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arXiv:quant-ph/0510222

Almost Global Stochastic Feedback Stabilization of Conditional Quantum Dynamics

Claudio Altafini, Francesco Ticozzi

Published Oct 28, 2005• 21 citations
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Abstract

We propose several parametrization-free solutions to the problem of quantum state reduction control by means of continuous measurement and smooth quantum feedback. In particular, we design a feedback law for which almost global stochastic feedback stabilization can be proved analytically by means of Lyapunov techinques. This synthesis arises very naturally from the physics of the problem, as it relies on the variance associated with the quantum filtering process.

DOI
10.48550/arxiv.quant-ph/0510222

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We propose several parametrization-free solutions to the problem of quantum state reduction control by means of continuous measurement and smooth quantum feedback. In particular, we design a feedback law for which almost global stochastic feedback stabilization can be proved analytically by means of Lyapunov techinques. This synthesis arises very naturally from the physics of the problem, as it relies on the variance associated with the quantum filtering process.
Feedback control of quantum mechanical systems must take into account the probabilistic nature of quantum measurement. We formulate quantum feedback control as a problem of stochastic nonlinear control by considering separately a quantum filtering problem and a state feedback control problem for the filter. We explore the use of stochastic Lyapunov techniques for the design of feedback controllers for quantum spin systems and demonstrate the possibility of stabilizing one outcome of a quantum measurement with unit probability.
We consider a class of pure-state preparation problems for stochastic quantum dynamics, by means of Hamiltonian control, continuous measurement and quantum feedback, in the presence of a Markovian environment. We prove that, whenever suitable dissipative effects are induced either by the unmonitored environment or by continuous-time measurements, open-loop time-invariant control is in principle sufficient to achieve stabilization of the target state (in probability). When this is not sufficient, we show that state stabilization can be attained for a wide class of models by the addition of a switching, filtering-based feedback control Hamiltonian.
We consider a class of pure-state preparation problems for stochastic quantum dynamics, by means of Hamiltonian control, continuous measurement and quantum feedback, in the presence of a Markovian environment. We prove that, whenever suitable dissipative effects are induced either by the unmonitored environment or by continuous-time measurements, open-loop time-invariant control is in principle sufficient to achieve stabilization of the target state (in probability). When this is not sufficient, we show that state stabilization can be attained for a wide class of models by the addition of a switching, filtering-based feedback control Hamiltonian.
In this paper, we investigate parametrization-free solutions of the problem of quantum pure state preparation and subspace stabilization by means of Hamiltonian control, continuous measurement, and quantum feedback, in the presence of a Markovian environment. In particular, we show that whenever suitable dissipative effects are induced either by the unmonitored environment, or by non-Hermitian measurements, there is no need for feedback, as open-loop time-invariant control is sufficient to achieve stabilization of the target set in probability. Constructive necessary and sufficient conditions on the form of the control Hamiltonian can be provided in this case. When time-invariant control is not sufficient, state stabilization can be attained by the addition of filtering-based feedback control.
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.
This paper presents a detailed Lyapunov-based theory to control and stabilize continuously-measured quantum systems, which are driven by Stochastic Schrodinger Equation (SSE). Initially, equivalent classes of states of a quantum system are defined and their properties are presented. With the help of equivalence classes of states, we are able to consider global phase invariance of quantum states in our mathematical analysis. As the second mathematical modelling tool, the conventional Ito formula is further extended to non-differentiable complex functions. Based on this extended Ito formula, a detailed stochastic stability theory is developed to stabilize the SSE. Main results of this proposed theory are sufficient conditions for stochastic stability and asymptotic stochastic stability of the SSE. Based on the main results, a solid mathematical framework is provided for controlling and analyzing quantum system under continuous measurement, which is the first step towards implementing weak continuous feedback control for quantum computing purposes.