# Schmidt decomposition

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

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 $\{ u_1, \ldots, u_m \} \subset H_1$ 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 real, 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

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

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

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

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

$M_v = \sum _{k=1} ^m \alpha_k u_k v_k ^\star ,$

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 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 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 $-\sum_i |\alpha_i|^2 \log|\alpha_i|^2$, and this is zero if and only if 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.