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arXiv:quant-ph/0201078 · Physical Review A

Evolution of a qubit under the influence of a succession of weak measurements with unitary feedback

Jürgen Audretsch, Lajos Diósi, Thomas Konrad

Published Aug 26, 2002• 25 citations
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Abstract

We investigate the evolution of a single qubit subject to a continuous unitary ``free'' dynamics and an additional interrupting influence which occurs periodically. One may imagine a dynamically evolving closed quantum system which becomes open at certain times. The interrupting influence is represented by an operation, which is assumed to equivalently describe a nonselective weak measurement. It may be decomposed into the action of a positive operator, which in the case of a measurement represents the pure measurement part, followed by a unitary backaction depending on the result of the measurement (feedback). Equations of motion for the state evolution are derived in the form of difference equations. In the lowest order, the stochastic feedbacks cause a modification of the ``free'' Hamiltonian by an additional appropriately averaged Hamiltonian. The positive operator specifies a decoherence rate and results in a decoherence term. Two higher-order terms are discussed. One shows decoherence induced by the stochastic feedbacks and the other represents generalized friction. The selective evolution is investigated. In order to bridge the gap between sequential and continuous measurements, the continuum limit to a master equation is performed. Additional correcting higher-order terms are worked out in the Appendixes.

DOI
10.1103/physreva.66.022310

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We investigate the evolution of a single qubit subject to a continuous unitary free'' dynamics and an additional interrupting influence which occurs periodically. One may imagine a dynamically evolving closed quantum system which becomes open at certain times. The interrupting influence is represented by an operation, which is assumed to equivalently describe a nonselective weak measurement. It may be decomposed into the action of a positive operator, which in the case of a measurement represents the pure measurement part, followed by a unitary backaction depending on the result of the measurement (feedback). Equations of motion for the state evolution are derived in the form of difference equations. In the lowest order, the stochastic feedbacks cause a modification of the free'' Hamiltonian by an additional appropriately averaged Hamiltonian. The positive operator specifies a decoherence rate and results in a decoherence term. Two higher-order terms are discussed. One shows decoherence induced by the stochastic feedbacks and the other represents generalized friction. The selective evolution is investigated. In order to bridge the gap between sequential and continuous measurements, the continuum limit to a master equation is performed. Additional correcting higher-order terms are worked out in the Appendixes.
We analyze the information that one can learn about the state of a quantum two-level system, i.e. a qubit, when probed weakly by a nearby detector. In particular, we focus on the case when the qubit Hamiltonian and the qubit's operator being probed by the detector do not commute. Because the qubit's state keeps evolving while being probed and because the measurement data is mixed with detector-related background noise, one might expect the detector to fail in this case. We show, however, that under suitable conditions and by proper analysis of the measurement data useful information about the state of the qubit can be extracted. It turns out that the measurement basis is stochastically determined every time the experiment is repeated. We analyze in detail the probability distributions that govern the choice of measurement bases. We also analyze the information acquisition rate and show that it is largely unaffected by the apparent conflict between the measurement and intrinsic qubit dynamics. We discuss the relation between our analysis and the stochastic master equation that describes the evolution of the qubit's state under the influence of measurement and decoherence. In particular, we write down a stochastic equation that encompasses the usual stochastic master equation for the evolution of the qubit's density matrix and additionally contains the measurement information that can be extracted from the observed signal.
One of the broadest concepts of measurement in quantum theory is the generalized measurement. Another paradigm of measurement---arising naturally in quantum optics, among other fields---is that of continuous-time measurements, which can be seen as the limit of a consecutive sequence of weak measurements. They are naturally described in terms of stochastic processes, or time-dependent random variables. We show that any generalized measurement can be decomposed as a sequence of weak measurements with a mathematical limit as a continuous stochastic process. We give an explicit construction for any generalized measurement, and prove that the resulting continuous evolution, in the long-time limit, collapses the state of the quantum system to one of the final states generated by the generalized measurement, being decomposed, with the correct probabilities. A prominent feature of the construction is the presence of a feedback mechanism---the instantaneous choice weak measurement at a given time depends on the outcomes of earlier measurements. For a generalized measurement with nnn outcomes, this information is captured by a real nnn-vector on an nnn-simplex, which obeys a simple classical stochastic evolution.
In quantum mechanics, standard or strong measurement approaches generally result in the collapse of an ensemble of wavefunctions into a stochastic mixture of eigenstates. On the other hand, continuous or weak measurements have the propensity to dynamically control the evolution of quantum states over time, guiding the trajectory of the state into non-trivial superpositions and maintaining state purity. This kind of measurement-induced state steering is of great theoretical and experimental interest for the harnessing of quantum bits or "qubits", which are the fundamental unit of the emerging quantum computer. We explore continuous measurement-based quantum state stabilization through linear feedback control for a single quantum bit. By applying a time-varying Rabi drive that includes a linear feedback term, we show that the fixed points of the continuous measurement may be relocated. Analytical derivations of Itô Stochastic Master Equations being ideal models, we employ them to derive the projected ensemble average while utilizing numerical simulations to characterize the stability of the set of possible fixed points, as well as their modified collapse time-scales. We include the effects of realistic experimental non-idealities, such as environmental energy relaxation, dephasing, time-delay, and inefficient measurement. Ultimately, we discuss potential experimental implementations from collaborating universities.
Suppose a single qubit is repeatedly prepared and evolved under imperfectly-controlled conditions. A drunk model represents uncontrolled interactions on each experimental trial as random or stochastic terms in the qubits Hamiltonian operator. Time evolution of states is generated by a stochastic differential equation whose sample paths evolve according to the Schrödinger equation. For models with Gaussian white noise which is independent of the qubits state, the expectation value of the solution obeys a master equation which is identical to the high-temperature limit of the Bloch equation. Drunk models predict that experimental data can appear consistent with decoherence even if qubit states evolve by unitary transformations. Examples are shown in which reversible evolution appears to cause irreversible information loss. This paradox is resolved by distinguishing between the true state of a system and the estimated state inferred from an experimental dataset.
It has been shown by Oreshkov and Brun that any two-outcome quantum measurement can be decomposed into a continuous stochastic process using a feedback loop. In this article, we characterize which of these decompositions are possible when each iteration of the feedback loop consists of a weak measurement caused by an interaction with a probe system. We restrict ourselves to the case when the probe is a qubit and the interaction Hamiltonian between the probe and system is constant. We find that even given the ability to perform arbitrary unitary pulses throughout the continuous decomposition, only generalized measurements with two distinct singular values are achievable. However, this is sufficient to decompose a generalized qubit measurement using a qubit probe and a simple interaction Hamiltonian.
We characterize the set of generalized quantum measurements that can be decomposed into a continuous measurement process using a stream of probe qubits and a tunable interaction Hamiltonian. Each probe in the stream interacts weakly with the target quantum system and then is measured projectively in a standard basis. This measurement result is used in a closed feedback loop to tune the interaction Hamiltonian for the next probe. The resulting evolution is a stochastic process with the structure of a one-dimensional random walk. To maintain this structure and require that at long times the measurement outcomes be independent of the path, the allowed interaction Hamiltonians must lie in a restricted set such that the Hamiltonian terms on the target system form a finite-dimensional Jordan algebra. This algebraic structure of the interaction Hamiltonians yields a large class of generalized measurements that can be continuously performed by our scheme and we fully describe this set.