Positive vs Completely Positive

A map that keeps single-system states valid may still fail when applied to part of an entangled pair; only complete positivity is safe.

Two notions of positivity

A linear map E on operators is positive if it sends positive semidefinite operators to positive semidefinite operators, i.e. valid density matrices to valid density matrices when acting alone. It is completely positive if E tensor I_n stays positive for an identity of any size n. Complete positivity is strictly stronger, and it is the condition required for a map to be a legitimate quantum channel.

The transpose counterexample

Matrix transposition is the classic map that is positive but not completely positive. Transposing a lone qubit's density matrix leaves a valid state. But apply transpose to just one half of a Bell state and the joint operator, the partial transpose, acquires a negative eigenvalue, an impossible state. Because the qubit might be secretly entangled with an unseen partner, transposition cannot represent any real physical process on that qubit.

import numpy as np
bell=np.array([1,0,0,1])/np.sqrt(2)
rho=np.outer(bell,bell)
pt=rho.reshape(2,2,2,2).transpose(0,3,2,1).reshape(4,4)
print(np.round(np.linalg.eigvalsh(pt),3))  # has -0.5 -> not CP

Why nature demands complete positivity

Any system we manipulate could be entangled with the rest of the universe. A physical operation must return a valid state no matter what correlations exist, so it must remain positive when extended by the identity on any spectator system. That is exactly complete positivity. A merely positive map would predict negative probabilities in some entangled context, so it cannot be a real evolution.

Consequences

This distinction is not academic. Positive-but-not-CP maps are exactly the tools that detect entanglement: the PPT criterion uses transposition, and the Horodecki theorem says a state is separable if and only if it survives every positive map on one side. Meanwhile the physical channels, characterized by the Choi positivity condition, are precisely the completely positive ones. The line between the two organizes both entanglement theory and the theory of open-system dynamics.