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 ${\textstyle w=\sum _{i=1}^{m}\alpha _{i}u_{i}\otimes v_{i}}$ , 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 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 states 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 ${\textstyle -\sum _{i}|\alpha _{i}|^{2}\log \left(|\alpha _{i}|^{2}\right)}$ , and this is zero if and only if ρ 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 reasos 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 $(4,2,2)$ .