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:cond-mat/0703344 · Physical Review B

Statistics of measurement of noncommuting quantum variables: Monitoring and purification of a qubit

Hongduo Wei, Yuli V. Nazarov

Published Jul 11, 2008• 33 citations
arXivHTMLPDF

Abstract

We address continuous weak linear quantum measurement and argue that it is best understood in terms of statistics of the outcomes of the linear detectors measuring a quantum system, for example, a qubit. We mostly concentrate on a setup consisting of a qubit and three independent detectors that simultaneously monitor three noncommuting operator variables, those corresponding to three pseudospin components of the qubit. We address the joint probability distribution of the detector outcomes and the qubit variables. When analyzing the distribution in the limit of big values of the outcomes, we reveal a high degree of correspondence between the three outcomes and three components of the qubit pseudospin after the measurement. This enables a high-fidelity monitoring of all three components. We discuss the relation between the monitoring described and the algorithms of quantum information theory that use the results of the partial measurement. We develop a proper formalism to evaluate the statistics of continuous weak linear measurement. The formalism is based on Feynman-Vernon approach, roots in the theory of full counting statistics, and boils down to a Bloch-Redfield equation augmented with counting fields.

DOI
10.1103/physrevb.78.045308

Related papers

Statistics of Measurement of Non-commuting Quantum Variables

Hongduo Wei, Yuli V. Nazarov, Vladimir Lebedev +1 more

We address continuous weak linear quantum measurement and argue that it is best understood in terms of statistics of the outcomes of the linear detectors measuring a quantum system, for example, a qubit. We develop a proper formalism to evaluate the statistics of such measurement. Generally, we are able to evaluate the joint probability distribution of the detector outcomes and the qubit variables. We concentrate on two setups. The application of our method to the setup where a single pseudospin component is measured gives a comphrehensive picture of quantum non‐demolition measurement. More interesting setup consists of a qubit and three independent detectors that simultaneously monitor three non‐commuting operator variables, those corresponding to three pseudo‐spin components of the qubit. When analyzing the distribution in the limit of big values of the outcomes, we reveal a high degree of correspondence between the three outcomes and three components of the qubit pseudo‐spin after the measurement. This enables a high‐fidelity monitoring of all three components. We discuss the relation between the monitoring described and the algorithms of quantum information theory that use the results of the partial measurement. The formalism is based on Feynman‐Vernon approach, roots in the theory of full counting statistics, and boils down to a Bloch‐Redfield equation augmented with counting fields.

AIP conference proceedings•Jan 1, 2009•0 citations

Probability distributions of continuous measurement results for two noncommuting variables and postselected quantum evolution

A. Franquet, Yuli V. Nazarov

We address the statistics of a simultaneous continuous weak linear measurement of two noncommuting variables on a few-state quantum system subject to a postselected evolution. The results of both postselected quantum measurement and simultaneous monitoring of two noncommuting variables differ drastically from the results of either classical or quantum projective measurement. We explore the peculiarities arising from the combination of the two. We concentrate on the distribution function of two measurement outcomes integrated over a time interval. We formulate a proper formalism for the evaluation of such distribution, and further compute and discuss the resulting statistics for idealized and experimentally relevant setups. We demonstrate the visibility and manifestations of the interference between initial and final states in the statistics of measurement outcomes for both variables in various regimes. We analytically predict the peculiarities at the circle ${\mathcal{O}}_{1}^{2}+{\mathcal{O}}_{2}^{2}=1$ in the distribution of measurement outcomes ${\mathcal{O}}_{1,2}$ in the limit of short measurement times and confirm this by numerical calculation at longer measurement times. We demonstrate analytically the anomalously large values of the time-integrated output cumulants in the limit of short measurement times and zero overlap between initial and final states, and give the detailed distributions for this case. We term this situation sudden jump. We present the numerical evaluation of the probability distributions for experimentally relevant parameters in several regimes and demonstrate that interference effects in the postselected measurement can be accurately predicted even if they are small.

arXiv:1708.05662•Physical Review A•Dec 6, 2019•1 citation

Probability distributions of continuous measurement results for conditioned quantum evolution

A. Franquet, Yuli V. Nazarov

We address the statistics of continuous weak linear measurement on a few-state quantum system that is subject to a conditioned quantum evolution. For a conditioned evolution, both the initial and final states of the system are fixed: the latter is achieved by the postselection in the end of the evolution. The statistics may drastically differ from the nonconditioned case, and the interference between initial and final states can be observed in the probability distributions of measurement outcomes as well as in the average values exceeding the conventional range of nonconditioned averages. We develop a proper formalism to compute the distributions of measurement outcomes, and evaluate and discuss the distributions in experimentally relevant setups. We demonstrate the manifestations of the interference between initial and final states in various regimes. We consider analytically simple examples of nontrivial probability distributions. We reveal peaks (or dips) at half-quantized values of the measurement outputs. We discuss in detail the case of zero overlap between initial and final states demonstrating anomalously big average outputs and sudden jump in time-integrated output. We present and discuss the numerical evaluation of the probability distribution aiming at extending the analytical results and describing a realistic experimental situation of a qubit in the regime of resonant fluorescence.

