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 ${\displaystyle H_{1}}$ 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 ${\displaystyle w=\sum _{i=1}^{m}\alpha _{i}u_{i}\otimes v_{i}}$, where the scalars ${\displaystyle \alpha _{i}}$ are real, non-negative, and, as a (multi-)set, uniquely determined by ${\displaystyle w}$.

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 ${\displaystyle e_{i}f_{j}^{T}}$, where ${\displaystyle f_{j}^{T}}$ 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 n × m matrix Σ such that

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

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

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

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 V, and ${\displaystyle \alpha _{1},\ldots ,\alpha _{m}}$ the diagonal elements of Σ. The previous expression is then

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

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

If w can be expressed as a product

${\displaystyle u\otimes v}$

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 ${\displaystyle -\sum _{i}|\alpha _{i}|^{2}\log(|\alpha _{i}|^{2})}$, 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 ${\displaystyle P=\mu \otimes \nu }$. 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.

See also

• Singular value decomposition
• Purification of quantum state

Further reading

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