Schmidt Decomposition Overview

Canonical treatment: Schmidt Decomposition maintains the proof, constructive algorithms, examples, exercises, and references.

This bridge is intentionally limited to finite-dimensional orientation and a prerequisite map.

The normal form

Every finite-dimensional bipartite pure state admits

$$\lvert\Psi\rangle = \sum_{r=1}^{R}s_r\, \lvert u_r\rangle_A\lvert v_r\rangle_B, \qquad s_r>0, \qquad \sum_r s_r^2=1.$$

The $s_r$ are Schmidt coefficients and $R$ is the Schmidt rank. The state is product exactly when $R=1$ and entangled when $R>1$. The numbers $s_r^2$ are the nonzero eigenvalues of either reduced state.

Readiness map

Before the canonical derivation, be comfortable with tensor-product bases, product versus entangled states, reduced density operators, spectral decomposition, and singular-value decomposition. For a calculation, place product-basis amplitudes in a matrix and compute its singular values.

The canonical page treats degeneracies, basis freedom, unequal subsystem dimensions, infinite-dimensional caveats, and worked coefficient-matrix examples.