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Concept
Schmidt decomposition
Summary
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 and be Hilbert spaces of dimensions n and m respectively. Assume . For any vector in the tensor product , there exist orthonormal sets and such that , where the scalars are real, non-negative, and unique up to re-ordering.
The Schmidt decomposition is essentially a restatement of the singular value decomposition in a different context. Fix orthonormal bases and . We can identify an elementary tensor with the matrix , where is the transpose of . 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 where is n × m and we have
Let be the m column vectors of , the column vectors of , and the diagonal elements of Σ. The previous expression is then
Then
which proves the claim.
Some properties of the Schmidt decomposition are of physical interest.
Consider a vector of the tensor product
in the form of Schmidt decomposition
Form the rank 1 matrix . Then the partial trace of , with respect to either system A or B, is a diagonal matrix whose non-zero diagonal elements are . In other words, the Schmidt decomposition shows that the reduced states of on either subsystem have the same spectrum.
The strictly positive values in the Schmidt decomposition of are its Schmidt coefficients, or Schmidt numbers. The total number of Schmidt coefficients of , counted with multiplicity, is called its Schmidt rank.
If can be expressed as a product
then is called a separable state. Otherwise, is said to be an entangled state. From the Schmidt decomposition, we can see that is entangled if and only if has Schmidt rank strictly greater than 1.
Official source
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