Concurrence

Concurrence is a closed-form entanglement measure for two qubits, valid even for mixed states through Wootters' formula.

Two-qubit entanglement in a number

For a pure two-qubit state |psi> = a|00> + b|01> + c|10> + d|11>, the concurrence is C = 2|ad - bc|. It ranges from 0 for product states to 1 for maximally entangled Bell states. The quantity ad - bc is the determinant of the coefficient matrix, so concurrence is zero exactly when that matrix has rank one, i.e. when the state factorizes.

Wootters' formula for mixed states

Concurrence extends to any two-qubit density matrix rho. Define the spin-flipped state rho-tilde = (Y tensor Y) rho* (Y tensor Y), let the eigenvalues of rho rho-tilde be sorted as l1 >= l2 >= l3 >= l4, and set C = max(0, sqrt(l1) - sqrt(l2) - sqrt(l3) - sqrt(l4)). This gives the entanglement of formation through a monotone function of C, making it one of the few mixed-state measures with a full closed form.

import numpy as np
Y = np.array([[0,-1j],[1j,0]])
def concurrence(rho):
    YY = np.kron(Y,Y)
    R = rho @ YY @ rho.conj() @ YY
    ev = np.sort(np.sqrt(np.abs(np.linalg.eigvals(R).real)))[::-1]
    return max(0.0, ev[0]-ev[1]-ev[2]-ev[3])
bell = np.array([1,0,0,1])/np.sqrt(2)
print(round(concurrence(np.outer(bell,bell)),3))  # 1.0

Relation to entanglement of formation

The entanglement of formation, the minimum average entanglement needed to build the state from Bell pairs, is E_F = h((1 + sqrt(1 - C^2))/2), where h is the binary entropy. Because this is monotone in C, concurrence and entanglement of formation carry the same ordering information for two qubits. Concurrence therefore serves as the practical everyday measure.

Scope

Concurrence is special to the two-qubit case; generalizations to higher dimensions and more parties exist but lose the clean closed form. For larger systems one typically resorts to negativity or numerically hard measures. Still, because two-qubit reduced states appear everywhere, concurrence remains a widely used diagnostic for pairwise entanglement in larger systems.