The PPT Criterion

The positive partial transpose test detects entanglement and is complete for the smallest bipartite systems.

Partial transpose as a probe

The PPT criterion, due to Peres and the Horodeckis, tests separability. Compute the partial transpose rho^{T_B} of a bipartite state and check whether it remains positive semidefinite. If rho is separable, rho^{T_B} is always positive. Therefore any negative eigenvalue certifies entanglement. The test is a one-line linear-algebra computation and requires no optimization.

When it is complete

For 2x2 (two qubits) and 2x3 (qubit-qutrit) systems, PPT is both necessary and sufficient: a state is separable if and only if its partial transpose is positive. In these dimensions the test never misses entanglement and never gives a false alarm. This completeness is why negativity is a faithful measure for small systems.

import numpy as np
def is_ppt(rho, dA, dB):
    r = rho.reshape(dA,dB,dA,dB).transpose(0,3,2,1).reshape(dA*dB,dA*dB)
    return np.all(np.linalg.eigvalsh(r) > -1e-10)
bell = np.array([1,0,0,1])/np.sqrt(2)
print(is_ppt(np.outer(bell,bell),2,2))  # False -> entangled

The bound-entangled gap

From 3x3 and 2x4 upward, positivity of the partial transpose no longer guarantees separability. There exist PPT states that are nonetheless entangled; their entanglement is bound, meaning no pure Bell pairs can be distilled from it. PPT therefore becomes a sufficient test for separability's failure only: negative eigenvalues still prove entanglement, but a positive result is no longer a clean bill of separability.

Why it works

Transposition is a positive map that is not completely positive. Separable states, being convex mixtures of products, stay positive under a positive map applied to one side. Entangled states can be exposed by such maps. The Horodecki theorem generalizes this: a state is separable if and only if it stays positive under every positive map applied to one subsystem. PPT is the special, computable case using transposition, and it captures all entanglement precisely when every relevant positive map decomposes in terms of transposition, which happens only in the smallest dimensions.