Schmidt decomposition

In linear algebra , the Schmidt decomposition (named after Erhard Schmidt ) describes a certain representation of a vector in the tensor product of two vector spaces with a scalar product as the sum of a few pairwise orthonormal product vectors . The Schmidt decomposition is used, for example, in quantum computing.

statement

Be and Hilbert dreams of dimension or and be . Then for each vector there are sets of pairwise orthonormal vectors and , such that ${\displaystyle H_{1}}$ ${\displaystyle H_{2}}$ ${\displaystyle n}$ ${\displaystyle m}$ ${\displaystyle n\geq m}$ ${\displaystyle v\in H_{1}\otimes H_{2}}$ ${\displaystyle \{u_{1},\ldots ,u_{m}\}\subset H_{1}}$ ${\displaystyle \{v_{1},\ldots ,v_{m}\}\subset H_{2}}$

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

applies, where the non-negative numbers are uniquely determined by. ${\displaystyle \alpha _{1}\geq \alpha _{2}\geq ...\geq \alpha _{m}\geq 0}$ ${\displaystyle v}$

proof

The Schmidt decomposition is essentially a consequence of the singular value decomposition . Fix orthonormal bases and . The elementary tensor can be identified with the matrix (here denotes the transposition of ). Any vector can be written in the base as ${\displaystyle \{e_{1},\ldots ,e_{n}\}\subset H_{1}}$ ${\displaystyle \{f_{1},\ldots ,f_{m}\}\subset H_{2}}$ ${\displaystyle e_{i}\otimes f_{j}}$ ${\displaystyle e_{i}f_{j}^{T}}$ ${\displaystyle f_{j}^{T}}$ ${\displaystyle f_{j}}$ ${\displaystyle v}$ ${\displaystyle e_{i}\otimes f_{j}}$

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

and can then use the matrix ${\displaystyle n\times m}$

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

be identified. After the singular value decomposition there are unitary matrices on and on and a positive-semidefinite diagonal matrix such that ${\displaystyle U}$ ${\displaystyle H_{1}}$ ${\displaystyle V}$ ${\displaystyle H_{2}}$ ${\displaystyle m\times m}$ ${\displaystyle \Sigma }$

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

If one writes , where is a matrix, then one obtains ${\displaystyle U={\begin{bmatrix}U_{1}&U_{2}\end{bmatrix}}}$ ${\displaystyle U_{1}}$ ${\displaystyle n\times m}$

${\displaystyle \;M_{v}=U_{1}\Sigma V^{T}.}$

If we now denote the first column vectors of with and with the column vectors of V and the diagonal elements of the matrix with then follows ${\displaystyle m}$ ${\displaystyle U_{1}}$ ${\displaystyle \{u_{1},\ldots ,u_{m}\}}$ ${\displaystyle \{v_{1},\ldots ,v_{m}\}}$ ${\displaystyle \Sigma }$ ${\displaystyle \alpha _{1},\ldots ,\alpha _{m}}$

${\displaystyle M_{v}=U_{1}\Sigma V^{T}=\sum _{i=1}^{m}\alpha _{i}u_{i}v_{i}^{T}=\sum _{i=1}^{m}\alpha _{i}u_{i}\otimes v_{i}}$,

what the claim proves.

Use in physics

The Schmidt decomposition takes place e.g. B. in quantum physics application.

Spectrum of reduced states

Consider a vector in Schmidt form

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

The matrix ( referred to the adjoint vector) is a one-dimensional projector on . The partial trace of with respect to either the subsystem or is then given by a diagonal matrix whose non-vanishing entries are. In other words, the Schmidt decomposition shows that the spectrum of the two partial traces and is the same. ${\displaystyle \rho =ww^{*}}$ ${\displaystyle w^{*}}$ ${\displaystyle w}$ ${\displaystyle H_{1}\otimes H_{2}}$ ${\displaystyle \rho }$ ${\displaystyle H_{1}}$ ${\displaystyle H_{2}}$ ${\displaystyle |\alpha _{i}|^{2}}$ ${\displaystyle {\text{tr}}_{1}(\rho )}$ ${\displaystyle {\text{tr}}_{2}(\rho )}$

In quantum mechanics (like every one-dimensional projector ) describes the pure state of a system composed of two parts and or describes the reduced state in subsystem 2 or 1. The spectrum of the reduced state determines, among other things, its Von Neumann entropy and various Entanglement dimensions of the pure state . ${\displaystyle \rho }$ ${\displaystyle H_{1}\otimes H_{2}}$ ${\displaystyle \rho _{2}:={\text{tr}}_{1}(\rho )}$ ${\displaystyle \rho _{1}:={\text{tr}}_{2}(\rho )}$ ${\displaystyle \rho }$

Schmidt rank and entanglement

For a vector , the strictly positive values in its Schmidt decomposition are called its Schmidt coefficients . The number of Schmidt coefficients is called the Schmidt rank of . ${\displaystyle w\in H_{1}\otimes H_{2}}$ ${\displaystyle \alpha _{i}>0}$ ${\displaystyle w}$

The following statements are equivalent:

• the Schmidt rank of is greater than one ${\displaystyle w}$
• ${\displaystyle w}$cannot be written as a product vector ${\displaystyle u\otimes v}$
• ${\displaystyle w}$is entangled
• the reduced states of are not pure ${\displaystyle w}$

All of its entanglement properties can be determined from the Schmidt coefficients of a pure state . The behavior of under local quantum operations is also determined by the Schmidt coefficients, in particular whether two states can be transformed locally into one another. ${\displaystyle w}$ ${\displaystyle w}$

literature

• Erhard Schmidt: On the theory of linear and non-linear integral equations , Mathematische Annalen 63 , 433-476 (1907).
• Asher Peres: Quantum Theory: Concepts and Methods , Kluwer (Dordrecht, 1993), Chapter 5.
• Artur Ekert and Peter L. Knight: Entangled quantum systems and the Schmidt decomposition . In: American Journal of Physics . 63, No. 5, May 1995, p. 415. doi : 10.1119 / 1.17904 .

Individual evidence

1. ↑ ^{a b} Guifre Vidal: Entanglement Monotones . In: . J. Mod Opt. . 47, 2000, p. 355. arxiv : quant-ph / 9807077 . doi : 10.1080 / 09500340008244048 .
2. ^ MA Nielsen: Conditions for a Class of Entanglement Transformations . In: Phys. Rev. Lett. . 83, 1999, p. 436. arxiv : quant-ph / 9811053 . doi : 10.1103 / PhysRevLett.83.436 .