arXiv:1608.06858•Physical review. B./Physical review. B•Feb 21, 2017•7 citations

Qubit State Monitoring by Measurement of Three Complementary Observables

Rusko Ruskov, Alexander N. Korotkov, Klaus Mølmer

We consider the evolution of a qubit (spin $1/2$) under the simultaneous continuous measurement of three noncommuting qubit operators ${\stackrel{^}{\ensuremath{\sigma}}}_{x}$, ${\stackrel{^}{\ensuremath{\sigma}}}_{y}$, and ${\stackrel{^}{\ensuremath{\sigma}}}_{z}$. For identical ideal detectors, the qubit state evolves by approaching a pure state with a random direction in the Bloch vector space and by undergoing locally isotropic diffusion in the perpendicular directions. The quantum state conditioned on the complete detector record is used to assess the fidelity of classically inspired estimates based on running time averages and discrete time bin detector outputs.

arXiv:1006.2013•Physical Review Letters•Sep 2, 2010•42 citations

Weak and strong measurement of a qubit using a switching-based detector

Sahel Ashhab, J. Q. You, Franco Nori

We analyze the operation of a switching-based detector that probes a qubit's observable that does not commute with the qubit's Hamiltonian, leading to a nontrivial interplay between the measurement and free-qubit dynamics. In order to obtain analytical results and develop intuitive understanding of the different possible regimes of operation, we use a theoretical model where the detector is a quantum two-level system that is constantly monitored by a macroscopic system. We analyze how to interpret the outcome of the measurement and how the state of the qubit evolves while it is measured. We find that the answers to the above questions depend on the relation between the different parameters in the problem. In addition to the traditional strong-measurement regime, we identify a number of regimes associated with weak qubit-detector coupling. An incoherent detector whose switching time is measurable with high accuracy can provide high-fidelity information, but the measurement basis is determined only upon switching of the detector. An incoherent detector whose switching time can be known only with low accuracy provides a measurement in the qubit's energy eigenbasis with reduced measurement fidelity. A coherent detector measures the qubit in its energy eigenbasis and, under certain conditions, can provide high-fidelity information.

arXiv:0810.4386•Physical Review A•Mar 16, 2009•34 citations

Extracting filtered signal statistics of continuously measured quantum systems

Anthony Kiely, Gabriel Teixeira Landi

The joint state of a continuously monitored quantum system and the classical filtered measurement record has recently been shown to be described by a quantum Fokker-Planck master equation []. We present a deterministic approach to compute the steady state of the system and detector. The method is shown to become particularly efficient in the absence of feedback, which we exploit to develop a perturbative approach valid for weak feedback. We show that through this method we can extract the full counting statistics of the signal, the quantum-classical mutual information between the system and signal, and the Fisher information of the signal, which can be used for sensing applications. Our results are illustrated with both single-qubit models and the spin chains governed by the one-dimensional transverse field Ising model or the Lipkin-Meshkov-Glick model.

