subunital

Subunital is a term used in operator algebras and quantum information to describe a type of linear map between C*-algebras or operator systems. A linear map φ: A → B is called subunital if it is positive (maps positive elements to positive elements) and satisfies φ(1_A) ≤ 1_B, i.e., the image of the identity is a positive contraction in B. If φ(1_A) = 1_B, the map is unital; subunital is a weaker condition.

Examples include corner or compression maps. For instance, φ(a) = p a p on a C*-algebra A, where p is a projection in B with p ≤ 1_B, is subunital; it is unital only if p = 1_B. Block-truncation maps in matrix algebras that keep a subblock and send the rest to zero are also subunital.

Properties and relations: For completely positive maps between C*-algebras, subunitality typically implies norm contraction, i.e., ||φ|| ≤ 1. In quantum information theory, subunital completely positive maps model trace-nonincreasing evolutions, meaning they describe processes that may lose information but do not create it, and they become trace-preserving when augmented with a suitable extension or conditioning.

The concept is closely tied to unital maps (which satisfy φ(1_A) = 1_B) and to Stinespring-style dilations. If φ is subunital and completely positive, there exists a Stinespring dilation φ(a) = V^* π(a) V with V^* V ≤ I, reflecting a dilation to a larger system where the evolution is implemented by an isometry followed by a projection.

Subunital maps thus provide a natural framework for describing positive, contractive evolutions in operator-algebraic and quantum contexts.

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