[1] Sometimes the term refers more specially to a completely positive map which also preserves or does not increase the trace of its argument.
This specialized meaning is used extensively in the field of quantum computing, especially quantum programming, as they characterise mappings between density matrices.
The use of the super- prefix here is in no way related to its other use in mathematical physics.
That is to say superoperators have no connection to supersymmetry and superalgebra which are extensions of the usual mathematical concepts defined by extending the ring of numbers to include Grassmann numbers.
Fix a choice of basis for the underlying Hilbert space
Defining the left and right multiplication superoperators by
respectively one can express the commutator as Next we vectorize the matrix
denotes a vector in the Fock-Liouville space.
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, expresses the time evolution of the state vector
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
It is defined by taking the commutator with respect to the Hamiltonian operator: where As commutator brackets are used extensively in QM this explicit superoperator presentation of the Hamiltonian's action is typically omitted.
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”
The Jacobian matrix is then an operator (at one higher level of abstraction) acting on that vector space (of operators).