Locally Optimal Control of Quantum Systems with Strong Feedback

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 ≤ 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. PACS numbers: 03.65. Yz, 87.19.lr, 02.30.Yy, 03.65.Ta Observation and control of coherent quantum behavior has been realized in a variety of mesoscopic devices [1, 2, 3, 4]. With further refinements, such devices may well form the basis of new technologies, for example in sensing [5] and communication [6,7]. Feedback, in which a system is continuously observed and the information used to control its behavior in the presence of noise, is an important element in the quantum engineer's toolbox [6,7,8,9,10,11]. In view of this, one would like to know the limits on such control, given any relevant limitations on the measurement and/or control forces. However, except in special cases [12], the dynamics of continuously observed quantum systems is nonlinear. Further, results on the quantum-to-classical transition show that this nonlinear dynamics, described by stochastic master equations (SME's), is necessarily every bit as complex (and chaotic) as that of nonlinear classical systems [13]. Because of this, fully general and exact results regarding optimal quantum feedback are unlikely to exist; certainly no such results have been found for nonlinear classical systems [14]. Nevertheless, one would like to obtain results that give insights applicable across a range of systems.Quantum feedback control is implemented by modifying a "control" Hamiltonian, H, that is some part of the system Hamiltonian. Here we will examine feedback protocols in the regime where the controls are able to keep the system close to a pure state. This is an important regime, both because it is where many quantum control systems will need to operate, and because it allows one to simplify the problem by using a power series expansion [15]. In addition to working in the regime of good control, we make two further simplifications. The first is that the control is strong -that is, that 1) the only constraint on H is that Tr[H 2 ] ≤ µ 2 for some constant µ, and 2) that H can induce dynamics much faster than both the dynamics of the system and the rate at which the measurement extracts information. This means that H is effectively unconstrained. We thus deal strictly with a subset of the regime of good control, defined by µ ≫ k and k ≫ β. Here k is the strength of the measurement (defined precisely below), and β is the noise strength, which we define as the rate of increase of the linear entropy due to the noise. The latter inequality is essential for good control. This regime is applicable, for example, to mesoscopic supercond...

Advances in quantum metrology

“…Several Authors [80][81][82][83][84][85][86][87][88][89][90][91][92][93][94][95][96] have recently considered the possibility of using nonlinear effects to go beyond the N −1 Heisenberg-like scalings in phase estimation problems. These new regimes have been called "superHeisenberg" scalings in Ref.…”

“…[97]. Ultimately the idea of these proposals is to consider settings where the unitary transformation that "writes" the unknown parameter x into the probing signals, is characterized by many-body Hamiltonian generators which are no longer extensive functions of the number of probes employed in the estimation [83][84][85][86][87][88][89][90][91][92] or, for the optical implementations which yielded the inequality (6), in the photon number operator of the input signals [80][81][82][83][93][94][95][96]. Consequently, in these setups, the mapping (e −ixH ) ⊗n which acts on the input states ρ (n) 0 , gets replaced by a transformations of the form e −ixH (n) which couples the probes non trivially.…”

Optimality of qubit purification protocols in the presence of imperfections

“…Subsequently, Refs. [7][8][9] generalized this result and showed that in the ideal case it is possible to utilize feedback to increase the instantaneous rate of purification for arbitrary finite dimensional quantum systems. Wiseman and Ralph [10] have noted that it is useful to separate two different goals in the task of quantum state purification: the first goal, which we refer to as max purity, is that of maximizing the average purity of the system at a given time, while the second goal, which we refer to as min time, is that of minimizing the average time taken to achieve a given purity.…”

“…Recall that the general expression for instantaneous average purification rate, which we want to maximize, is given in Eqn. (8). For general γ 1 , γ φ , and η, in order to maximize f (u) we require:…”

Continuous Measurement of a Non-Markovian Open Quantum System

“…Introduction -Generalized or weak quantum measurement has become increasingly important in the last decade due to its application in quantum feedback control [1,2], quantum metrology [1], quantum information [3][4][5], and the study of quantum-classical transitions [6,7]. The existing theories consider continuous weak measurement of simple open quantum systems with Born-Markov decoherence models [5,[8][9][10].…”