Notably density matrices representing quantum states are given by linear operators and a quantum operation is a superoperator that preserves the defining properties of density matrices. (Beware that sometimes different or no distinctions between “quantum operation” and “superoperator” are made.)

Related entries

The Geometry of Interaction-construction is a genral abstract construction of a category of superoperators. See there for more.

References

Greg Kuperberg, section 1.4 of A concise introduction to quantum probability, quantum mechanics, and quantum computation, 2005 (pdf)