The Schmidt Decomposition

Any bipartite pure state can be written with a single sum over matched local bases, exposing its entanglement structure at a glance.

A canonical form for bipartite states

For any pure state |psi_AB> there exist orthonormal bases {|u_i>_A} and {|v_i>_B} such that |psi_AB> = sum_i lambda_i |u_i>_A |v_i>_B, with real nonnegative Schmidt coefficients lambda_i satisfying sum_i lambda_i^2 = 1. The single index i is the striking feature: no cross terms are needed. The number of nonzero lambda_i is the Schmidt rank.

Where it comes from

Reshape the state's amplitudes into a matrix M with rows indexed by A and columns by B, then take the singular value decomposition M = U S V-dagger. The singular values are the Schmidt coefficients and the left and right singular vectors give the local bases. The Schmidt decomposition is thus the SVD of the coefficient matrix, which is why it is so robust and easy to compute.

import numpy as np
psi = np.array([1,0,0,1])/np.sqrt(2)  # Bell
M = psi.reshape(2,2)
_, s, _ = np.linalg.svd(M)
print(np.round(s,3))   # [0.707, 0.707]  -> rank 2

Reading off entanglement

A state is a product state exactly when its Schmidt rank is one. It is entangled when the rank exceeds one, and maximally entangled when all lambda_i are equal. The squared coefficients lambda_i^2 are the eigenvalues of both reduced density matrices, so the entanglement entropy is S = -sum_i lambda_i^2 log lambda_i^2. Every bipartite entanglement measure for pure states is a function of the Schmidt spectrum.

Consequences and limits

The Schmidt form makes many results immediate: the reduced states of the two halves have identical nonzero spectra, purifications are essentially unique up to reference unitaries, and local unitaries cannot change the Schmidt coefficients. The decomposition is special to two parties, however. For three or more subsystems no single-index canonical form exists in general, which is one reason multipartite entanglement is qualitatively richer, as the GHZ and W states show.