Schmidt decomposition

In linear algebra, the Schmidt decomposition (named after its originator Erhard Schmidt) refers to a particular way of expressing a vector as 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 formula_1 and formula_2 be Hilbert spaces of dimensions "n" and "m" respectively. Assume formula_3. For any vector formula_4 in the tensor product formula_5, there exist orthonormal sets formula_6 and formula_7 such that formula_8, where the scalars formula_9 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 formula_10 and formula_11. We can identify an elementary tensor formula_12 with the matrix formula_13, where formula_14 is the transpose of formula_15. 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 formula_19 where formula_20 is "n" × "m" and we have
Let formula_22 be the "m" column vectors of formula_20, formula_24 the column vectors of "formula_25", and formula_26 the diagonal elements of Σ. The previous expression is then
Then
which proves the claim.
Some observations.
Some properties of the Schmidt decomposition are of physical interest.
Spectrum of reduced states.
Consider a vector "w" of the tensor product
in the form of Schmidt decomposition
Form the rank 1 matrix "ρ" = "w w*". Then the partial trace of "ρ", with respect to either system "A" or "B", is a diagonal matrix whose non-zero diagonal elements are |"αi"|2. In other words, the Schmidt decomposition shows that the reduced state of "ρ" on either subsystem have the same spectrum.
Schmidt rank and entanglement.
The strictly positive values "formula_9" in the Schmidt decomposition of "w" are its Schmidt coefficients. The number of Schmidt coefficients of formula_4, counted with multiplicity, is called its Schmidt rank, or Schmidt number.
If "w" can be expressed as a product
then "w" is called a separable state. Otherwise, "w" is said to be an entangled state. From the Schmidt decomposition, we can see that "w" is entangled if and only if "w" has Schmidt rank strictly greater than 1. Therefore, two subsystems that partition a pure state are entangled if and only if their reduced states are mixed states.
Von Neumann entropy.
A consequence of the above comments is that, for pure states, the von Neumann entropy of the reduced states is a well-defined measure of entanglement. For the von Neumann entropy of both reduced states of "ρ" is formula_34, and this is zero if and only if "ρ" is a product state (not entangled).
Crystal plasticity.
In the field of plasticity, crystalline solids such as metals deform plastically primarily along crystal planes. Each plane, defined by its normal vector "ν" can "slip" in one of several directions, defined by a vector "μ". Together a slip plane and direction form a slip system which is described by the Schmidt tensor formula_35. The velocity gradient is a linear combination of these across all slip systems where the scaling factor is the rate of slip along the system.