nLab superoperator

Contents

Context

Linear algebra

homotopy theory, (∞,1)-category theory, homotopy type theory

flavors: stable, equivariant, rational, p-adic, proper, geometric, cohesive, directed

models: topological, simplicial, localic, …

see also algebraic topology

Introductions

Definitions

Paths and cylinders

Homotopy groups

Basic facts

Theorems

Monoidal categories

monoidal categories

With braiding

With duals for objects

With duals for morphisms

With traces

Closed structure

Special sorts of products

Semisimplicity

Morphisms

Internal monoids

Examples

Theorems

In higher category theory

Contents

Idea

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.)

Details

Specifically, in a symmetric monoidal closed category (𝒞,,𝟙)(\mathcal{C}, \otimes, \mathbb{1}) with internal hom denoted (-)(-)(\text{-})\multimap (\text{-}), a superoperator is a morphism of a form like

(1)()(𝒦𝒦). (\mathscr{H} \multimap \mathscr{H}) \longrightarrow (\mathscr{K} \multimap \mathscr{K}) \,.

If (𝒞,,𝟙)(\mathcal{C}, \otimes, \mathbb{1}) is also compact closed then (1) is isomorphic to a morphism of the form

(2) *𝒦𝒦 * \mathscr{H} \otimes \mathscr{H}^\ast \longrightarrow \mathscr{K} \otimes \mathscr{K}^\ast

(whence one also speaks of operators on spaces of (density) matrices)

which in turn is adjunct, in particular, to a morphism of the form

(3)𝒦 *𝒦 *. \mathscr{K}^\ast \otimes \mathscr{H} \longrightarrow \mathscr{K}^\ast \otimes \mathscr{H} \mathrlap{\,.}

Sometimes (3) is the more transparent incarnation of superoperators:

For example, if (𝒞,,𝟙)(\mathcal{C}, \otimes, \mathbb{1}) 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].

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

References

Traditional discussion:

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.