 
Summary: Controllability properties for finite dimensional quantum Markovian master
equations
Claudio Altafini
SISSAISAS
International School for Advanced Studies
via Beirut 24, 34014 Trieste, Italy
Various notions from geometric control theory are used to characterize the behavior of the Marko
vian master equation for Nlevel quantum mechanical systems driven by unitary control and to
describe the structure of the sets of reachable states. It is shown that the system can be accessible
but neither smalltime controllable nor controllable in finite time. In particular, if the generators of
quantum dynamical semigroups are unital, then the reachable sets admit easy characterizations as
they monotonically grow in time. The two level case is treated in detail.
PACS numbers: 03.65.w, 02.60.Cb, 02.30.Mv, 02.70.c
I. INTRODUCTION
The main question that we discuss in this work is the following: to which density operators can we drive the
quantum Markovian master equation by means of coherent control? This problem is of relevance whenever
one is interested in quantum state manipulation in presence of nonunitary evolution, for example in the
context of quantum information processing [5, 17, 19] and of molecular control [23]. The ultimate goal is
obviously to know when and how the state of a quantum mechanical system can be arbitrarily manipulated
by means of unitary (reversible) control operations or at least to what extent this is possible.
