmission level of 84%, we show a 62% reduction over the optimal classical protocol in the variance in transmission estimation when probing with a bright two-mode squeezed state with -8 dB of intensity-difference squeezing. Given that transmission estimation is an integral part of many sensing protocols, such as plasmonic sensing, spectroscopy, calibration of the quantum efficiency of detectors, etc., the results presented promise to have a significant impact on a number of applications in various fields of research.

arXiv:2201.08902•EPJ Quantum Technology•Dec 1, 2022•12 citations

Phase Estimators with Predetection Feedback for Optical Communication

Stanley R. Robinson

Causal, minimum mean-square error (MMSE) estimators of a Gauss-Markov process observed through a conditional Poisson process whose rate parameter is a linear function of the estimation error are presented. Although the conditional estimation performance is data dependent, precomputable upper bounds on the average estimation performance are obtained. Approximate expressions are also presented for the Cramer-Rao lower bound, and it is shown that the estimator performance achieves that bound with equality when the estimator is operating considerably "above threshold." The estimator structure is applied to the problem of phase-tracking receivers for optical communication. Two receiver structures that use predetector phase feedback are considered: one uses local reference fields to allow the detector to observe phase error (homodyne/heterodyne), while the other is a novel direct detection receiver that depends explicitly on the closed-loop nature of the phase estimator. It is concluded that a large local oscillator amplitude is desirable to improve the phase-tracking performance in the homodyne/heterodyne case, and that as few as 4/8 detected signal photons per phase coherence time are required to keep the estimator above threshold. The direct detection scheme achieves the same performance as the homodyne system only in the limit of no dark current-background noise counts, and in general may require considerably more signal photons to keep the estimator "locked."

IRE Transactions on Communications Systems•Nov 1, 1976•2 citations

Optimal measurements for quantum multiparameter estimation with general states

Jing Yang, Shengshi Pang, Yiyu Zhou +1 more

We generalize the approach by Braunstein and Caves, [Phys. Rev. Lett. 72, 3439 (1994)]. to quantum multiparameter estimation with general states. We derive a matrix bound of the classical Fisher information matrix due to each measurement operator. The saturation of all these bounds results in the saturation of the matrix Helstrom Cram\'er-Rao bound. Remarkably, the saturation of the matrix bound is equivalent to the saturation of the scalar bound with respect to any given positive definite weight matrix. Necessary and sufficient conditions are obtained for the optimal measurements that give rise to the Helstrom Cram\'er-Rao bound associated with a general quantum state. To saturate the Helstrom bound with separable measurements or collective measurement entangling only a small number of identical states, we find it is necessary for the symmetric logarithmic derivatives to commute on the support of the state. As an important application of our results, we construct several local optimal measurements for the problem of estimating the three-dimensional separation of two incoherent optical point sources.

