Complete positivity and the system

Fabio Benatti, Roberto Floreanini B

Jason Dominy, Daniel A. Lidar

James M. McCracken

Complete positivity is a ubiquitous assumption in the study of quantum systems interacting with the environment. It will be shown that Hamiltonian evolution of a quantum system and its environment can be negative (i.e., not completely positive) in the energy basis, by showing that such evolution is almost always negative for given initial conditions. As such, ignoring or ``correcting'' experimental data that is not completely positive may cause the loss of important information regarding system-environment correlations and coupling. Complete positivity assumptions are an important part of many quantum information theories and it is important to understand how (and why) it appears to be possible to violate the complete positivity requirement in the examples presented here. A relationship between the negativity of an evolution and the eigenvalues of the Hamiltonian will be shown and experimental verification of negative reduced dynamics will be proposed.

Fabio Benatti, Roberto Floreanini B, Marco Piani +1 more

After reviewing the main properties of time-evolutions of open quantum systems, some considerations about the positivity of factorized Markovian dynamics for bipartite systems are made. In particular, it is shown that the positivity of the whole time-evolution in general does not ask for the complete positivity of the single system time-evolutions, if they are allowed to differ. However, they must be completely positive if one is a small perturbation of the other, which is the typical situation for open systems in a heat bath.

Guillaume Théret, Dominique Sugny

We study the concepts of complete positivity and positivity in a two-level open quantum system whose dynamics are governed by a time-local quantum master equation. We establish necessary and sufficient conditions on the time-dependent relaxation rates to ensure complete positivity and positivity of the dynamical map. We discuss their relations with the non-Markovian behavior of the open system. We also analyze the long-time asymptotic behavior of the dynamics as a function of the rates. We show under which conditions on the rates the system tends to an equilibrium state. Different examples illustrate this general study.