© 2026 Aqora Quantum S.A.S.

Terms Privacy Legal Notice

arXiv:1503.04887 · Journal of Physics A Mathematical and Theoretical

Quantum filtering for multiple diffusive and Poissonian measurements

Muhammad F. Emzir, Matthew J. Woolley, Ian R. Petersen

Published Sep 1, 2015• 16 citations

Abstract

We provide a rigorous derivation of a quantum filter for the case of multiple measurements being made on a quantum system. We consider a class of measurement processes which are functions of bosonic field operators, including combinations of diffusive and Poissonian processes. This covers the standard cases from quantum optics, where homodyne detection may be described as a diffusive process and photon counting may be described as a Poissonian process. We obtain a necessary and sufficient condition for any pair of such measurements taken at different output channels to satisfy a commutation relationship. Then, we derive a general, multiple-measurement quantum filter as an extension of a single-measurement quantum filter. As an application we explicitly obtain the quantum filter corresponding to homodyne detection and photon counting at the output ports of a beam splitter.

DOI: 10.1088/1751-8113/48/38/385302

Related papers

Quantum filtering for multiple measurements driven by fields in single-photon states

Zhiyuan Dong, Guofeng Zhang, Nina H. Amini

In this paper, we derive the stochastic master equations for quantum systems driven by a single-photon input state which is contaminated by quantum vacuum noise. To improve estimation performance, quantum filters based on multiple-channel measurements are designed. Two cases, namely diffusive plus Poissonian measurements and two diffusive measurements, are considered.

arXiv:1511.08335•Proceedings of the ... American Control Conference/Proceedings of the American Control Conference•Jul 1, 2016•10 citations

Single‐photon quantum filtering with multiple measurements

Zhiyuan Dong, Guofeng Zhang, Nina H. Amini

Summary The single‐photon quantum filtering problems have been investigated recently with applications in quantum computing. In practice, the detector responds with a quantum efficiency of less than unity since there exists some mode mismatch between the detector and the system and the single‐photon signal may be corrupted by quantum white noise. Consequently, quantum filters based on multiple measurements are designed in this paper to improve estimation performance. More specifically, the filtering equations for a 2‐level quantum system driven by a single‐photon input state and under multiple measurements are presented in this paper. Four scenarios, ie, (1) 2 diffusive measurements with Q‐P quadrature form, (2) 2 diffusive measurements with Q‐Q quadrature form, (3) diffusive plus Poissonian measurements, and (4) 2 Poissonian measurements, are considered. It is natural to compare the filtering results, ie, measuring a single channel or both channels, which one is better? By the simulation where we use a single photon to excite an atom, it seems that multiple measurements enable us to excite the atom with higher probability than only measuring a single channel. In addition, a measurement back‐action phenomenon is revealed by the simulation results.

arXiv:1801.03267•International Journal of Adaptive Control and Signal Processing•Feb 7, 2018•14 citations

Quantum filtering for multiple measurements driven by two single-photon states

Zhiyuan Dong, Guofeng Zhang, Nina H. Amini

In this paper, the problem of quantum filtering with two homodyne detection measurements for a two-level system has been considered. The quantum system is driven by two input light field channels, each of which contains a single photon. A quantum filter based on multiple measurements is designed; both the master equations and stochastic master equations are derived. In addition, numerical simulations for master equations with various pulse shape parameters are also compared.

World Congress on Intelligent Control and Automation•Jun 1, 2016•5 citations

Simultaneous continuous measurement of photon-counting and homodyne detection on a free photon field: dynamics of state reduction and the mutual influence of measurement backaction

Yui Kuramochi, Yu Watanabe, Masahito Ueda

We analyze a simultaneous continuous measurement of photon-counting and homodyne detection. The stochastic master equation or stochastic Schrödinger equation describing the measurement process includes both jump-type and diffusive-type stochastic increments. Analytic expressions of the wave function conditioned on homodyne and photon-counting records are obtained, yielding the probability density distributions and generating functions of the measurement records. Formula for the expectation values of the homodyne records conditioned on a photon-counting event is also derived which quantitatively describes the measurement backaction of photon-counting on the homodyne output. The obtained results are applied to typical initial states --- coherent, number, thermal, and squeezed states. Monte Carlo simulations of the measurement processes are also presented to demonstrate the dynamics of the combined measurement process.

arXiv:1212.0968•Journal of Physics A Mathematical and Theoretical•Oct 7, 2013•5 citations

