CPTP Maps

Completely positive, trace-preserving maps are exactly the transformations a quantum state can physically undergo.

Two conditions define physicality

A map E on density matrices qualifies as a physical evolution if it is trace-preserving and completely positive. Trace preservation, Tr(E(rho)) = Tr(rho), keeps total probability at one. Complete positivity is the subtle condition: not only must E map positive operators to positive operators, but E tensor I must do so on any enlarged system. A map with both properties is CPTP, and CPTP maps are precisely the quantum channels.

Why positivity is not enough

Some maps are positive but not completely positive; transposition is the standard example. Transposing a single qubit keeps that qubit's state valid, but applied to half of an entangled pair it can produce negative eigenvalues, an unphysical result. Since any subsystem might secretly be entangled with something we cannot see, only complete positivity guarantees validity in every context. This distinction is the basis of the PPT entanglement criterion.

Structure theorem

Every CPTP map admits a Kraus decomposition E(rho) = sum_k K_k rho K_k-dagger with the completeness relation sum_k K_k-dagger K_k = I. Equivalently, by the Stinespring dilation, it can be realized as a unitary on the system plus an ancilla, followed by discarding the ancilla. Trace preservation corresponds to the completeness relation, and complete positivity is automatic in both forms.

import numpy as np
K0 = np.array([[1,0],[0,np.sqrt(0.9)]])
K1 = np.array([[0,np.sqrt(0.1)],[0,0]])
print(np.round(K0.conj().T@K0 + K1.conj().T@K1, 3))  # I -> valid CPTP

Composition and reversibility

CPTP maps are closed under composition: a sequence of channels is a channel, which lets circuits with noise be modeled end to end. Most channels are irreversible; only unitary channels have CPTP inverses. This asymmetry is the mathematical shadow of the second law inside quantum information: information leaks to the environment and cannot generally be pulled back by any physical operation.