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:1508.04412 · Physical Review A

Adaptive identification of coherent states

Markku P. V. Stenberg, Kevin Pack, Frank K. Wilhelm

Published Dec 31, 2015• 7 citations
arXivHTMLPDF

Abstract

We present methods for efficient characterization of an optical coherent state $|\ensuremath{\alpha}\ensuremath{\rangle}$. We choose measurement settings adaptively and stochastically, based on data while it is collected. Our algorithm divides the estimation into two distinct steps: (i) before the first detection of a vacuum state, the probability of choosing a measurement setting is proportional to detecting vacuum with the setting, which makes using too similar measurement settings twice unlikely; and (ii) after the first detection of vacuum, we focus measurements in the region where vacuum is most likely to be detected. In step (i) [(ii)] the detection of vacuum (a photon) has a significantly larger effect on the shape of the posterior probability distribution of $\ensuremath{\alpha}$. Compared to nonadaptive schemes, our method makes the number of measurement shots required to achieve a certain level of accuracy smaller approximately by a factor proportional to the area describing the initial uncertainty of $\ensuremath{\alpha}$ in phase space. While this algorithm is not directly robust against readout errors, we make it such by introducing repeated measurements in step (i).

DOI
10.1103/physreva.92.063852

Related papers

Characterization of decohering quantum systems: Machine learning approach

Markku P. V. Stenberg, Oliver Köhn, Frank K. Wilhelm

Adaptive data collection and analysis, where data are being fed back to update the measurement settings, can greatly increase speed, precision, and reliability of the characterization of quantum systems. However, decoherence tends to make adaptive characterization difficult. As an example, we consider two coupled discrete quantum systems. When one of the systems can be controlled and measured, the standard method to characterize another, with an unknown frequency ${\ensuremath{\omega}}_{\mathrm{r}}$, is swap spectroscopy. Here, adapting measurements can provide estimates whose error decreases exponentially in the number of measurement shots rather than as a power law in conventional swap spectroscopy. However, when the decoherence time is so short that an excitation oscillating between the two systems can only undergo less than a few tens of vacuum Rabi oscillations, this approach can be marred by a severe limit on accuracy unless carefully designed. We adopt machine learning techniques to search for efficient policies for the characterization of decohering quantum systems. We find, for instance, that when the system undergoes more than 2 Rabi oscillations during its relaxation time ${T}_{1}, O({10}^{3})$ measurement shots are sufficient to reduce the squared error of the Bayesian initial prior of the unknown frequency ${\ensuremath{\omega}}_{\mathrm{r}}$ by a factor $O({10}^{4})$ or larger. We also develop policies optimized for extreme initial parameter uncertainty and for the presence of imperfections in the readout.

arXiv:1510.05655•Physical Review A•Jan 26, 2016•22 citations

Adaptive optimal measurement for the squeezed vacuum and coherent state

Kun Chen, Shuxin Chen, Wu Dewei +5 more

The output phase of the Sagnac interferometer has been measured with quantum balanced homodyne technique when coherent light and squeezed vacuum light are fed into the Sagnac interferometer simultaneously [Chen Kun et al., Acta Phys. Sin. 65 054203(2016)]. Nevertheless, there exist two deficiencies: 1) the phase sensitivity is related to the phase itself; 2) there are strict requirements for the phases of local oscillator light, coherent light and squeezed vacuum light. For overcoming these deficiencies, an adaptive optimal measurement scheme is suggested for the phase estimation. Firstly, we calculate that the quantum Fisher information (QFI) of the squeezed vacuum and coherent state is sinh2r+||2e2r by treating them as a quantum pure state, for they satisfy a condition of the quantum pure state, namely ()=()2. The QFI is related to quantum Cramer-Rao lower bound which can be used to evaluate the performance of the estimator. Secondly, we make an analysis of positive operator-valued measure (POVM) and design a set of the optimal measurement operators for reaching the quantum Cramer-Rao lower bound, whereas the optimal measurement operators depend on the true value of the phase which is what we want to estimate. In order to solve the problem and estimate the parameter effectively, an adaptive method is suggested. We set an initial value of the phase parameter to obtain a set of measurement operators which are not optimal at the first step. And then the initial measurement operators are used for POVM and to obtain a conditional probability function, from which we can obtain a new value of the phase with maximum likelihood estimator. Therefore, the measurement operators and conditional probability function will be updated with the new value. As the measurement operators and conditional probability function are updated step by step, we can estimate the value adaptively. In fact, the results of the maximum likelihood estimator will converge at the true value of the phase parameter gradually, which is then proved with the theoretical analysis. All in all, an adaptive measurement method of estimating the phase parameter of the squeezed vacuum and coherent state in Sagnac interferometer is suggested, and is proved theoretically to be that the scheme will converge at the true value of the phase with a probability of 1 and can reach the quantum Cramer-Rao lower bound.

