Decoherent Histories and Quantum Maps
by Artur Scherer
Abstract:
Discrete maps play an important role in the investigation
of dynamical features of complex classical systems, especially
within the theory of chaos. Similarly, quantum maps have proven
to be a very useful mathematical tool within the study of complex
quantum dynamical systems. Using the decoherent histories formulation
of quantum mechanics we consider a particular framework for studying
quantum maps which is motivated by the method of classical symbolic
dynamics. Symbolic dynamics is known to be a very powerful method
specifically invented for the purpose of representing classical dynamical
systems by a discrete model that is suitable for information theoretic studies.
Our framework uses the decoherent histories formalism which,
similarly to classical symbolic dynamics, allows one to introduce information
theoretic quantities with respect to system dynamics.
Our research within this framework can be viewed as a contribution
towards the development of a general theory of "quantum
symbolic dynamics".
We start by considering a special but very important example
for a quantum map: the quantum baker's map, invented for
the theoretical investigation of quantum chaos.
Here we use the decoherent histories formalism to examine
the coarse-grained evolution of this map
with regard to the question of how classical predictability of
the evolution depends on the character of coarse-graining.
%is affected by the choice of coarse-graining.
We demonstrate that hierarchical coarse-grainings display
interesting features with respect to this question.
We proceed with the investigation of decoherence properties
of arbitrary unitary quantum maps. A number of interesting results
is obtained within the framework of arbitrarily long histories
constructed from a fixed projective partition of a finite
dimensional Hilbert space. In particular, we derive simple
necessary decoherence conditions, which employ only a single
iteration of a given unitary quantum map. Furthermore,
a surprising result is obtained with regard to the
fundamental question of how the choice of the
initial state affects decoherence of histories.
Within the considered framework we show that if decoherence
is established for arbitrary history lengths and all initial
states from the smallest natural set of states that
can be associated with the framework, then we get decoherence
of such histories for arbitrary initial states.
Finally, we make first steps towards proving analogous
results for approximate decoherence and suggest
various questions for future research.