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 ${\displaystyle H_{1}}$ and ${\displaystyle H_{2}}$ be Hilbert spaces of dimensions n and m respectively. Assume ${\displaystyle n\geq m}$. For any vector ${\displaystyle w}$ in the tensor product ${\displaystyle H_{1}\otimes H_{2}}$, there exist orthonormal sets ${\displaystyle \{u_{1},\ldots ,u_{m}\}\subset H_{1}}$ and ${\displaystyle \{v_{1},\ldots ,v_{m}\}\subset H_{2}}$ such that ${\textstyle w=\sum _{i=1}^{m}\alpha _{i}u_{i}\otimes v_{i}}$, where the scalars ${\displaystyle \alpha _{i}}$ 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 ${\displaystyle \{e_{1},\ldots ,e_{n}\}\subset H_{1}}$ and ${\displaystyle \{f_{1},\ldots ,f_{m}\}\subset H_{2}}$. We can identify an elementary tensor ${\displaystyle e_{i}\otimes f_{j}}$ with the matrix ${\displaystyle e_{i}f_{j}^{\mathsf {T}}}$, where ${\displaystyle f_{j}^{\mathsf {T}}}$ is the transpose of ${\displaystyle f_{j}}$. A general element of the tensor product

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

can then be viewed as the n × m matrix

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

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

${\displaystyle M_{w}=U{\begin{bmatrix}\Sigma \\0\end{bmatrix}}V^{*}.}$

Write ${\displaystyle U={\begin{bmatrix}U_{1}&U_{2}\end{bmatrix}}}$ where ${\displaystyle U_{1}}$ is n × m and we have

${\displaystyle \;M_{w}=U_{1}\Sigma V^{*}.}$

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

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

Then

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

which proves the claim.

Some observations

Some properties of the Schmidt decomposition are of physical interest.

Spectrum of reduced states

Consider a vector ${\displaystyle w}$ of the tensor product

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

in the form of Schmidt decomposition

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

Form the rank 1 matrix ${\displaystyle \rho =ww^{*}}$. Then the partial trace of ${\displaystyle \rho }$, with respect to either system A or B, is a diagonal matrix whose non-zero diagonal elements are ${\displaystyle |\alpha _{i}|^{2}}$. In other words, the Schmidt decomposition shows that the reduced states of ${\displaystyle \rho }$ on either subsystem have the same spectrum.

Schmidt rank and entanglement

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

If ${\displaystyle w}$ can be expressed as a product

${\displaystyle u\otimes v}$

then ${\displaystyle w}$ is called a separable state. Otherwise, ${\displaystyle w}$ is said to be an entangled state. From the Schmidt decomposition, we can see that ${\displaystyle w}$ is entangled if and only if ${\displaystyle w}$ 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 ${\displaystyle \rho }$ is ${\textstyle -\sum _{i}|\alpha _{i}|^{2}\log \left(|\alpha _{i}|^{2}\right)}$, and this is zero if and only if ${\displaystyle \rho }$ is a product state (not entangled).

Schmidt-rank vector

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

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

The concept of Schmidt rank can be extended to quantum systems made up of more than two subsystems.^[1]

Consider the tripartite quantum system:

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

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

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

Each of the systems obtained is a bipartite system and therefore can be characterized by one number (its Schmidt rank), respectively ${\displaystyle r_{A},r_{B}}$ and ${\displaystyle r_{C}}$. 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

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

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 ^[2]

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

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 ${\displaystyle (4,2,2)}$.

See also

• Singular value decomposition
• Purification of quantum state

References

Further reading

• Pathak, Anirban (2013). Elements of Quantum Computation and Quantum Communication. London: Taylor & Francis. pp. 92–98. ISBN 978-1-4665-1791-2.