Tracenonincreasing

Trace-nonincreasing is a term used in linear algebra and quantum information theory to describe a property of linear maps acting on operators. A map Λ is trace-nonincreasing if for every positive semidefinite operator X, the trace does not increase under the map: Tr(Λ(X)) ≤ Tr(X). If X is a density operator with Tr(X) = 1, this means the output state has trace at most 1.

In the context of quantum operations, trace-nonincreasing maps are often considered together with complete positivity. A completely positive trace-nonincreasing (CPTNI) map Λ can be represented in Kraus form as Λ(ρ) = ∑i Mi ρ Mi†, where the Kraus operators Mi satisfy ∑i Mi† Mi ≤ I. When the inequality is saturated (∑i Mi† Mi = I), the map is trace-preserving, also called a completely positive trace-preserving (CPTP) map. If the inequality is strict, the map describes a probabilistic or post-selected process, where the overall probability of obtaining any outcome is less than or equal to 1.

Physically, trace-nonincreasing maps model processes where some outcomes are discarded or where a measurement only sometimes succeeds. For example, a measurement with a subset of accepted outcomes corresponds to a CPTNI map with Kraus operators associated to the accepted outcomes, and the total probability of success is Tr(Λ(ρ)). These maps are fundamental in describing probabilistic quantum operations, quantum filtering, and post-selected quantum computing.

Related concepts include CPTP maps (trace-preserving), positive maps, and Kraus representations. Trace-nonincreasing maps generalize the notion of physical quantum evolutions to probabilistic or conditional scenarios.