# 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

Let $H_{1}$ and $H_{2}$ be Hilbert spaces of dimensions n and m respectively. Assume $n\geq m$ . For any vector $w$ in the tensor product $H_{1}\otimes H_{2}$ , there exist orthonormal sets $\{u_{1},\ldots ,u_{m}\}\subset H_{1}$ and $\{v_{1},\ldots ,v_{m}\}\subset H_{2}$ such that $w=\sum _{i=1}^{m}\alpha _{i}u_{i}\otimes v_{i}$ , where the scalars $\alpha _{i}$ are real, non-negative, and unique up to re-ordering.

### 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

$w=\sum _{1\leq i\leq n,1\leq j\leq m}\beta _{ij}e_{i}\otimes f_{j}$ can then be viewed as the n × m matrix

$\;M_{w}=(\beta _{ij}).$ 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

$M_{w}=U{\begin{bmatrix}\Sigma \\0\end{bmatrix}}V^{*}.$ Write $U={\begin{bmatrix}U_{1}&U_{2}\end{bmatrix}}$ where $U_{1}$ is n × m and we have

$\;M_{w}=U_{1}\Sigma V^{*}.$ Let $\{u_{1},\ldots ,u_{m}\}$ be the m column vectors of $U_{1}$ , $\{v_{1},\ldots ,v_{m}\}$ the column vectors of ${\overline {V}}$ , and $\alpha _{1},\ldots ,\alpha _{m}$ the diagonal elements of Σ. The previous expression is then

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

$w=\sum _{k=1}^{m}\alpha _{k}u_{k}\otimes v_{k},$ which proves the claim.

## Some observations

Some properties of the Schmidt decomposition are of physical interest.

### Spectrum of reduced states

Consider a vector w of the tensor product

$H_{1}\otimes H_{2}$ in the form of Schmidt decomposition

$w=\sum _{i=1}^{m}\alpha _{i}u_{i}\otimes v_{i}.$ 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

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

If w can be expressed as a product

$u\otimes v$ 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

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 $-\sum _{i}|\alpha _{i}|^{2}\log(|\alpha _{i}|^{2})$ , and this is zero if and only if ρ is a product state (not entangled).

## 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.