arXiv:1806.07337•Physical Review A•Sep 3, 2019•86 citations
Recent inquiries into optical communication have raised questions as to the validity of classical detection and estimation theory for weak light fields. Helstrom [1] proposed that the axioms of quantum mechanics be incorporated into a quantum approach to optical estimation and detection. In this paper, we discuss two important results, the quantum equivalent of the minimum-mean-square-error (MMSE) estimator and the quantum Cramér-Rao bound for estimation of the parameters of an electromagnetic field. The first result, a new one, is applied to linear modulation. We show that homodyning is the optimal demodulation scheme in that case. Parallels to the classical MMSE estimator are drawn. The Cramér-Rao bound, first derived in the quantum ease by Helstrom [1], is applied to specific estimation problems. Details are left to the references, but interesting results are presented.
A quantum mechanical form of the Cramér-Rao inequality and a minimum-mcan-square-error quantum estimator for multiple parameters are derived, allowing all possible quantum measurements of the received field. The role of nonselfadjoint operators is emphasized in the formulation. Relations of our results to previous work on quantum estimation are discussed. For the estimation of complex mode amplitudes of coherent signals in Gaussian noise, it is shown that the optimal receiver measures the photon annihilation operator, which corresponds to optical heterodyning. This demonstrates the possible optimality of nonselfadjoint operators and clearly indicates the importance of considering more general quantum measurements in quantum signal detection.
Abstract We derive fundamental lower bounds on the performance of optical metrology and communication systems in a Bayesian framework. The derivation uses classical rate-distortion theory in conjunction with bounds on the capacity to transmit classical information of various optical channels specified by the system design. The bounds are expressed in terms of the system parameters, the prior probability distribution of the parameter, and the average energy, i.e. number of photons E in the probe state. For phase estimation, the bounds pertain to a cyclic mean squared error criterion incorporating the cyclic nature of the phase. In the absence of optical loss, our bounds are applicable to multimode linear phase modulation schemes (including ancilla-assisted ones), and to nonlinear modulations on a single mode. The bounds display inverse-quadratic Heisenberg-limit scaling of the cyclic mean square error with respect to E . In the presence of any finite amount of loss, a lower bound on ancilla-assisted phase estimation with standard quantum limit scaling is derived, which is shown to be little different from a similar bound for coherent-state probes. For systems involving a single optical mode, we also obtain lower bounds on the mean squared error of estimating any classical parameter, and on the average error probability of any M -ary communication system under an average energy constraint. The bounds are valid for arbitrary quantum measurements, for any prior probability distribution, and do not rely on unbiasedness assumptions.
In order to treat the covariant approach and the Cramer-Rao approach in a unified viewpoint, we compare the Cramer-Rao bound and the local asymptotic mini-max bound, which coincides the limit of Bayesian bound in the group covariant case. it is shown that the local asymptotic mini-max bound is strictly larger than the Cramer-Rao bound in the phase estimation case while the both bounds coincide when the minimum mean square error decreases with the order O(1/n). We also derive a sufficient condition for that the minimum mean square error decreases with the order O(1/n).
The field of quantum metrology seeks to apply quantum techniques and/or resources to classical sensing approaches with the goal of enhancing the precision in the estimation of a parameter beyond what can be achieved with classical resources. Theoretically, the fundamental minimum uncertainty in the estimation of a parameter for a given probing state is bounded by the quantum Cramér-Rao bound. From a practical perspective, it is necessary to find physical measurements that can saturate this fundamental limit and to show experimentally that it is possible to perform measurements with the required precision to do so. Here we perform experiments that saturate the quantum Cramér-Rao bound for transmission estimation over a wide range of transmissions when probing the system under study with a continuous wave bright two-mode squeezed state. To properly take into account the imperfections in the generation of the quantum state, we extend our previous theoretical results to incorporate the measured properties of the generated quantum state. For our largest transmission level of 84%, we show a 62% reduction over the optimal classical protocol in the variance in transmission estimation when probing with a bright two-mode squeezed state with -8 dB of intensity-difference squeezing. Given that transmission estimation is an integral part of many sensing protocols, such as plasmonic sensing, spectroscopy, calibration of the quantum efficiency of detectors, etc., the results presented promise to have a significant impact on a number of applications in various fields of research.
Causal, minimum mean-square error (MMSE) estimators of a Gauss-Markov process observed through a conditional Poisson process whose rate parameter is a linear function of the estimation error are presented. Although the conditional estimation performance is data dependent, precomputable upper bounds on the average estimation performance are obtained. Approximate expressions are also presented for the Cramer-Rao lower bound, and it is shown that the estimator performance achieves that bound with equality when the estimator is operating considerably "above threshold." The estimator structure is applied to the problem of phase-tracking receivers for optical communication. Two receiver structures that use predetector phase feedback are considered: one uses local reference fields to allow the detector to observe phase error (homodyne/heterodyne), while the other is a novel direct detection receiver that depends explicitly on the closed-loop nature of the phase estimator. It is concluded that a large local oscillator amplitude is desirable to improve the phase-tracking performance in the homodyne/heterodyne case, and that as few as 4/8 detected signal photons per phase coherence time are required to keep the estimator above threshold. The direct detection scheme achieves the same performance as the homodyne system only in the limit of no dark current-background noise counts, and in general may require considerably more signal photons to keep the estimator "locked."
We generalize the approach by Braunstein and Caves, [Phys. Rev. Lett. 72, 3439 (1994)]. to quantum multiparameter estimation with general states. We derive a matrix bound of the classical Fisher information matrix due to each measurement operator. The saturation of all these bounds results in the saturation of the matrix Helstrom Cram\'er-Rao bound. Remarkably, the saturation of the matrix bound is equivalent to the saturation of the scalar bound with respect to any given positive definite weight matrix. Necessary and sufficient conditions are obtained for the optimal measurements that give rise to the Helstrom Cram\'er-Rao bound associated with a general quantum state. To saturate the Helstrom bound with separable measurements or collective measurement entangling only a small number of identical states, we find it is necessary for the symmetric logarithmic derivatives to commute on the support of the state. As an important application of our results, we construct several local optimal measurements for the problem of estimating the three-dimensional separation of two incoherent optical point sources.