arXiv:2507.06303•Physical Review A•Mar 9, 2026•1 citation
We address continuous weak linear quantum measurement and argue that it is best understood in terms of statistics of the outcomes of the linear detectors measuring a quantum system, for example, a qubit. We mostly concentrate on a setup consisting of a qubit and three independent detectors that simultaneously monitor three noncommuting operator variables, those corresponding to three pseudospin components of the qubit. We address the joint probability distribution of the detector outcomes and the qubit variables. When analyzing the distribution in the limit of big values of the outcomes, we reveal a high degree of correspondence between the three outcomes and three components of the qubit pseudospin after the measurement. This enables a high-fidelity monitoring of all three components. We discuss the relation between the monitoring described and the algorithms of quantum information theory that use the results of the partial measurement. We develop a proper formalism to evaluate the statistics of continuous weak linear measurement. The formalism is based on Feynman-Vernon approach, roots in the theory of full counting statistics, and boils down to a Bloch-Redfield equation augmented with counting fields.
We address continuous weak linear quantum measurement and argue that it is best understood in terms of statistics of the outcomes of the linear detectors measuring a quantum system, for example, a qubit. We develop a proper formalism to evaluate the statistics of such measurement. Generally, we are able to evaluate the joint probability distribution of the detector outcomes and the qubit variables. We concentrate on two setups. The application of our method to the setup where a single pseudospin component is measured gives a comphrehensive picture of quantum non‐demolition measurement. More interesting setup consists of a qubit and three independent detectors that simultaneously monitor three non‐commuting operator variables, those corresponding to three pseudo‐spin components of the qubit. When analyzing the distribution in the limit of big values of the outcomes, we reveal a high degree of correspondence between the three outcomes and three components of the qubit pseudo‐spin after the measurement. This enables a high‐fidelity monitoring of all three components. We discuss the relation between the monitoring described and the algorithms of quantum information theory that use the results of the partial measurement. The formalism is based on Feynman‐Vernon approach, roots in the theory of full counting statistics, and boils down to a Bloch‐Redfield equation augmented with counting fields.
We address the statistics of a simultaneous continuous weak linear measurement of two noncommuting variables on a few-state quantum system subject to a postselected evolution. The results of both postselected quantum measurement and simultaneous monitoring of two noncommuting variables differ drastically from the results of either classical or quantum projective measurement. We explore the peculiarities arising from the combination of the two. We concentrate on the distribution function of two measurement outcomes integrated over a time interval. We formulate a proper formalism for the evaluation of such distribution, and further compute and discuss the resulting statistics for idealized and experimentally relevant setups. We demonstrate the visibility and manifestations of the interference between initial and final states in the statistics of measurement outcomes for both variables in various regimes. We analytically predict the peculiarities at the circle O12+O22=1{\mathcal{O}}_{1}^{2}+{\mathcal{O}}_{2}^{2}=1O12​+O22​=1 in the distribution of measurement outcomes O1,2{\mathcal{O}}_{1,2}O1,2​ in the limit of short measurement times and confirm this by numerical calculation at longer measurement times. We demonstrate analytically the anomalously large values of the time-integrated output cumulants in the limit of short measurement times and zero overlap between initial and final states, and give the detailed distributions for this case. We term this situation sudden jump. We present the numerical evaluation of the probability distributions for experimentally relevant parameters in several regimes and demonstrate that interference effects in the postselected measurement can be accurately predicted even if they are small.
We address the statistics of continuous weak linear measurement on a few-state quantum system that is subject to a conditioned quantum evolution. For a conditioned evolution, both the initial and final states of the system are fixed: the latter is achieved by the postselection in the end of the evolution. The statistics may drastically differ from the nonconditioned case, and the interference between initial and final states can be observed in the probability distributions of measurement outcomes as well as in the average values exceeding the conventional range of nonconditioned averages. We develop a proper formalism to compute the distributions of measurement outcomes, and evaluate and discuss the distributions in experimentally relevant setups. We demonstrate the manifestations of the interference between initial and final states in various regimes. We consider analytically simple examples of nontrivial probability distributions. We reveal peaks (or dips) at half-quantized values of the measurement outputs. We discuss in detail the case of zero overlap between initial and final states demonstrating anomalously big average outputs and sudden jump in time-integrated output. We present and discuss the numerical evaluation of the probability distribution aiming at extending the analytical results and describing a realistic experimental situation of a qubit in the regime of resonant fluorescence.
We consider the evolution of a qubit (spin 1/21/21/2) under the simultaneous continuous measurement of three noncommuting qubit operators ${\stackrel{^}{\ensuremath{\sigma}}}_{x}$, ${\stackrel{^}{\ensuremath{\sigma}}}_{y}$, and ${\stackrel{^}{\ensuremath{\sigma}}}_{z}$. For identical ideal detectors, the qubit state evolves by approaching a pure state with a random direction in the Bloch vector space and by undergoing locally isotropic diffusion in the perpendicular directions. The quantum state conditioned on the complete detector record is used to assess the fidelity of classically inspired estimates based on running time averages and discrete time bin detector outputs.
We analyze the operation of a switching-based detector that probes a qubit's observable that does not commute with the qubit's Hamiltonian, leading to a nontrivial interplay between the measurement and free-qubit dynamics. In order to obtain analytical results and develop intuitive understanding of the different possible regimes of operation, we use a theoretical model where the detector is a quantum two-level system that is constantly monitored by a macroscopic system. We analyze how to interpret the outcome of the measurement and how the state of the qubit evolves while it is measured. We find that the answers to the above questions depend on the relation between the different parameters in the problem. In addition to the traditional strong-measurement regime, we identify a number of regimes associated with weak qubit-detector coupling. An incoherent detector whose switching time is measurable with high accuracy can provide high-fidelity information, but the measurement basis is determined only upon switching of the detector. An incoherent detector whose switching time can be known only with low accuracy provides a measurement in the qubit's energy eigenbasis with reduced measurement fidelity. A coherent detector measures the qubit in its energy eigenbasis and, under certain conditions, can provide high-fidelity information.
The joint state of a continuously monitored quantum system and the classical filtered measurement record has recently been shown to be described by a quantum Fokker-Planck master equation []. We present a deterministic approach to compute the steady state of the system and detector. The method is shown to become particularly efficient in the absence of feedback, which we exploit to develop a perturbative approach valid for weak feedback. We show that through this method we can extract the full counting statistics of the signal, the quantum-classical mutual information between the system and signal, and the Fisher information of the signal, which can be used for sensing applications. Our results are illustrated with both single-qubit models and the spin chains governed by the one-dimensional transverse field Ising model or the Lipkin-Meshkov-Glick model.