Schmidt decomposition
In linear algebra, the Schmidt decomposition (named after its originator Erhard Schmidt) refers to a particular way of expressing a vector as 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 unique up to re-ordering.
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 n × m matrix Σ such that
Write where is n × m and we have
Let be the m column vectors of , the column vectors of , 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 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]
- Pathak, Anirban (2013). Elements of Quantum Computation and Quantum Communication. London: Taylor & Francis. pp. 92–98. ISBN 978-1-4665-1791-2.