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arXiv:0803.2705 · Physical Review Letters

Locally Optimal Control of Quantum Systems with Strong Feedback

Alireza Shabani, Kurt A. Jacobs

Published Dec 3, 2008• 35 citations
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Abstract

For quantum systems with high purity, we find all observables that, when continuously monitored, maximize the instantaneous reduction in the average linear entropy. This allows us to obtain all locally optimal feedback protocols with strong feedback, and explicit expressions for the best such protocols for systems of size N < or =4. We also show that for a qutrit the locally optimal protocol is the optimal protocol for observables with equispaced eigenvalues, providing the first fully optimal feedback protocol for a 3-state system.

DOI
10.1103/physrevlett.101.230403

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For quantum systems with high purity, we find all observables that, when continuously monitored, maximize the instantaneous reduction in the average linear entropy. This allows us to obtain all locally optimal feedback protocols with strong feedback, and explicit expressions for the best such protocols for systems of size N < or =4. We also show that for a qutrit the locally optimal protocol is the optimal protocol for observables with equispaced eigenvalues, providing the first fully optimal feedback protocol for a 3-state system.
It was first shown by Jacobs, in 2003, that the process of qubit state purification by continuous measurement of one observable can be enhanced, on average, by unitary feedback control. Here, we quantify this by the reduction in any one of the family of Rényi entropies , with , at some terminal time, revealing the rich structure of stochastic quantum control even for this simple problem. We generalize Jacobs' original argument, which was for the (unique) impurity measure with a linear evolution map under his protocol, by replacing linearity with convexity, thereby making it applicable to Rényi entropies for α in a finite interval about one. Even with this generalization, Jacobs' argument fails to identify the surprising fact, which we prove by Bellmanʼs principle of dynamic programming, that his protocol is globally optimal for all Rényi entropies whose decrease is locally maximized by that protocol. Also surprisingly, even though there is a range of Rényi entropies whose decrease is always locally maximized by the null-control protocol, that null-control protocol cannot be shown to be globally optimal in any instance. These results highlight the non-intuitive relation between local and global optimality in stochastic quantum control.
We present a general approach to measurement-based quantum feedback that employs proportional and quantum state--(PaQS-) based feedback components to obtain locally optimal protocols. To demonstrate the power of the method, we first show that it reproduces many known feedback protocols and then apply it to generation of multipartite entanglement with an emphasis on remote entanglement, which requires spatially local feedback Hamiltonians. The symmetry of both measurement and feedback operators is found to be essential for construction of effective protocols. We show that under perfect measurement efficiency, entangled states can be reached with fidelity approaching unity under non-Markovian feedback control protocols, while Markovian protocols resulting from optimizing the feedback unitaries on ensemble-averaged states still yield fidelities above 94%. Application of the PaQS approach to generation of NNN-qubit WWW, general Dicke, and Greenberger-Horne-Zeilinger states shows that such entangled states can be efficiently generated with high fidelity, for up to N=100N=100N=100 in some cases.
Feedback control of quantum systems via continuous measurement involves complex nonlinear dynamics. Except in very special cases, even for a single qubit optimal feedback protocols are unknown. Not even do intuitive candidates exist for choosing the measurement basis, which is the primary non-trivial ingredient in the feedback control of a qubit. Here we present a series of arguments that suggest a particular form for the optimal protocol for a broad class of noise sources in the regime of good control. This regime is defined as that in which the control is strong enough to keep the system close to the desired state. With the assumption of this form the remaining parameters can be determined via a numerical search. The result is a non-trivial feedback protocol valid for all feedback strengths in the regime of good control. We conjecture that this protocol is optimal to leading order in the small parameters that define this regime. The protocol can be described relatively simply, and as a notable feature contains a discontinuity as a function of the feedback strength.
Abstract. The fundamental mathematical definitions of the controlled feedback Markov dynamics of quantum-mechanical systems are introduced with regard to the dynamical reduction and filtering of quantum states in the course of quantum measurement in either discrete or continuous real time. The concept of sufficient coordinates for the description of a posteriori quantum states in a given class is introduced and it is proved that they form a classical Markov process with values in either state operators or state vector space. The general problem of optimal control of a quantum-mechanical system is discussed and the corresponding Bellman equation in the space of sufficient coordinates is derived. The results are illustrated in the example of control of the semigroup dynamics of a quantum system that is instantaneously observed at discrete times and evolves between measurement times according to the Schrődinger equation. 1.
This paper reconsiders the claimed rapidity of a scheme for the purification of the quantum state of a qubit, proposed recently in Jacobs (2003 Phys. Rev. A 67 030301(R)). The qubit starts in a completely mixed state, and information is obtained by a continuous measurement. Jacobs' rapid purification protocol uses Hamiltonian feedback control to maximize the average purity of the qubit for a given time , with a factor of two increase in the purification rate over the no-feedback protocol. However, by re-examining the latter approach, we show that it minimizes the average time taken for a qubit to reach a given purity . In fact, the average time taken for the no-feedback protocol beats that for Jacobs' protocol by a factor of two. We discuss how this is compatible with Jacobs' result and the usefulness of the different approaches.