Schmidt decomposition

In linear algebra, the Schmidt decomposition (named after its originator Erhard Schmidt) refers to a particular way of expressing a vector in the tensor product of two inner product spaces. It has numerous applications in quantum information theory, for example in entanglement characterization and in state purification, and plasticity. Let H 1 {displaystyle H_{1}} and H 2 {displaystyle H_{2}} be Hilbert spaces of dimensions n and m respectively. Assume n ≥ m {displaystyle ngeq m} . For any vector w {displaystyle w} in the tensor product H 1 ⊗ H 2 {displaystyle H_{1}otimes H_{2}} , there exist orthonormal sets { u 1 , … , u m } ⊂ H 1 {displaystyle {u_{1},ldots ,u_{m}}subset H_{1}} and { v 1 , … , v m } ⊂ H 2 {displaystyle {v_{1},ldots ,v_{m}}subset H_{2}} such that w = ∑ i = 1 m α i u i ⊗ v i {displaystyle w=sum _{i=1}^{m}alpha _{i}u_{i}otimes v_{i}} , where the scalars α i {displaystyle alpha _{i}} are real, non-negative, and, as a (multi-)set, uniquely determined by w {displaystyle w} . The Schmidt decomposition is essentially a restatement of the singular value decomposition in a different context. Fix orthonormal bases { e 1 , … , e n } ⊂ H 1 {displaystyle {e_{1},ldots ,e_{n}}subset H_{1}} and { f 1 , … , f m } ⊂ H 2 {displaystyle {f_{1},ldots ,f_{m}}subset H_{2}} . We can identify an elementary tensor e i ⊗ f j {displaystyle e_{i}otimes f_{j}} with the vector e i f j T {displaystyle e_{i}f_{j}^{T}} , where f j T {displaystyle f_{j}^{T}} is the transpose of f j {displaystyle f_{j}} . A general element of the tensor product can then be viewed as the n × m matrix By the singular value decomposition, there exist an n × n unitary U, m × m unitary V, and a positive semidefinite diagonal m × m matrix Σ such that Write U = [ U 1 U 2 ] {displaystyle U={egin{bmatrix}U_{1}&U_{2}end{bmatrix}}} where U 1 {displaystyle U_{1}} is n × m and we have Let { u 1 , … , u m } {displaystyle {u_{1},ldots ,u_{m}}} be the first m column vectors of U 1 {displaystyle U_{1}} , { v 1 , … , v m } {displaystyle {v_{1},ldots ,v_{m}}} the column vectors of V, and α 1 , … , α m {displaystyle alpha _{1},ldots ,alpha _{m}} the diagonal elements of Σ. The previous expression is then