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.
Contents
Theorem
Let
Proof
The Schmidt decomposition is essentially a restatement of the singular value decomposition in a different context. Fix orthonormal bases
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
Let
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
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 bipartite 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
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