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arXiv:1608.06858 · Physical review. B./Physical review. B

Probability distributions of continuous measurement results for conditioned quantum evolution

A. Franquet, Yuli V. Nazarov

Published Feb 21, 2017• 7 citations
arXivHTMLPDF

Abstract

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.

DOI
10.1103/physrevb.95.085427

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Quantum walks have been studied under several regimes. Motivated by experimental results on quantum weak measurements and weak values as well as by the need to develop new insights for quantum algorithm development, we are extending our knowledge by studying the behavior of quantum walks under the regime of quantum weak measurements and weak values of pre- and postselected measurements (QWWM hereinafter). In particular, we investigate the limiting position probability distribution and several statistical measures (such as standard deviation) of a QWWM on an infinite line, and compare such results with corresponding classical and quantum walks position probability distributions and statistical measures, stressing the differences provided by weak measurements and weak values with respect to results computed by using canonical observables. We start by producing a concise introduction to quantum weak values and quantum weak measurements. We then introduce definitions as well as both analytical and numerical results for a QWWM under Hadamard evolution and extend our analysis to quantum evolution ruled by general unitary operators. Moreover, we propose a definition and focus on the properties of mixing time of QWWM on an infinite line, followed by a comparison of known corresponding results for classical and quantum walks mixing times. We finish this paper by presenting a plausible experimental implementation of a QWWM.

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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 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 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 peculiarities of the quantum measurement process in the course of a continuous weak linear measurement (CWLM). As a tool, we implement an efficient numerical simulation scheme that allows us to generate single quantum trajectories of the measured system state as well as the recorded detector signal, and study statistics of these trajectories with and without post-selection. In this scheme, a linear detector is modelled with a qubit that is weakly coupled to the quantum system measured and is subject to projective measurement and re-initialization after a time interval at each simulation step. We explain the conditions under which the scheme provides an accurate description of CWLM. We restrict ourselves to a qubit non-demolition measurement. The qubit is initially in an equal-weight superposition of two quantum states. In the course of time, the detector signal is accumulated and the superposition is destroyed. The times required to resolve the quantum states and to destroy the superposition are of the same order. We prove numerically a rather counter intuitive fact: the average detector output conditioned on the final state does not depend on time. It looks like from the very beginning, the qubit knows in which state it is. We study statistics of decision times where the decision time is defined as time required for the density matrix along a certain trajectory to reach a threshold where it is close to one of the resulting states. This statistics is useful to estimate how fast a decisive CWLM can be. Basing on this, we devise and study a simple feedback scheme that attempts to keep the qubit in the equal-weight superposition. The detector readings are used to decide in which state the qubit is and which correction rotation to apply to bring it back to the superposition. We show how to optimize the feedback parameters and move towards more efficient feedback schemes.
Abstract The dynamics of a quantum system, undergoing unitary evolution and continuous monitoring, can be described in term of quantum trajectories. Although the averaged state fully characterizes expectation values, the entire ensemble of stochastic trajectories goes beyond simple linear observables, keeping a more attentive description of the entire dynamics. Here we go beyond the Lindblad dynamics and study the probability distribution of the expectation value of a given observable over the possible quantum trajectories. The measurements are applied to the entire system, having the effect of projecting the system into a product state. We develop an analytical tool to evaluate this probability distribution at any time t. We illustrate our approach by analyzing two paradigmatic examples: a single qubit subjected to magnetization measurements, and a free hopping particle subjected to position measurements.
Quantum walks have been studied under several regimes. Motivated by experimental results on quantum weak measurements and weak values as well as by the need to develop new insights for quantum algorithm development, we are extending our knowledge by studying the behavior of quantum walks under the regime of quantum weak measurements and weak values of pre- and postselected measurements (QWWM hereinafter). In particular, we investigate the limiting position probability distribution and several statistical measures (such as standard deviation) of a QWWM on an infinite line, and compare such results with corresponding classical and quantum walks position probability distributions and statistical measures, stressing the differences provided by weak measurements and weak values with respect to results computed by using canonical observables. We start by producing a concise introduction to quantum weak values and quantum weak measurements. We then introduce definitions as well as both analytical and numerical results for a QWWM under Hadamard evolution and extend our analysis to quantum evolution ruled by general unitary operators. Moreover, we propose a definition and focus on the properties of mixing time of QWWM on an infinite line, followed by a comparison of known corresponding results for classical and quantum walks mixing times. We finish this paper by presenting a plausible experimental implementation of a QWWM.