Emergence of order in quasi-species evolution: classical and quantum

Harald A. Posch

Faculty of Physics, University of Vienna,

Boltzmanngasse 5, 1090 Vienna, Austria

Harald.Posch@univie.ac.at

We study evolution equations which model selection and mutation, both

classically and within the framework of quantum mechanics. The main question

is to what extent order is achieved for an ensemble of typical systems. As an

indicator for mixing or purification, a quadratic entropy is used, which

assumes values between zero, for pure states, and (d-1)/d, for

fully mixed states. Here, d is the dimension. Classically, a state of the

system is given by d positive numbers which sum to unity and which are

interpreted as probabilities, as relative populations, or as financial assets.

A pure state is obtained, if one of the p is unity and all the others vanish.

In quantum mechanics a pure state is characterized by a density matrix which

projects onto a general vector in Hilbert space.

Whereas the classical quasi-species model is found to be predominantly

mixing. the quantum quasi-species (QS) evolution, surprisingly, is found to

be strictly purifying for all dimensions. This is also typically true for

an alternative formulation (AQS) of this quantum mechanical flow. We compare

this also to analogous results for the Lindblad evolution. Although the

latter may be viewed as a simple linear superposition of the purifying QS and

AQS evolutions, it is found to be predominantly mixing. The reason for this

behavior may be explained by the fact that the two subprocesses by themselves

converge to different pure states, such that the combined process is mixing.

These results apply also to high-dimensional systems.