Negativity

Negativity measures entanglement by how much the partial transpose of a state fails to be positive, computable for any dimension.

From partial transpose to a measure

Take a bipartite state rho_AB and transpose only the B indices to form rho^{T_B}. This partial transpose is still Hermitian with unit trace, but for entangled states some of its eigenvalues can go negative. The negativity is N(rho) = (||rho^{T_B}||_1 - 1)/2, the sum of the absolute values of the negative eigenvalues. It is zero for states with positive partial transpose and grows with entanglement.

Logarithmic negativity

A closely related quantity is the logarithmic negativity E_N = log2 ||rho^{T_B}||_1. Unlike most entanglement measures it is additive and gives an upper bound on distillable entanglement. Both quantities are entanglement monotones: they cannot increase under local operations and classical communication, which is the defining requirement of a legitimate measure.

import numpy as np
def negativity(rho, dA, dB):
    r = rho.reshape(dA,dB,dA,dB).transpose(0,3,2,1).reshape(dA*dB,dA*dB)
    ev = np.linalg.eigvalsh(r)
    return np.sum(np.abs(ev[ev<0]))
bell = np.array([1,0,0,1])/np.sqrt(2)
print(round(negativity(np.outer(bell,bell),2,2),3))  # 0.5

Strengths and blind spots

Negativity is easy to compute for arbitrary dimensions, needing only an eigenvalue calculation, which makes it far more practical than measures defined by hard optimizations. Its weakness is inherited from the PPT criterion: in systems larger than two qubits by a qutrit, there exist entangled states with positive partial transpose, so-called bound entangled states. Negativity assigns these zero and thus misses them.

Practical role

Because of its computability, negativity is the standard entanglement quantifier in numerical studies of noisy circuits, many-body ground states, and channel benchmarking. When a state is a two-qubit or qubit-qutrit system, PPT detects all entanglement, so negativity being zero really does certify separability. In larger systems it remains a sufficient witness: positive negativity always proves entanglement, even if a zero value cannot fully rule it out.