Quantum Information Processing, Vol. 7, Nos. 2/3, June 2008 (© 2008)
Optimality of Feedback Control Strategies for Qubit
Howard M. Wiseman
and Luc Bouten
Received November 16, 2007; accepted February 2, 2008; Published online: April 5, 2008
Recently two papers [K. Jacobs, Phys. Rev. A 67, 030301(R)(2003); H. M. Wise-
man and J. F. Ralph, New J. Physics 8,90(2006)] have derived a number of con-
trol strategies for rapid puriﬁcation of qubits, optimized with respect to various
goals. In the former paper the proof of optimality was not mathematically rigor-
ous, while the latter gave only heuristic arguments for optimality. In this paper
we provide rigorous proofs of optimality in all cases, by applying simple con-
cepts from optimal control theory, including Bellman equations and veriﬁcation
KEY WORDS: quantum state preparation; quantum feedback control; optimal
control theory; Bellman equation.
PACS: 03.67.-a; 03.65.Ta; 02.50.Tt; 02.50.Ey.
In the absence of decoherence, monitoring (that is, continuous-in-time
weak measurement) of a qubit observable such as σ
will eventually purify
the qubit. However, the process of puriﬁcation for ﬁnite times can be
affected by applying Hamiltonian controls. Based upon results for dis-
crete (but non-projective) measurements, Jacobs
derived a strategy to
maximize the average purity of the qubit at any given time. His strategy
requires the controller to maintain the qubit state with
= 0. Wiseman
considered two different goals: ﬁrst to maximize the ﬁdelity
Centre for Quantum Computer Technology, Centre for Quantum Dynamics, Grifﬁth
University, Brisbane, QLD 4111, Australia. E-mail: h.wiseman@grifﬁth.edu.au
Physical Measurement and Control 266-33, California Institute of Technology, Pasadena,
CA 91125, USA.
To whom correspondence should be addressed.
1570-0755/08/0600-0071/0 © 2008 Springer Science+Business Media, LLC