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arXiv:quant-ph/0701117 · Physical Review A

Decomposing generalized measurements into continuous stochastic processes

Martin Varbanov, Todd A. Brun

Published Sep 5, 2007• 10 citations
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Abstract

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 $n$ outcomes, this information is captured by a real $n$-vector on an $n$-simplex, which obeys a simple classical stochastic evolution.

DOI
10.1103/physreva.76.032104

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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.
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.
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.
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.
The topics presented in this thesis have continuous quantum measurements and quantum walks at their core. The first topic being discussed centers around simulating a generalize measurement with a finite number of outcomes using a continuous measurement process with a continuous measurement history. We provide conditions under which it is possible to prove that such a process exists and that at long times it simulates faithfully the generalized measurement. We give the stochastic equations governing the feedback between the measurement history and the instantaneous weak measurements. The second topic examines a definition of hitting time for continuous-time quantum walks. A crucial component for such a definition is the use of weak measurements. Several methods using alternative but equivalent definitions of weak, continuous measurements are employed to derive a formula for the hitting time. The behavior of the thus defined hitting time is studied subsequently, in general and for specific graphs. The last topic explores continuous-time quantum walks on graphs with infinite tails. The equations for propagating and bound states are derived and the S-matrix is defined. Their properties, such as orthogonality of the propagating and bound states, unitarity of the S-matrix, are discussed. Formulas for the S-matrix under operations of cutting, adding or connecting tails are derived.
It has been shown that any generalized measurement can be decomposed into a sequence of weak measurements corresponding to a stochastic process. However, the weak measurements may require almost arbitrary unitaries, which are unlikely to be realized by any real measurement device. Furthermore, many measurement processes are destructive, like photon counting procedures that terminate once all photons are consumed. One cannot expect to have full control of the evolution of a state under such destructive measurements, and the possible unitaries allow only a limited set of weak measurements. In this paper, we consider a qubit model of destructive weak measurements, which is a toy version of an optical cavity, in which the state of an electromagnetic field mode inside the cavity leaks out and is measured destructively while the vacuum state $|0\ensuremath{\rangle}$ leaks into the cavity. At long times, the state of the qubit inevitably evolves to be $|0\ensuremath{\rangle}$, and the only available control is the choice of measurement on the external ancilla system. Surprisingly, this very limited model can still perform an arbitrary projective measurement on the qubit in any basis, where the probability of getting an outcome satisfies the usual Born rule. Combining this method with probabilistic postprocessing, the result can be extended to any generalized measurement with commuting positive-operator-valued measurement (POVM) elements. This implies, among other results, that any two-outcome POVM on a qubit can be decomposed into a sequence of destructive weak measurements by this restricted measurement device.