Open Systems & Information Dynamics

I review here a small part of Andrzej Kossakowski scientific activity focusing on his pioneering contribution to the evolution of open quantum systems (celebrated Gorini-Kossakowski-Lindblad-Sudarshan (GKLS) master equation) and positive maps in operator algebras. These particular topics turned out to be of primary importance for fundamentals of quantum physics and the rapid development of modern quantum information theory.

When someone disappears, where do their ideas go?

Some achievements of the late Andrzej Kossakowski in the field of statistical physics and quantum theory are presented. We recall also his attempt to find an analytical solution of the 3-dimensional Ising model.

We propose a quantum extension of the Markov-Dobrushin inequality. As an application, we estimate the Markov-Dobrushin constant for some classes of quantum Markov channels, in particular for the Pauli channel, widely studied in quantum information theory.

Starting from the canonical symmetroid ${𝒮}(G)$ associated with a groupoid $G$, the issue of describing dynamical maps in the groupoidal approach to quantum mechanics is addressed. After inducing a Haar measure on the canonical symmetroid ${𝒮}(G)$, the associated von Neumann groupoid algebra is constructed. It is shown that the left-regular representation allows to define linear maps on the groupoid-algebra of the groupoid $G$ and given subsets of functions are associated with completely positive maps. Some simple examples are also presented.

We extend a procedure exploiting spatial indistinguishability of identical particles to recover the spoiled entanglement between two qubits interacting with Markovian noisy environments. Here, the spatially localized operations and classical communication (sLOCC) operational framework is used to activate the entanglement restoration from the indistinguishable constituents. We consider the realistic scenario where noise acts for the whole duration of the process. Three standard types of noises are considered: a phase damping, a depolarizing, and an amplitude damping channel. Within this general scenario, we find the entanglement to be restored in an amount proportional to the degree of spatial indistinguishability. These results elevate sLOCC to a practical framework for accessing and utilizing quantum state protection within a quantum network of spatially indistinguishable subsystems.