 Schmidt decomposition

Every finite-dimensional bipartite pure state has a decomposition $|\psi ⟩={\sum }_{j=1}^r{s}_j|a_j⟩|b_j⟩$ with positive Schmidt coefficients $s_j$ and orthonormal families on each subsystem.

The Schmidt decomposition is a mathematical technique used in quantum mechanics and quantum information theory to express a bipartite quantum state in a particularly useful form. It is analogous to the singular value decomposition in linear algebra. For a bipartite quantum system, which consists of two subsystems (commonly referred to as systems A and B), the Schmidt decomposition allows us to write a pure state \(|\psi\rangle\) in such a way that it identifies the correlations between the two subsystems.