In finite-dimensional quantum mechanics, density matrices provide the most general mathematical description of a quantum physical system. One of the fundamental postulates of quantum mechanics states that the time evolution of a single closed quantum system is unitary.

What happens when we consider composite systems consisting of multiple particles, or when the system is not isolated and can interact with its environment? The most general (physically relevant) evolutions of quantum systems are described by a particular class of linear maps called completely positive trace-preserving maps (also known as quantum channels).

The aim of this talk is to motivate and introduce the concept of completely positive trace-preserving maps, and eventually explore their real counterpart, and discuss some open problems such as the PPT-squared conjecture.