Mismatched quantum filtering and entropic information

Mankei Tsang

Quantum filtering is a signal processing technique that estimates the posterior state of a quantum system under continuous measurements and has become a standard tool in quantum information processing, with applications in quantum state preparation, quantum metrology, and quantum control. If the filter assumes a nominal model that differs from reality, however, the estimation accuracy is bound to suffer. Here I derive identities that relate the excess error caused by quantum filter mismatch to the relative entropy between the true and nominal observation probability measures, with one identity for Gaussian measurements, such as optical homodyne detection, and another for Poissonian measurements, such as photon counting. These identities generalize recent seminal results in classical information theory and provide new operational meanings to relative entropy, mutual information, and channel capacity in the context of quantum experiments.

arXiv:1310.0291•IEEE International Symposium on Information Theory (ISIT)•Jun 1, 2014•3 citations

Filtering equation for measurement of a coherent channel

Anita Magdalena Dąbrowska, P. Staszewski

We propose a stochastic model for continuous photon counting and heterodyne measurement of a coherent source. A nonlinear filtering equation for the posterior state of a single-mode field in a cavity is derived by using the methods of the quantum stochastic calculus. The posterior dynamics is found for the observation of a Bose field being initially in a coherent state. The filtering equations for the counting and diffusion processes are given.

arXiv:1207.1196•Journal of the Optical Society of America B•Apr 25, 2011•3 citations

We provide a rigorous derivation of a quantum filter for the case of multiple measurements being made on a quantum system. We consider a class of measurement processes which are functions of bosonic field operators, including combinations of diffusive and Poissonian processes. This covers the standard cases from quantum optics, where homodyne detection may be described as a diffusive process and photon counting may be described as a Poissonian process. We obtain a necessary and sufficient condition for any pair of such measurements taken at different output channels to satisfy a commutation relationship. Then, we derive a general, multiple-measurement quantum filter as an extension of a single-measurement quantum filter. As an application we explicitly obtain the quantum filter corresponding to homodyne detection and photon counting at the output ports of a beam splitter.

Summary The single‐photon quantum filtering problems have been investigated recently with applications in quantum computing. In practice, the detector responds with a quantum efficiency of less than unity since there exists some mode mismatch between the detector and the system and the single‐photon signal may be corrupted by quantum white noise. Consequently, quantum filters based on multiple measurements are designed in this paper to improve estimation performance. More specifically, the filtering equations for a 2‐level quantum system driven by a single‐photon input state and under multiple measurements are presented in this paper. Four scenarios, ie, (1) 2 diffusive measurements with Q‐P quadrature form, (2) 2 diffusive measurements with Q‐Q quadrature form, (3) diffusive plus Poissonian measurements, and (4) 2 Poissonian measurements, are considered. It is natural to compare the filtering results, ie, measuring a single channel or both channels, which one is better? By the simulation where we use a single photon to excite an atom, it seems that multiple measurements enable us to excite the atom with higher probability than only measuring a single channel. In addition, a measurement back‐action phenomenon is revealed by the simulation results.

We analyze a simultaneous continuous measurement of photon-counting and homodyne detection. The stochastic master equation or stochastic Schrödinger equation describing the measurement process includes both jump-type and diffusive-type stochastic increments. Analytic expressions of the wave function conditioned on homodyne and photon-counting records are obtained, yielding the probability density distributions and generating functions of the measurement records. Formula for the expectation values of the homodyne records conditioned on a photon-counting event is also derived which quantitatively describes the measurement backaction of photon-counting on the homodyne output. The obtained results are applied to typical initial states --- coherent, number, thermal, and squeezed states. Monte Carlo simulations of the measurement processes are also presented to demonstrate the dynamics of the combined measurement process.

Quantum filtering is a signal processing technique that estimates the posterior state of a quantum system under continuous measurements and has become a standard tool in quantum information processing, with applications in quantum state preparation, quantum metrology, and quantum control. If the filter assumes a nominal model that differs from reality, however, the estimation accuracy is bound to suffer. Here I derive identities that relate the excess error caused by quantum filter mismatch to the relative entropy between the true and nominal observation probability measures, with one identity for Gaussian measurements, such as optical homodyne detection, and another for Poissonian measurements, such as photon counting. These identities generalize recent seminal results in classical information theory and provide new operational meanings to relative entropy, mutual information, and channel capacity in the context of quantum experiments.