Schmidt decomposition

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In linear algebra, the Schmidt decomposition refers to a particular way of expressing a vector in the tensor product of two linear spaces. It has applications in quantum information theory and plasticity.

1 Statement
- 1.1 Proof
2 Some observations
- 2.1 Spectrum of reduced states
- 2.2 Schmidt rank and entanglement
3 Crystal plasticity

[edit] Statement

Let $H 1$ and $H 2$ be Hilbert spaces of dimensions n and m respectively. Assume $n \geq m$ . For any vector v in the tensor product $H_1 \otimes H_2$ , there exist orthonormal sets $\{ u_1, \cdots, u_m \} \subset H_1$ and $\{ v_1, \cdots, v_m \} \subset H_2$ such that $v = \sum_{i =1} ^m \alpha _i u_i \otimes v_i$ , where the scalars $α i$ are non-negative.

[edit] Proof

The Schmidt decomposition is essentially a restatement of the singular value decomposition in a different context. Fix orthonormal bases $\{ e_1, \cdots, e_n \} \subset H_1$ and $\{ f_1, \cdots, f_m \} \subset H_2$ . We can identify an elementary tensor $e_i \otimes e_j$ with the matrix $e_i e_j ^T$ , where $e_j ^T$ is the transpose of $e j$ . A general element of the tensor product

$v = \sum _{1 \leq i \leq n, 1 \leq j \leq m} \beta _{ij} e_i \otimes e_j$

can then be viewed as the n X m matrix

$\; M_v = (\beta_{ij})_{ij} .$

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

$M_v = U \begin{bmatrix} \Sigma \\ 0 \end{bmatrix} V^T .$

Write $U =\begin{bmatrix} U_1 & U_2 \end{bmatrix}$ where $U 1$ is n X m and we have

$\; M_v = U_1 \Sigma V^T .$

Let $\{ u_1, \cdots, u_m \}$ be the column vectors of $U 1$ , $\{ v_1, \cdots, v_m \}$ the column vectors of V, and $\sigma_1, \cdots, \sigma_m$ the diagonal elements of Σ, the previous expression is then

$M_v = \sum _{i=1} ^m \sigma_i u_i v_i ^T.$

In other words,

$v = \sum _{i=1} ^m \sigma_i u_i \otimes v_i$

and this proves the claim.

[edit] Some observations

Some properties of the Schmidt decomposition are of physical interest.

[edit] Spectrum of reduced states

Consider a vector in the form of Schmidt decomposition

$v = \sum_{i =1} ^m \alpha _i u_i \otimes v_i.$

Form the rank 1 matrix ρ = v v*. Then the partial trace 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 the above comments is that, for bipartite pure states, the von Neumann entropy of either reduced state is a well defined measure of entanglement.

[edit] Schmidt rank and entanglement

For an element v of the tensor product

$H_1 \otimes H_2$

the number m in its Schmidt decomposition is called its Schmidt rank. The strictly positive values σ_i are its Schmidt coefficients.

If v can not be expressed as

$u \otimes v$

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

[edit] Crystal plasticity

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In the field of plasticity, crystaline solids such as metals deform plasticly 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 $P=\mu\otimes \nu$ . 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.