Quantum Channels — CP Maps and Their Representations

Corresponds to Lectures 17–19. A quantum channel generalizes unitary dynamics to allow noise, measurement back-action, and discarding part of a system — formalized as a completely positive, trace-preserving (CPTP) linear map on density matrices; “completely positive” (positive even when tensored with the identity on an arbitrary ancilla) is the correct generalization of “positive,” ruling out maps that only look physical in isolation. Three equivalent descriptions make different properties easy to check: the Choi matrix (apply the map to half of a maximally entangled state) turns complete positivity into an ordinary positive-semidefiniteness check; Kraus operators $\{K_i\}$ with $\rho\mapsto\sum_iK_i\rho K_i^\dagger$ give an explicit operational form; and the Stinespring dilation shows every channel arises from a unitary on a larger system followed by discarding (partial-tracing out) an environment — the channel version of purification. Closes by classifying the channels a single qubit can undergo (depolarizing, dephasing, amplitude damping, …), the testbed for quantum-error-correction-and-povms.

Outline (TODO — flesh out each)

• Completely positive maps
• Quantum channels — the CPTP definition
• The Choi matrix
• Kraus operators
• The Stinespring dilation
• Qubit channels — depolarizing, dephasing, amplitude damping