CPTPMap

CPTPMap, short for completely positive trace-preserving map, is a mathematical formalism used to describe the most general physical evolution of quantum states. In quantum information theory, it represents a linear map that takes density operators (states) on one Hilbert space to density operators on another, while respecting the probabilistic interpretation of quantum mechanics.

Formally, let Φ be a linear map from operators on a input Hilbert space H_in to operators on an output Hilbert space H_out. Φ is called completely positive and trace-preserving if:

- Trace-preserving: Tr[Φ(ρ)] = Tr[ρ] for all ρ.

- Completely positive: Φ ⊗ I_k is positive for all k, meaning it maps positive semidefinite operators to positive semidefinite operators even when extended to a larger system.

In finite dimensions, CPTP maps admit Kraus representations: there exist operators {K_i} such that Φ(ρ) = Σ_i K_i ρ K_i† and Σ_i K_i† K_i = I_in. If the map is not necessarily trace-preserving but trace-non-increasing, the condition becomes Σ_i K_i† K_i ≤ I_in. The Choi–Jamiołkowski isomorphism provides an alternative view: the Choi matrix J(Φ) = (Φ ⊗ I)(|Ω⟩⟨Ω|) is positive semidefinite, and Φ is trace-preserving if Tr_out J(Φ) = I_in. A related representation is the Stinespring dilation: Φ(ρ) = Tr_E(V ρ V†) for some isometry V from H_in to H_out ⊗ H_E.

Applications include modelling noise in quantum communication and computation, open-system dynamics, and quantum error correction. Examples include unitary channels, depolarizing channels, and amplitude-damping channels.