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arXiv:1504.03772 · Physical Review A

Continuous decomposition of quantum measurements via Hamiltonian feedback

Jan Florjanczyk, Todd A. Brun

Published Dec 9, 2015• 3 citations
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Abstract

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.

DOI
10.1103/physreva.92.062113

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One of the abstract concepts of measurement in quantum information theory is the \\textit{generalized measurement}, a special subclass of which is the \\textit{positive-operator value measurements}. A different paradigm of measurement comes from Quantum Optics, when one considers measurements continuous in time. Their natural description involves the concept of \\textit{a stochastic process}--a time-dependent random variable. In this paper we construct a \\textit{continuous} stochastic process to simulate any generalized measurement--in the long time limit it gives the same outcomes with the right probabilities as the measurement being simulated. This stochastic evolution takes the form of a random walk in a space of measurement parameters, and the continuous measurement process must in general take into account the outcomes of earlier measurements.

•Jan 17, 2007•0 citations
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.
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.
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.
Quantum Markov models have been successfully used to accurately model various physical quantum systems in fields such as quantum optics, optomechanics and superconducting circuits and they provide the basis for (measurement-based) quantum feedback control. However, the quantum Markov assumption is a strong one and it is not expected to hold for general quantum systems of interest. The projection operator approach is one approach that has been developed to model non-Markovian quantum systems by considering its embedding in a larger Markovian quantum system, but mainly in the context of quantum master equations for the dynamics of the unmonitored reduced quantum state of a quantum system. This approach was recently adapted for continuously measured non-Markovian quantum systems, which enables open-loop control but did not yet consider the presence of feedback of the stochastic measurement record, deriving non-Markovian SDEs for the evolution of the projected state of the Markovian embedding. This paper generalizes these stochastic equations to the setting of stochastic feedback based on the continuous-measurement record and shows that the equations take the same form but that previously deterministic terms become stochastic ones which depend on the measurement record, as would be intuitively expected. The stochastic equations are obtained for a generalized class of measurements that includes continuous (possibly adaptive) homodyne and photon counting measurements.
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.
One of the abstract concepts of measurement in quantum information theory is the \\textit{generalized measurement}, a special subclass of which is the \\textit{positive-operator value measurements}. A different paradigm of measurement comes from Quantum Optics, when one considers measurements continuous in time. Their natural description involves the concept of \\textit{a stochastic process}--a time-dependent random variable. In this paper we construct a \\textit{continuous} stochastic process to simulate any generalized measurement--in the long time limit it gives the same outcomes with the right probabilities as the measurement being simulated. This stochastic evolution takes the form of a random walk in a space of measurement parameters, and the continuous measurement process must in general take into account the outcomes of earlier measurements.