Superoperator

Fix a choice of basis for the underlying Hilbert space ${\displaystyle \{|i\rangle \}_{i}}$.

Defining the left and right multiplication superoperators by ${\displaystyle {\mathcal {L}}(A)[\rho ]=A\rho }$ and ${\displaystyle {\mathcal {R}}(A)[\rho ]=\rho A}$ respectively one can express the commutator as

${\displaystyle [A,\rho ]={\mathcal {L}}(A)[\rho ]-{\mathcal {R}}(A)[\rho ].}$

Next we vectorize the matrix ${\displaystyle \rho }$ which is the mapping

${\displaystyle \rho =\sum _{i,j}\rho _{ij}|i\rangle \langle j|\mapsto |\rho \rangle \!\rangle =\sum _{i,j}\rho _{ij}|i\rangle \otimes |j\rangle ,}$

where ${\displaystyle |\cdot \rangle \!\rangle }$ denotes a vector in the Fock-Liouville space. The matrix representation of ${\displaystyle {\mathcal {L}}(A)}$ is then calculated by using the same mapping

${\displaystyle A\rho =\sum _{i,j}\rho _{ij}A|i\rangle \langle j|\mapsto \sum _{i,j}\rho _{ij}(A|i\rangle )\otimes |j\rangle =\sum _{i,j}\rho _{ij}(A\otimes I)(|i\rangle \otimes |j\rangle )=(A\otimes I)|\rho \rangle \!\rangle ={\mathcal {L}}(A)[\rho ],}$

indicating that ${\displaystyle {\mathcal {L}}(A)=A\otimes I}$. Similarly one can show that ${\displaystyle {\mathcal {R}}(A)=(I\otimes A^{T})}$. These representations allows us to calculate things like eigenvalues associated to superoperators. These eigenvalues are particularly useful in the field of open quantum systems, where the real parts of the Lindblad superoperator's eigenvalues will indicate whether a quantum system will relax or not.

In quantum mechanics the Schrödinger equation,

${\displaystyle i\hbar {\frac {\partial }{\partial t}}\Psi ={\hat {H}}\Psi }$,

expresses the time evolution of the state vector ${\displaystyle \psi }$ by the action of the Hamiltonian ${\displaystyle {\hat {H}}}$ which is an operator mapping state vectors to state vectors.

In the more general formulation of John von Neumann, statistical states and ensembles are expressed by density operators rather than state vectors. In this context the time evolution of the density operator is expressed via the von Neumann equation in which density operator is acted upon by a superoperator ${\displaystyle {\mathcal {H}}}$ mapping operators to operators. It is defined by taking the commutator with respect to the Hamiltonian operator:

${\displaystyle i\hbar {\frac {\partial }{\partial t}}\rho ={\mathcal {H}}[\rho ]}$

where

${\displaystyle {\mathcal {H}}[\rho ]=[{\hat {H}},\rho ]\equiv {\hat {H}}\rho -\rho {\hat {H}}}$

As commutator brackets are used extensively in quantum mechanics this explicit superoperator presentation of the Hamiltonian's action is typically omitted.

When considering an operator valued function of operators ${\displaystyle {\hat {H}}={\hat {H}}({\hat {P}})}$ as for example when we define the quantum mechanical Hamiltonian of a particle as a function of the position and momentum operators, we may (for whatever reason) define an “Operator Derivative” ${\displaystyle {\frac {\Delta {\hat {H}}}{\Delta {\hat {P}}}}}$ as a superoperator mapping an operator to an operator.

For example, if ${\displaystyle H(P)=P^{3}=PPP}$ then its operator derivative is the superoperator defined by:

${\displaystyle {\frac {\Delta H}{\Delta P}}[X]=XP^{2}+PXP+P^{2}X}$

This “operator derivative” is simply the Jacobian matrix of the function (of operators) where one simply treats the operator input and output as vectors and expands the space of operators in some basis. The Jacobian matrix is then an operator (at one higher level of abstraction) acting on that vector space (of operators).