Acta Physica Sinica•Jan 1, 2016•3 citations

Single-Shot Non-Gaussian Measurements for Optical Phase Estimation

M. T. DiMario, F. E. Becerra

Estimation of the properties of a physical system with minimal uncertainty is a central task in quantum metrology. Optical phase estimation is at the center of many metrological tasks where the value of a physical parameter is mapped to the phase of an electromagnetic field and single-shot measurements of this phase are necessary. While there are measurements able to estimate the phase of light in a single shot with small uncertainties, demonstrations of near-optimal single-shot measurements for an unknown phase of a coherent state remain elusive. Here, we propose and demonstrate strategies for single-shot measurements for ab initio phase estimation of coherent states that surpass the sensitivity limit of heterodyne measurement and approach the Cramer-Rao lower bound for coherent states. These single-shot estimation strategies are based on real-time optimization of coherent displacement operations, single photon counting with photon number resolution, and fast feedback. We show that our demonstration of these optimized estimation strategies surpasses the heterodyne limit for a wide range of optical powers without correcting for detection efficiency with a moderate number of adaptive measurement steps. This is, to our knowledge, the most sensitive single-shot measurement of an unknown phase encoded in optical coherent states.

arXiv:2008.08185•Physical Review Letters•Sep 18, 2020•11 citations

Adaptive quantum measurements of a continuously varying phase

Dominic William Berry, Howard M. Wiseman

We analyze the problem of quantum-limited estimation of a stochastically varying phase of a continuous beam (rather than a pulse) of the electromagnetic field. We consider both nonadaptive and adaptive measurements, and both dyne detection (using a local oscillator) and interferometric detection. We take the phase variation to be $\stackrel{\ifmmode \dot{}\else \.{}\fi{}}{\ensuremath{\varphi}}=\sqrt{\ensuremath{\kappa}}\ensuremath{\xi}(t),$ where $\ensuremath{\xi}(t)$ is \ensuremath{\delta}-correlated Gaussian noise. For a beam of power P, the important dimensionless parameter is $N=P/\ensuremath{\Elzxh}\ensuremath{\omega}\ensuremath{\kappa},$ the number of photons per coherence time. For the case of dyne detection, both continuous-wave (cw) coherent beams and cw (broadband) squeezed beams are considered. For a coherent beam a simple feedback scheme gives good results, with a phase variance $\ensuremath{\simeq}{N}^{\ensuremath{-}1/2}/2.$ This is $\sqrt{2}$ times smaller than that achievable by nonadaptive (heterodyne) detection. For a squeezed beam a more accurate feedback scheme gives a variance scaling as ${N}^{\ensuremath{-}2/3},$ compared to ${N}^{\ensuremath{-}1/2}$ for heterodyne detection. For the case of interferometry only a coherent input into one port is considered. The locally optimal feedback scheme is identified, and it is shown to give a variance scaling as ${N}^{\ensuremath{-}1/2}.$ It offers a significant improvement over nonadaptive interferometry only for N of order unity.

arXiv:quant-ph/0111013•Physical Review A•Mar 14, 2002•79 citations

Adaptive Optical Phase Estimation Using Time-Symmetric Quantum Smoothing

Trevor A. Wheatley, Dominic William Berry, Hidehiro Yonezawa +7 more

Quantum parameter estimation has many applications, from gravitational wave detection to quantum key distribution. The most commonly used technique for this type of estimation is quantum filtering, using only past observations. We present the first experimental demonstration of quantum smoothing, a time-symmetric technique that uses past and future observations, for quantum parameter estimation. We consider both adaptive and nonadaptive quantum smoothing, and show that both are better than their filtered counterparts. For the problem of estimating a stochastically varying phase shift on a coherent beam, our theory predicts that adaptive quantum smoothing (the best scheme) gives an estimate with a mean-square error up to 2sqrt[2] times smaller than nonadaptive filtering (the standard quantum limit). The experimentally measured improvement is 2.24+/-0.14.

arXiv:0912.1162•Physical Review Letters•Mar 3, 2010•95 citations

Complete Characterization of Quantum Optical Processes with a Focus on Quantum Memory

Connor Kupchak

This thesis introduces and implements a characterization procedure called coherent state quantum process tomography and applies it to a selection of quantum operations. The procedure holds advantages over previous quantum process tomography methods, a pri- mary one is that a process can be characterized by measuring its effect on a set of coherent states which are readily available from a laser source. After introduction of the characterization procedure, the method is tested on a simple process of an electro-optical modulator and polarizing beam splitter. The accuracy of the characterization is verified by comparison of the predicted action of the reconstructed process tensor versus the actual experiment involving a squeezed vacuum state. The algorithm is then implemented and verified on a quantum memory system based on electromagnetically induced transparency, a system that was previously shown capable of storing a squeezed vacuum state. Lastly, a new optical storage system based on a gradient echo memory scheme is con- structed and optimized to achieve memory retrieval efficiencies of >80%. To characterize this system, a modified coherent state quantum process tomography algorithm based on the method of maximum likelihood estimation is employed. Despite the high efficiency values, the presence of excess noise resulted in the degradation of the storage device per- formance to be below the benchmarks of a quantum memory system. This estimation algorithm is also successfully implemented on the non-deterministic processes of photon creation and annihilation.

