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AIP conference proceedings

Statistics of Measurement of Non-commuting Quantum Variables

Hongduo Wei, Yuli V. Nazarov, Vladimir Lebedev, M. V. Feigel’man

Published Jan 1, 2009• 0 citations
Publisher

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 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.

DOI
10.1063/1.3149480

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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 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 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 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.
In this paper, we establish a general theoretical framework for the description of continuous quantum measurements and the statistics of the results of such measurements. The framework concerns the measurement of an arbitrary quantum system with arbitrary number of detectors under realistic assumption of instant detector reactions and white noise sources. We attend various approaches to the problem showing their equivalence. The approaches include the full counting statistics (FCS) evolution equation a for pseudo-density matrix, the drift-diffusion equation for a density matrix in the space of integrated outputs, and discrete stochastic updates. We provide the derivation of the underlying equations from a microscopic approach based on full counting statistics method, a phenomenological approach based on Lindblad construction, and interaction with auxiliary quantum systems representing the detectors. We establish the necessary conditions on the phenomenological susceptibilities and noises that guarantee the unambiguous interpretation of the measurement results and the positivity of the density matrix. Our results can be easily extended to describe various quantum feedback schemes where the manipulation decision is based on the values of detector outputs.
Recently, weak measurements have attracted a lot of interest as an experimental method for the investigation of non-classical correlations between observables that cannot be measured jointly. Here, I explain how the complex valued statistics observed in weak measurements relate to the operator algebra of the conventional Hilbert space formalism and show that the algebra of operators originates from more fundamental relations between the physical properties of a quantum system that can be expressed in terms of complex conditional probabilities. In particular, commutation relations can be identified with fundamental imaginary correlations that characterize the relations between physical properties in terms of their transformation dynamics. Non-commutativity thus originates from a definition of relations between physical properties that replaces the assumption of joint reality with a complex-valued probability reflecting the dynamical response of the system to external forces, e.g. in measurement interactions.