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In higher category theory
In quantum physics and in particular in the context of quantum computing a linear map between spaces of linear operators (hence an “operator on operators”) is sometimes called a superoperator.
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.)
Specifically, in a symmetric monoidal closed category with internal hom denoted , a superoperator is a morphism of a form like
If is also compact closed then (1) is isomorphic to a morphism of the form
(whence one also speaks of operators on spaces of (density) matrices)
which in turn is adjunct, in particular, to a morphism of the form
Sometimes (3) is the more transparent incarnation of superoperators:
For example, if is even a dagger-compact category, then a superoperator in the form (1) or (2) is a completely positive map (and hence restricts to a “quantum operation” on density matrices) iff its adjunct (3) is a positive operator [Selinger 2005 Cor. 4.13].
Traditional discussion:
Karol Życzkowski, Ingemar Bengtsson, Section 4 of: On Duality between Quantum Maps and Quantum States, Open Systems & Information Dynamics 11 01 (2004) 3-42 [doi:10.1023/B:OPSY.0000024753.05661.c2]
Greg Kuperberg, Section 1.4 of: A concise introduction to quantum probability, quantum mechanics, and quantum computation (2005) [pdf, pdf]
See also:
In terms of string diagrams in dagger-compact categories (cf. quantum information theory via dagger-compact categories):
Formalizing superoperators as arrows (in computer science):
Last revised on September 23, 2023 at 15:05:10. See the history of this page for a list of all contributions to it.