•Jan 1, 2013•1 citation
We present methods for efficient characterization of an optical coherent state ∣\ensuremathα\ensuremath⟩|\ensuremath{\alpha}\ensuremath{\rangle}∣\ensuremathα\ensuremath⟩. We choose measurement settings adaptively and stochastically, based on data while it is collected. Our algorithm divides the estimation into two distinct steps: (i) before the first detection of a vacuum state, the probability of choosing a measurement setting is proportional to detecting vacuum with the setting, which makes using too similar measurement settings twice unlikely; and (ii) after the first detection of vacuum, we focus measurements in the region where vacuum is most likely to be detected. In step (i) [(ii)] the detection of vacuum (a photon) has a significantly larger effect on the shape of the posterior probability distribution of \ensuremathα\ensuremath{\alpha}\ensuremathα. Compared to nonadaptive schemes, our method makes the number of measurement shots required to achieve a certain level of accuracy smaller approximately by a factor proportional to the area describing the initial uncertainty of \ensuremathα\ensuremath{\alpha}\ensuremathα in phase space. While this algorithm is not directly robust against readout errors, we make it such by introducing repeated measurements in step (i).
Adaptive data collection and analysis, where data are being fed back to update the measurement settings, can greatly increase speed, precision, and reliability of the characterization of quantum systems. However, decoherence tends to make adaptive characterization difficult. As an example, we consider two coupled discrete quantum systems. When one of the systems can be controlled and measured, the standard method to characterize another, with an unknown frequency ${\ensuremath{\omega}}_{\mathrm{r}}$, is swap spectroscopy. Here, adapting measurements can provide estimates whose error decreases exponentially in the number of measurement shots rather than as a power law in conventional swap spectroscopy. However, when the decoherence time is so short that an excitation oscillating between the two systems can only undergo less than a few tens of vacuum Rabi oscillations, this approach can be marred by a severe limit on accuracy unless carefully designed. We adopt machine learning techniques to search for efficient policies for the characterization of decohering quantum systems. We find, for instance, that when the system undergoes more than 2 Rabi oscillations during its relaxation time T1,O(103){T}_{1}, O({10}^{3})T1​,O(103) measurement shots are sufficient to reduce the squared error of the Bayesian initial prior of the unknown frequency ${\ensuremath{\omega}}_{\mathrm{r}}$ by a factor O(104)O({10}^{4})O(104) or larger. We also develop policies optimized for extreme initial parameter uncertainty and for the presence of imperfections in the readout.
The output phase of the Sagnac interferometer has been measured with quantum balanced homodyne technique when coherent light and squeezed vacuum light are fed into the Sagnac interferometer simultaneously [Chen Kun et al., Acta Phys. Sin. 65 054203(2016)]. Nevertheless, there exist two deficiencies: 1) the phase sensitivity is related to the phase itself; 2) there are strict requirements for the phases of local oscillator light, coherent light and squeezed vacuum light. For overcoming these deficiencies, an adaptive optimal measurement scheme is suggested for the phase estimation. Firstly, we calculate that the quantum Fisher information (QFI) of the squeezed vacuum and coherent state is sinh2r+||2e2r by treating them as a quantum pure state, for they satisfy a condition of the quantum pure state, namely ()=()2. The QFI is related to quantum Cramer-Rao lower bound which can be used to evaluate the performance of the estimator. Secondly, we make an analysis of positive operator-valued measure (POVM) and design a set of the optimal measurement operators for reaching the quantum Cramer-Rao lower bound, whereas the optimal measurement operators depend on the true value of the phase which is what we want to estimate. In order to solve the problem and estimate the parameter effectively, an adaptive method is suggested. We set an initial value of the phase parameter to obtain a set of measurement operators which are not optimal at the first step. And then the initial measurement operators are used for POVM and to obtain a conditional probability function, from which we can obtain a new value of the phase with maximum likelihood estimator. Therefore, the measurement operators and conditional probability function will be updated with the new value. As the measurement operators and conditional probability function are updated step by step, we can estimate the value adaptively. In fact, the results of the maximum likelihood estimator will converge at the true value of the phase parameter gradually, which is then proved with the theoretical analysis. All in all, an adaptive measurement method of estimating the phase parameter of the squeezed vacuum and coherent state in Sagnac interferometer is suggested, and is proved theoretically to be that the scheme will converge at the true value of the phase with a probability of 1 and can reach the quantum Cramer-Rao lower bound.
