Super-operators

Let's now consider mappings from the space of mappings $\mathcal{H} \rightarrow \mathcal{H}$ to itself, i.e. $(\mathcal{H} \rightarrow \mathcal{H}) \rightarrow (\mathcal{H} \rightarrow \mathcal{H})$. In operator notation we also call these objects super-operators. With the operators $A,B$ and the super-operator $S$ the basis independent expression is denoted by

\[A = S B\]

In contrast, for the basis specific version we have to choose two possibly different bases for A which we denote as $\{|u\rangle\}$ and $\{|v\rangle\}$ and additionally two, also possibly different bases for B, $\{|m\rangle\}$ and $\{|n\rangle\}$.

\[A &= \sum_{uv} A_{uv} |v \rangle \langle u| \\ B &= \sum_{mn} B_{mn} |n \rangle \langle m| \\ S &= \sum_{uvmn} S_{uvmn} |v \rangle \langle u| \otimes |n \rangle \langle m|\]

The coefficients are then connected by

\[A_{uv} = \sum_{mn} S_{uvmn} B_{mn}\]

The implementation of super-operators in QuantumOptics.jl therefore has to know about four, possibly different, bases. The two basis choices for the codomain (output) are stored in the basis_l field and the two basis choices for the domain (input) are stored in the basis_r field. At the moment there is one concrete SuperOperator type implemented.

Besides the expected algebraic operations there are a few additional functions that help creating and working with super-operators:

• spre
• spost
• liouvillian
• exp(::SuperOperator)