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.

Theorem

be Hilbert spaces of dimensions n and m are real, non-negative, and unique up to re-ordering.

Proof

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

is n × m and we have

be the m the column vectors of 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

of the tensor product

in the form of Schmidt decomposition

. Then the partial trace, with respect to either system A or B, is a diagonal matrix on either subsystem have the same spectrum.

Schmidt rank and entanglement

The strictly positive values are its Schmidt coefficients, or Schmidt numbers, counted with multiplicity, is called its Schmidt rank.

can be expressed as a product

is called a separable state is said to be an entangled state 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 is a product state (not entangled).

Schmidt-rank vector

The Schmidt rank is defined for bipartite systems, namely quantum states

The concept of Schmidt rank can be extended to quantum systems made up of more than two subsystems.

Consider the tripartite quantum system:

There are three ways to reduce this to a bipartite system by performing the partial trace

. These numbers capture the "amount of entanglement" in the bipartite system when respectively A, B or C are discarded. For these reasons the tripartite system can be described by a vector, namely the Schmidt-rank vector

Multipartite entanglementMultipartite systems

The concept of Schmidt-rank vector can be likewise extended to systems made up of more than three subsystems through the use of tensors.

Example

Source:

This kind of system is made possible by encoding the value of a qudit into the orbital angular momentum (OAM) of a photon rather than its spin, since the latter can only take two values.

.

See also

• Singular value decomposition

• Purification of quantum state

References

Further reading