Estimation of the properties of a physical system with minimal uncertainty is a central task in quantum metrology. Optical phase estimation is at the center of many metrological tasks where the value of a physical parameter is mapped to the phase of an electromagnetic field and single-shot measurements of this phase are necessary. While there are measurements able to estimate the phase of light in a single shot with small uncertainties, demonstrations of near-optimal single-shot measurements for an unknown phase of a coherent state remain elusive. Here, we propose and demonstrate strategies for single-shot measurements for ab initio phase estimation of coherent states that surpass the sensitivity limit of heterodyne measurement and approach the Cramer-Rao lower bound for coherent states. These single-shot estimation strategies are based on real-time optimization of coherent displacement operations, single photon counting with photon number resolution, and fast feedback. We show that our demonstration of these optimized estimation strategies surpasses the heterodyne limit for a wide range of optical powers without correcting for detection efficiency with a moderate number of adaptive measurement steps. This is, to our knowledge, the most sensitive single-shot measurement of an unknown phase encoded in optical coherent states.
We analyze the problem of quantum-limited estimation of a stochastically varying phase of a continuous beam (rather than a pulse) of the electromagnetic field. We consider both nonadaptive and adaptive measurements, and both dyne detection (using a local oscillator) and interferometric detection. We take the phase variation to be $\stackrel{\ifmmode \dot{}\else \.{}\fi{}}{\ensuremath{\varphi}}=\sqrt{\ensuremath{\kappa}}\ensuremath{\xi}(t),$ where $\ensuremath{\xi}(t)$ is \ensuremath{\delta}-correlated Gaussian noise. For a beam of power P, the important dimensionless parameter is $N=P/\ensuremath{\Elzxh}\ensuremath{\omega}\ensuremath{\kappa},$ the number of photons per coherence time. For the case of dyne detection, both continuous-wave (cw) coherent beams and cw (broadband) squeezed beams are considered. For a coherent beam a simple feedback scheme gives good results, with a phase variance $\ensuremath{\simeq}{N}^{\ensuremath{-}1/2}/2.$ This is 2\sqrt{2}2​ times smaller than that achievable by nonadaptive (heterodyne) detection. For a squeezed beam a more accurate feedback scheme gives a variance scaling as ${N}^{\ensuremath{-}2/3},$ compared to ${N}^{\ensuremath{-}1/2}$ for heterodyne detection. For the case of interferometry only a coherent input into one port is considered. The locally optimal feedback scheme is identified, and it is shown to give a variance scaling as ${N}^{\ensuremath{-}1/2}.$ It offers a significant improvement over nonadaptive interferometry only for N of order unity.
Quantum parameter estimation has many applications, from gravitational wave detection to quantum key distribution. The most commonly used technique for this type of estimation is quantum filtering, using only past observations. We present the first experimental demonstration of quantum smoothing, a time-symmetric technique that uses past and future observations, for quantum parameter estimation. We consider both adaptive and nonadaptive quantum smoothing, and show that both are better than their filtered counterparts. For the problem of estimating a stochastically varying phase shift on a coherent beam, our theory predicts that adaptive quantum smoothing (the best scheme) gives an estimate with a mean-square error up to 2sqrt[2] times smaller than nonadaptive filtering (the standard quantum limit). The experimentally measured improvement is 2.24+/-0.14.
This thesis introduces and implements a characterization procedure called coherent state quantum process tomography and applies it to a selection of quantum operations. The procedure holds advantages over previous quantum process tomography methods, a pri- mary one is that a process can be characterized by measuring its effect on a set of coherent states which are readily available from a laser source. After introduction of the characterization procedure, the method is tested on a simple process of an electro-optical modulator and polarizing beam splitter. The accuracy of the characterization is verified by comparison of the predicted action of the reconstructed process tensor versus the actual experiment involving a squeezed vacuum state. The algorithm is then implemented and verified on a quantum memory system based on electromagnetically induced transparency, a system that was previously shown capable of storing a squeezed vacuum state. Lastly, a new optical storage system based on a gradient echo memory scheme is con- structed and optimized to achieve memory retrieval efficiencies of >80%. To characterize this system, a modified coherent state quantum process tomography algorithm based on the method of maximum likelihood estimation is employed. Despite the high efficiency values, the presence of excess noise resulted in the degradation of the storage device per- formance to be below the benchmarks of a quantum memory system. This estimation algorithm is also successfully implemented on the non-deterministic processes of photon creation and annihilation.