Schmidt decomposition

In linear algebra, the Schmidt decomposition called ( which is named after Erhard Schmidt) a particular representation of a vector in the tensor product of two vector spaces with scalar product as the sum of a few pairs of orthonormal product vectors. The Schmidt decomposition is used for example in quantum computer science application.

Statement

Let and be Hilbert spaces of dimension respectively, and was. Then for each vector quantities of pairwise orthonormal vectors and so

Holds, where the non- negative numbers are determined by unique.

Evidence

Schmidt decomposition is essentially a consequence of the singular value decomposition. Fix orthonormal bases and. The Elementartensor can with the matrix ( here denotes the transpose of ) are identified. Any vector can be written in the base as

And can then use the matrix

Be identified. According to the singular value decomposition, there are unitary matrices and and a positive semi-definite diagonal matrix such that

If we write, in which a matrix is then obtained

Is called now the first column of vectors with and with the column vectors of V, and the diagonal elements of the matrix is followed by

Which proves the assertion.

Use in Physics

The Schmidt decomposition takes place, for example, in quantum physics application.

Spectrum of reduced states

Consider a vector in the Schmidt form

The matrix ( called the adjoint vector to ) is a one-dimensional projector. The partial trace of respect either the subsystem or is given by a diagonal matrix whose non-zero entries. In other words shows the Schmidt decomposition, that the spectrum of the two partial tracks and is the same.

In quantum mechanics ( like any one-dimensional projector ) describes the pure state of a composite of two parts system, and describes the reduced state in the subsystem 2 or 1 The spectrum of the reduced state determines, among other things whose von Neumann entropy and various entanglement measures the pure state.

Schmidt- rank and entanglement

For a vector strictly positive values ​​in its Schmidt decomposition are called its Schmidt coefficients. The number of Schmidt coefficients is called Schmidt rank of.

The following statements are equivalent:

• The Schmidt rank of greater than one
• Can not be written as a product vector
• Is crossed
• The reduced states of are not purely

From the Schmidt coefficients of a pure state, all its entanglement properties can be determined. The behavior of under local quantum operations is determined by the Schmidt coefficients, in particular, whether it is possible two states transform into each other locally.