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IEEE Conference on Decision and Control (CDC)

Feedback control of quantum entanglement in a two-spin system

Naoki Yamamoto, Koji Tsumura, Shinji Hara

Published Oct 4, 2006• 13 citations
Publisher

Abstract

A pair of spins is the most simple quantum system that can possess entanglement, a non-classical property playing an essential role in quantum information technologies. In this paper, feedback control problems of the two-spin system conditioned on a continuous measurement are considered. In order to make some useful formulas in stochastic control theory directly applicable, we first derive a two-dimensional description of the system. We then prove that a feedback controller stabilizes an entangled state of the two spins almost globally with probability one. Furthermore, it is shown that another entangled state, which corresponds to a non-equilibrium point of the dynamics, is stabilized via feedback in the sense that the expectation of the distance from the target can be made arbitrarily small.

DOI
10.1109/cdc.2005.1582651

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•Jan 1, 2005•184 citations
A pair of spins is the most simple quantum system that can possess entanglement, a non-classical property playing an essential role in quantum information technologies. In this paper, feedback control problems of the two-spin system conditioned on a continuous measurement are considered. In order to make some useful formulas in stochastic control theory directly applicable, we first derive a two-dimensional description of the system. We then prove that a feedback controller stabilizes an entangled state of the two spins almost globally with probability one. Furthermore, it is shown that another entangled state, which corresponds to a non-equilibrium point of the dynamics, is stabilized via feedback in the sense that the expectation of the distance from the target can be made arbitrarily small.
In this paper, a stabilization problem of quantum spin systems in general dimension under continuous measurement is considered and it is shown that the global stabilization at an eigenstate by continuous inputs is possible. Quantum states of spin systems under continuous measurement by mutual interference with laser beams can be continuously estimated by quantum filtering and with its information, the intension of magnetic field, which is applied to atoms, can be controlled. Because of the structural symmetry of the nonlinear dynamics, this stabilization problem by continuous feedback has been considered to be hard and recently a switching rule was introduced which attains the global stability. Our proposing control rule is the sum of two terms: a term which attracts the quantum states to a target and the other term which draw apart from the other equilibrium points. The proof is done by the strict analysis on the sample paths of solutions of a stochastic differential equation. We also show numerical examples to demonstrate the efficiency of the proposing control rule.
No quantum measurement can give full information on the state of a quantum system; hence any quantum feedback control problem is necessarily one with partial observations and can generally be converted into a completely observed control problem for an appropriate quantum filter as in classical stochastic control theory. Here we study the properties of controlled quantum filtering equations as classical stochastic differential equations. We then develop methods, using a combination of geometric control and classical probabilistic techniques, for global feedback stabilization of a class of quantum filters around a particular eigenstate of the measurement operator.
In this paper, a stabilization problem for a quantum 1/2 spin system is discussed. The system is assumed to be observed and controlled with a sampled data system. We introduce a discrete time stochastic model of the dynamics and propose a switching feedback controller. We show that the controller attains global stabilization at the target states under a condition on a sampling time by using a stochastic Lyapunov function for the discrete time systems. Finally, we demonstrate numerical simulations to show the effectiveness of our proposing controller.
In this paper, feedback control of quantum spin systems is investigated. The quantum control scenario follows the formalism of continuous quantum measurement and quantum state estimation using the so-called stochastic master equation. In quantum feedback control implementations the state estimation is delayed, which is known to degrade the resulting control performance. As yet there are no theoretical means to take such estimation delays into consideration. In this paper we derive a globally stabilizing control law for two-dimensional quantum spin systems taking estimation delays into explicit account. From a control theoretic viewpoint, the obtained result is a delay-dependent stability criterion for a nonlinear stochastic switching system.
Abstract. No quantum measurement can give full information on the state of a quantum system; hence any quantum feedback control problem is necessarily one with partial observations and can generally be converted into a completely observed control problem for an appropriate quantum filter as in classical stochastic control theory. Here we study the properties of controlled quantum filtering equations as classical stochastic differential equations. We then develop methods, using a combination of geometric control and classical probabilistic techniques, for global feedback stabilization of a class of quantum filters around a particular eigenstate of the measurement operator.