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.