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arXiv:quant-ph/0503206 · Physical Review A

Markovian feedback to control continuous-variable entanglement

Stefano Mancini

Published Jan 20, 2006• 52 citations
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Abstract

We present a model to realize quantum feedback control of continuous variable entanglement. It consists of two interacting bosonic modes subject to amplitude damping and achieving an entangled Gaussian steady state. The possibility to greatly improve the degree of entanglement by means of Markovian (direct) feedback is then shown.
DOI
10.1103/physreva.73.010304

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We determine a general upper bound for the steady-state entanglement achievable by continuous feedback for a system of any number of bosonic degrees of freedom. We apply such a bound to the specific case of parametric interactions--the most common practical way to generate entanglement in quantum optics--and single out optimal feedback strategies that achieve the maximal entanglement. We also consider the case of feedback schemes entirely restricted to local operations and compare their performance to the optimal, generally nonlocal, schemes.
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We consider the system consisting of two qubits collectively damped, with the output being unit-efficiency measured and subsequently fed back to control the system state. Our primary goal in this paper is (i) to solve the feedback-modified master equation, (ii) to demonstrate the ability of feedback control based on the solutions, and (iii) to pick out different steady states by choosing different driving strengths and feedback strengths to counteract the effects of both damping and the measurement back-action on the system. We further investigate some properties of the equilibrium steady state, its distribution probability and entanglement vs. the driving and feedback amplitudes. We find that in our feedback model feedback plays a negative role in producing entanglement.
It has recently been shown that finding the optimal measurement on the environment for stationary linear quadratic Gaussian control problems is a semidefinite program. We apply this technique to the control of the Einstein-Podolsky-Rosen correlations between two bosonic modes interacting via a parametric Hamiltonian at steady state. The optimal measurement turns out to be nonlocal homodyne measurement---the outputs of the two modes must be combined before measurement. We also find the optimal local measurement and control technique. This gives the same degree of entanglement but a higher degree of purity than the local technique previously considered [S. Mancini, Phys. Rev. A 73, 010304(R) (2006)].
We consider the system consisting of two qubits collectively damped,with the output being unit-efficiencymeasured and subsequently fed back to control the system state.Our primary goal in this paper is (ⅰ) to solve thefeedback-modified master equation,(ⅱ) to demonstrate the ability of feedback control based on the solutions,and (ⅲ) topick out different steady states by choosing different driving strengths and feedback strengths to counteract the effects ofboth damping and the measurement back-action on the system.We further investigate some properties of the equilibriumsteady state,its distribution probability and entanglement vs.the driving and feedback amplitudes.We find that in ourfeedback model feedback plays a negative role in producing entanglement.
We develop a feedback strategy based on optimal quantum feedback control for Gaussian systems to maximize the likelihood of steady-state entanglement detection between two directly interacting masses. We employ linear quadratic Gaussian control to engineer the phase-space dynamics of the two masses and propose Einstein-Podolsky-Rosen-type variance minimization constraints for the feedback to facilitate unconditional entanglement generation. This scheme allows for stationary entanglement in parameter regimes where strategies based on total energy minimization () would fail.