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arXiv:cond-mat/0606676 · Physical Review B

Qubit feedback and control with kicked quantum nondemolition measurements: A quantum Bayesian analysis

Andrew N. Jordan, Alexander N. Korotkov

Published Aug 9, 2006• 71 citations
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Abstract

The informational approach to continuous quantum measurement is derived from positive operator-valued measure formalism for a mesoscopic scattering detector measuring a charge qubit. Quantum Bayesian equations for the qubit density matrix are derived, and cast into the form of a stochastic conformal map. Measurement statistics are derived for kicked quantum nondemolition measurements, combined with conditional unitary operations. These results are applied to derive a feedback protocol to produce an arbitrary pure state after a weak measurement, as well as to investigate how an initially mixed state becomes purified with and without feedback.

DOI
10.1103/physrevb.74.085307

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The informational approach to continuous quantum measurement is derived from positive operator-valued measure formalism for a mesoscopic scattering detector measuring a charge qubit. Quantum Bayesian equations for the qubit density matrix are derived, and cast into the form of a stochastic conformal map. Measurement statistics are derived for kicked quantum nondemolition measurements, combined with conditional unitary operations. These results are applied to derive a feedback protocol to produce an arbitrary pure state after a weak measurement, as well as to investigate how an initially mixed state becomes purified with and without feedback.
The quantum evolution of an individual solid-state qubit during the process of its continuous measurement can be described by the recently developed Bayesian formalism. In contrast to the conventional ensemble-averaged formalism, it takes into account the measurement record (in a way similar to the standard Bayesian analysis) and therefore is able to consider individual realizations of the measurement process. The formalism provides testable experimental predictions and can be used for the analysis of a quantum feedback control of solid-state qubits. The Bayesian formalism can be also applied to the continuous measurement of entangled qubits; in particular, it shows how to create a fully entangled pair of qubits without their direct interaction, just by measuring them with an equally coupled detector.
We consider a two-level quantum system (qubit) which is continuously measured by a detector. The conventional formalism, which implies the ensemble averaging, describes the gradual decoherence of the qubit state due to measurement. However, in each particular realization of the measurement process we can have the opposite effect: gradual purification of the qubit density matrix. This can be described by the recently developed Bayesian formalism suitable for individual quantum systems. The purification effect may be verified experimentally using present day technology and can be useful for quantum computing. In particular, the decoherence of a single qubit can be suppressed using continuous measurement and the feedback loop.
We generalize and extend the stochastic path-integral formalism and action principle for continuous quantum measurement introduced in Chantasri, Dressel, and Jordan [Phys. Rev. A 88, 042110 (2013)], where the optimal dynamics, such as the most likely paths, is obtained by extremizing the action of the stochastic path integral. In this work, we apply exact functional methods as well as develop a perturbative approach to investigate the statistical behavior of continuous quantum measurement. Examples are given for the qubit case. For qubit measurement with zero-qubit Hamiltonian, we find analytic solutions for average trajectories and their variances while conditioning on fixed initial and final states. For qubit measurement with unitary evolution, we use the perturbation method to compute expectation values, variances, and multi-time-correlation functions of qubit trajectories in the short-time regime. Moreover, we consider continuous qubit measurement with feedback control, using the action principle to investigate the global dynamics of its most likely paths, and finding that in an ideal case, qubit state stabilization at any desired pure state is possible with linear feedback. We also illustrate the power of the functional method by computing correlation functions for the qubit trajectories with a feedback loop to stabilize the qubit Rabi frequency.
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.
Quantum mechanical systems exhibit an inherently probabilistic nature upon measurement which excludes in principle the singular direct observability continual case. Quantum theory of time continuous measurements and quantum prediction theory, developed by the author on the basis of an independent-increment model for quantum noise and nondemolition causality principle in the 80's, solves this problem allowing continual quantum predictions and reducing many quantum information problems like problems of quantum feedback control to the classical stochastic ones. Using explicit indirect observation models for diffusive and counting measurements we derive quantum filtering (prediction) equations to describe the stochastic evolution of the open quantum system under the continuous partial observation. Working in parallel with classical indeterministic control theory, we show the Markov Bellman equations for optimal feedback control of the a posteriori stochastic quantum states conditioned upon these two kinds of measurements. The resulting filtering and Bellman equation for the diffusive observation is then applied to the explicitly solvable quantum linear-quadratic-Gaussian (LQG) problem which emphasizes many similarities and differences with the corresponding classical nonlinear filtering and control problems and demonstrates microduality between quantum filtering and classical control.
Measurement and feedback control are essential features of quantum science, with applications ranging from quantum technology protocols to information-to-work conversion in quantum thermodynamics. Theoretical descriptions of feedback control are typically given in terms of stochastic equations requiring numerical solutions, or are limited to linear feedback protocols. Here we present a formalism for continuous quantum measurement and feedback, both linear and nonlinear. Our main result is a quantum Fokker-Planck master equation describing the joint dynamics of a quantum system and a detector with finite bandwidth. For fast measurements, we derive a Markovian master equation for the system alone, amenable to analytical treatment. We illustrate our formalism by investigating two basic information engines, one quantum and one classical.