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, as a set, uniquely determined by .
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 first m column vectors of , the column vectors of V, and the diagonal elements of Σ. The previous expression is then
which proves the claim.
Some properties of the Schmidt decomposition are of physical interest.
Spectrum of reduced states
Consider a vector in the form of Schmidt decomposition
Form the rank 1 matrix ρ = v v*. 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.
In the language of quantum mechanics, a rank 1 projection ρ is called a pure state. A consequence of the above comments is that, for bipartite pure states, the von Neumann entropy of either reduced state is a well defined measure of entanglement.
Schmidt rank and entanglement
For an element w of the tensor product
the strictly positive values in its Schmidt decomposition are its Schmidt coefficients. The number of Schmidt coefficients of is called its Schmidt rank, or Schmidt number.
If w cannot be expressed as
then 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.
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 . 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.