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, and trace preservation. Complete positivity means that for every auxiliary system of dimension n, the extended map Phi ⊗ I_n takes positive operators to positive operators. Trace preservation means Tr(Phi(ρ)) = Tr(ρ) for all input states ρ, ensuring total probability is conserved.

In finite dimensions, CPTP maps admit Kraus representations: Phi(ρ) = ∑_k E_k ρ E_k† with ∑_k E_k† E_k = I. The Kraus operators {E_k} are not unique. A unitary channel is a special case with a single Kraus operator E = U, since U†U = I.

The Choi–Jamiołkowski isomorphism provides a one-to-one correspondence between CPTP maps and positive semidefinite operators, the Choi matrix J(Phi). Phi is CP iff J(Phi) ≥ 0, and Phi is TP iff Tr_out J(Phi) = I_in. This framework also leads to Stinespring dilation, which represents Phi(ρ) as a partial trace over an environment: Phi(ρ) = Tr_E(V ρ V†) for some isometry V.

CPTP maps model noise processes such as depolarizing, amplitude-damping, and phase-damping channels, as well as general quantum operations used in computation and communication. They form the mathematically standard description of physically realizable state evolution.