Quantum Information Processing, Vol. 6, No. 1, February 2007 (© 2006)
Feedback Control of Spin Systems
Received March 17, 2006; accepted June 9, 2006; Published online December 7, 2006
The feedback stabilization problem for ensembles of coupled spin 1/2 systems is
discussed from a control theoretic perspective. The noninvasive nature of the bulk
measurement allows in principle for a fully unitary and deterministic closed loop.
The Lyapunov-based feedback design presented does not require spins that are selec-
tively addressable. With this method, it is possible to obtain control inputs also for
difﬁcult tasks, like suppressing undesired couplings in identical spin systems.
KEY WORDS: Quantum control; spin systems; feedback stabilization; spin
PACS: 03.65.-w; 02.60.Cb; 02.30.Mv; 02.70.-c.
NMR spectroscopy deals with the manipulation of nuclear spins of
quantum ensembles (see Refs. 2 and 12). These systems exhibit most of
the essential features of quantum mechanical systems, like the state space
of tensorial type (providing exponential growth of the degrees of freedom
available) and natural coupling mechanisms between spins, which guar-
antee the nonclassical nonlocality characteristic of quantum evolutions.
For the purposes of state manipulation, over the last 40 years the ﬁeld of
NMR has developed an extremely versatile and universally accepted set of
tools, in the form of sequences of electromagnetic pulses.
of classical control theory, these would be classiﬁed as open loop control
methods. From the perspective of control theory, open-loop methods are
by far more complicated and less robust than closed-loop methods, in which
one or more functions of the state variables are measured on-line and used
to impose a behavior to the system through a feedback algorithm, or sim-
ply to reject errors and disturbances deviating the state from its desired
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