Schmidt decomposition

In linear algebra

Linear algebra

Linear algebra is a branch of mathematics that studies vector spaces, also called linear spaces, along with linear functions that input one vector and output another. Such functions are called linear maps and can be represented by matrices if a basis is given. Thus matrix theory is often...

, the Schmidt decomposition (named after its originator Erhard Schmidt

Erhard Schmidt

Erhard Schmidt was a German mathematician whose work significantly influenced the direction of mathematics in the twentieth century. He was born in Tartu, Governorate of Livonia . His advisor was David Hilbert and he was awarded his doctorate from Georg-August University of Göttingen in 1905...

) refers to a particular way of expressing a vector

Coordinate vector

In linear algebra, a coordinate vector is an explicit representation of a vector in an abstract vector space as an ordered list of numbers or, equivalently, as an element of the coordinate space Fn....

 in the tensor product

Tensor product

In mathematics, the tensor product, denoted by ⊗, may be applied in different contexts to vectors, matrices, tensors, vector spaces, algebras, topological vector spaces, and modules, among many other structures or objects. In each case the significance of the symbol is the same: the most general...

 of two inner product space

Inner product space

In mathematics, an inner product space is a vector space with an additional structure called an inner product. This additional structure associates each pair of vectors in the space with a scalar quantity known as the inner product of the vectors...

s. It has applications in quantum information theory and plasticity

Plasticity (physics)

In physics and materials science, plasticity describes the deformation of a material undergoing non-reversible changes of shape in response to applied forces. For example, a solid piece of metal being bent or pounded into a new shape displays plasticity as permanent changes occur within the...

.

Theorem

Let and be Hilbert spaces of dimension

Dimension

In physics and mathematics, the dimension of a space or object is informally defined as the minimum number of coordinates needed to specify any point within it. Thus a line has a dimension of one because only one coordinate is needed to specify a point on it...

s n and m respectively. Assume . For any vector in the tensor product , there exist orthonormal sets and such that , where the scalars are non-negative and, as a set, uniquely determined by .

Proof

The Schmidt decomposition is essentially a restatement of the singular value decomposition

Singular value decomposition

In linear algebra, the singular value decomposition is a factorization of a real or complex matrix, with many useful applications in signal processing and statistics....

 in a different context. Fix orthonormal bases and . We can identify an elementary tensor with the matrix , where is the transpose

Transpose

In linear algebra, the transpose of a matrix A is another matrix AT created by any one of the following equivalent actions:...

 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

Positive semidefinite

In mathematics, positive semidefinite may refer to:* positive-semidefinite matrix* positive-semidefinite function...

 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.

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

Partial trace

In linear algebra and functional analysis, the partial trace is a generalization of the trace. Whereas the trace is a scalar valued function on operators, the partial trace is an operator-valued function...

 of ρ, with respect to either system A or B, is a diagonal matrix whose non-zero diagonal elements are |α_i |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

Von Neumann entropy

In quantum statistical mechanics, von Neumann entropy, named after John von Neumann, is the extension of classical entropy concepts to the field of quantum mechanics....

 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.

If w can not be expressed as
then w is said to be an entangled state

Quantum entanglement

Quantum entanglement occurs when electrons, molecules even as large as "buckyballs", photons, etc., interact physically and then become separated; the type of interaction is such that each resulting member of a pair is properly described by the same quantum mechanical description , which is...

. From the Schmidt decomposition, we can see that w is entangled if and only if w has Schmidt rank strictly greater than 1. Therefore, a bipartite pure state is entangled if and only if its reduced states are mixed states.

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 . 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.