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 ${\displaystyle H_{1}}$ and ${\displaystyle H_{2}}$ be Hilbert spaces of dimensions n and m respectively. Assume ${\displaystyle n\geq m}$. For any vector ${\displaystyle w}$ in the tensor product ${\displaystyle H_{1}\otimes H_{2}}$, there exist orthonormal sets ${\displaystyle \{u_{1},\ldots ,u_{n}\}\subset H_{1}}$ and ${\displaystyle \{v_{1},\ldots ,v_{m}\}\subset H_{2}}$ such that ${\displaystyle w=\sum _{i=1}^{m}\alpha _{i}u_{i}\otimes v_{i}}$, where the scalars ${\displaystyle \alpha _{i}}$ are real, non-negative, and, as a (multi-)set, uniquely determined by ${\displaystyle w}$.

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