CPTPmaps
CPTP maps, short for completely positive trace-preserving maps, are linear transforms that describe the physical evolution of quantum states. They map density operators on an input Hilbert space to density operators on an output Hilbert space and are central to quantum information theory as models of quantum channels and open-system dynamics.
A map is CPTP if it satisfies three properties: complete positivity, positivity when extended to larger systems,
In finite dimensions, CPTP maps admit Kraus representations: Phi(ρ) = ∑_k E_k ρ E_k† with ∑_k E_k† E_k
The Choi–Jamiołkowski isomorphism provides a one-to-one correspondence between CPTP maps and positive semidefinite operators, the Choi
CPTP maps model noise processes such as depolarizing, amplitude-damping, and phase-damping channels, as well as general