Kraus representation

The Kraus representation , named after the physicist Karl Kraus , is a form of representation of quantum channels that describe the dynamics of a quantum system. Through Kraus' theorem, which says that a mapping is completely positive and retains a trace if and only if it can be written in Kraus representation, the Kraus representation is of particular importance in the theory of open quantum systems and quantum informatics .

definition

Let be Hilbert spaces and be a mapping between these Hilbert spaces. The Kraus representation of the figure is then given by ${\displaystyle {\mathcal {H}},{\mathcal {G}}}$ ${\displaystyle \Lambda \colon {\mathcal {H}}\to {\mathcal {G}}}$ ${\displaystyle \Lambda }$

${\displaystyle \Lambda [\rho ]=\sum _{\alpha }K_{\alpha }\rho K_{\alpha }^{\dagger },}$

where are the Kraus operators. Is trace preserving the Kraus operators satisfy the completeness relation: . Where is the identity operator. ${\displaystyle \{K_{\alpha }\}}$ ${\displaystyle \Lambda }$ ${\displaystyle \textstyle \sum _{\alpha }K_{\alpha }^{\dagger }K_{\alpha }=1}$ ${\displaystyle 1}$

motivation

In general, a quantum mechanical state is represented by the density operator . This has the following properties: ${\displaystyle \rho }$

• Hermitesch: ${\displaystyle \rho ^{\dagger }=\rho }$
• Normalized: ${\displaystyle {\rm {{Spur}\{\rho \}=1}}}$
• Positive semidefinite:, or ${\displaystyle \rho \geq 0}$ ${\displaystyle \langle \psi \left|\rho \right|\psi \rangle \geq 0\quad \forall \left|\psi \right\rangle \in {\mathcal {H}}}$

If an image transfers a density operator to another density operator , the following must apply: ${\displaystyle \Lambda \colon {\mathcal {H}}\to {\mathcal {G}}}$ ${\displaystyle \rho \in {\mathcal {H}}}$ ${\displaystyle \Lambda [\rho ]\in {\mathcal {G}}}$ ${\displaystyle \Lambda }$

• Preservation of Hermiticity: ${\displaystyle \rho ^{\dagger }=\rho \Rightarrow \Lambda [\rho ^{\dagger }]=\Lambda [\rho ],}$
• Track maintenance: ${\displaystyle {\rm {{Spur}\{\Lambda [\rho ]\}={\rm {{Spur}\{\rho \},}}}}}$
• Preservation of positivity: ${\displaystyle \rho \geq 0\Rightarrow \Lambda [\rho ]\geq 0.}$

Track maintenance

Every figure in Kraus's representation is trace-retaining, there

${\displaystyle {\rm {{Spur}\{\Lambda [\rho ]\}=\sum _{\alpha }{\rm {{Spur}\{K_{\alpha }\rho .K_{\alpha }^{\dagger }\}=\sum _{\alpha }{\rm {{Spur}\{K_{\alpha }^{\dagger }K_{\alpha }\rho \}={\rm {{Spur}\{\rho \}.}}}}}}}}}$

The fact that the track is linear and invariant with cyclical interchanging of the elements was used here.

Preservation of positivity

The terms of the form are positive, since a new state can be defined and then the following applies: ${\displaystyle K_{\alpha }\rho K_{\alpha }^{\dagger }}$ ${\displaystyle |\phi \rangle =K_{\alpha }^{\dagger }|\psi \rangle }$

${\displaystyle {\big \langle }\psi {\big |}K_{\alpha }\rho K_{\alpha }^{\dagger }{\big |}\psi {\big \rangle }=\left\langle \phi \right|\rho \left|\phi \right\rangle \geq 0\quad \forall \left|\psi \right\rangle \in {\mathcal {H}}.}$

This means that the sum of positive-semidefinite terms is also positive-semidefinite. ${\displaystyle \textstyle \Lambda [\rho ]=\sum _{\alpha }K_{\alpha }\rho K_{\alpha }^{\dagger }}$

Individual evidence

1. ^ Angel Rivas, Susana F. Huelga: Open quantum systems . Berlin: Springer, 2012. doi : 10.1007 / 978-3-642-23354-8