Schmidt decomposition

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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[edit]

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 (multi-)set, uniquely determined by .

Proof[edit]

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 row vectors of , the column vectors of V, and the diagonal elements of Σ. The previous expression is then

Then

which proves the claim.

Some observations[edit]

Some properties of the Schmidt decomposition are of physical interest.

Spectrum of reduced states[edit]

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[edit]

The strictly positive values in the Schmidt decomposition of w are its Schmidt coefficients. The number of Schmidt coefficients of , counted with multiplicity, is called its Schmidt rank, or Schmidt number.

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[edit]

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 , and this is zero if and only if ρ is a product state (not entangled).

Crystal plasticity[edit]

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.

See also[edit]

Further reading[edit]