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 applications in quantum information theory and plasticity.

Theorem

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 $\{u_{1},\ldots ,u_{n}\}\subset H_{1}$ and $\{v_{1},\ldots ,v_{m}\}\subset H_{2}$ such that $v=\sum _{{i=1}}^{m}\alpha _{i}u_{i}\otimes v_{i}$ , where the scalars $\alpha _{i}$ are non-negative and, as a set, uniquely determined by $v$ .

Proof

The Schmidt decomposition is essentially a restatement of the singular value decomposition in a different context. Fix orthonormal bases $\{e_{1},\ldots ,e_{n}\}\subset H_{1}$ and $\{f_{1},\ldots ,f_{m}\}\subset H_{2}$ . We can identify an elementary tensor $e_{i}\otimes f_{j}$ with the matrix $e_{i}f_{j}^{T}$ , where $f_{j}^{T}$ is the transpose of $f_{j}$ . A general element of the tensor product

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

can then be viewed as the n × 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 × m and we have

$\;M_{v}=U_{1}\Sigma V^{T}.$

Let $\{u_{1},\ldots ,u_{m}\}$ be the first m column vectors of $U_{1}$ , $\{v_{1},\ldots ,v_{m}\}$ the column vectors of V, and $\alpha _{1},\ldots ,\alpha _{m}$ the diagonal elements of Σ. The previous expression is then

$M_{v}=\sum _{{k=1}}^{m}\alpha _{k}u_{k}v_{k}^{T},$

Then

$v=\sum _{{k=1}}^{m}\alpha _{k}u_{k}\otimes v_{k},$

which proves the claim.

Some observations

Some properties of the Schmidt decomposition are of physical interest.

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 above comments is that, for bipartite pure states, the von Neumann entropy of either reduced state is a well defined measure of entanglement.

Schmidt rank and entanglement

For an element w of the tensor product

$H_{1}\otimes H_{2}$

the strictly positive values $\alpha _{i}$ in its Schmidt decomposition are its Schmidt coefficients. The number of Schmidt coefficients of $w$ is called its Schmidt rank.

If w cannot be expressed as

$u\otimes v$

then 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, 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 $P=\mu \otimes \nu$ . 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.

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