Schmidt decomposition
# 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 $w$ 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 $w=∑_{i=1}^mα_iu_i⊗v_i{\textstyle$ 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 $e_{i}f_{j}^{\mathsf {T}}$, where $f_{j}^{\mathsf {T}}$ 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 m × 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

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

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 $\rho =ww^{*}$. Then the partial trace of $\rho$, with respect to either system A or B, is a diagonal matrix whose non-zero diagonal elements are $|\alpha _{i}|^{2}$. In other words, the Schmidt decomposition shows that the reduced states of $\rho$ 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, or Schmidt numbers. The total number of Schmidt coefficients of $w$, counted with multiplicity, is called its Schmidt rank.

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 $\rho$ is $−∑_i|α_i|^2log⁡(|α_i|^2){\textstyle$ -\sum _{i}|\alpha _{i}|^{2}\log \left(|\alpha _{i}|^{2}\right)}, and this is zero if and only if $\rho$ is a product state (not entangled).

## Schmidt-rank vector

The Schmidt rank is defined for bipartite systems, namely quantum states

$|\psi \rangle \in H_{A}\otimes H_{B}$

The concept of Schmidt rank can be extended to quantum systems made up of more than two subsystems.

Consider the tripartite quantum system:

$|\psi \rangle \in H_{A}\otimes H_{B}\otimes H_{C}$

There are three ways to reduce this to a bipartite system by performing the partial trace with respect to $H_{A},H_{B}$ or $H_{C}$

${\begin{cases}{\hat {\rho }}_{A}=Tr_{A}(|\psi \rangle \langle \psi |)\\{\hat {\rho }}_{B}=Tr_{B}(|\psi \rangle \langle \psi |)\\{\hat {\rho }}_{C}=Tr_{C}(|\psi \rangle \langle \psi |)\end{cases}}$

Each of the systems obtained is a bipartite system and therefore can be characterized by one number (its Schmidt rank), respectively $r_{A},r_{B}$ and $r_{C}$. These numbers capture the "amount of entanglement" in the bipartite system when respectively A, B or C are discarded. For these reasons the tripartite system can be described by a vector, namely the Schmidt-rank vector

${\vec {r}}=(r_{A},r_{B},r_{C})$

### Multipartite systems

The concept of Schmidt-rank vector can be likewise extended to systems made up of more than three subsystems through the use of tensors.

### Example

Take the tripartite quantum state $|\psi _{4,2,2}\rangle ={\frac {1}{2}}{\big|0,0,0\rangle +|1,0,1\rangle +|2,1,0\rangle +|3,1,1\rangle {\big )}$

This kind of system is made possible by encoding the value of a qudit into the orbital angular momentum (OAM) of a photon rather than its spin, since the latter can only take two values.

The Schmidt-rank vector for this quantum state is $(4,2,2)$.

