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.
Theorem 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 n\geq 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 {\textstyle 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 unique up to re-ordering.
Proof 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 matrix e i f j T {\displaystyle e_{i}f_{j}^{\mathsf {T}}} , where f j T {\displaystyle f_{j}^{\mathsf {T}}} is the transpose of f j {\displaystyle f_{j}} . A general element of the tensor product
w = ∑ 1 ≤ i ≤ n , 1 ≤ j ≤ m β i j e i ⊗ f j {\displaystyle w=\sum _{1\leq i\leq n,1\leq j\leq m}\beta _{ij}e_{i}\otimes f_{j}}
can then be viewed as the n × m matrix
M w = ( β i j ) . {\displaystyle \;M_{w}=(\beta _{ij}).}
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
M w = U [ Σ 0 ] V ∗ . {\displaystyle M_{w}=U{\begin{bmatrix}\Sigma \\0\end{bmatrix}}V^{*}.}
Write U = [ U 1 U 2 ] {\displaystyle U={\begin{bmatrix}U_{1}&U_{2}\end{bmatrix}}} where U 1 {\displaystyle U_{1}} is n × m and we have
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