Dimension-independent approximation of η-idempotent UCP maps by idempotent UCP maps

Determine whether every η-idempotent unital completely positive map Φ: B(H) → B(H), meaning ||Φ^2 − Φ||_cb ≤ η, can be approximated in completely bounded norm by an idempotent unital completely positive map Ψ with a universal, dimension-independent error bound ||Ψ − Φ||_cb ≤ f(η), for example f(η) = O(√η). Establish such a bound uniformly over all Hilbert space dimensions, or otherwise characterize the optimal universal dependence on η.

Background

The paper studies unital completely positive (UCP) maps Φ on operator algebras that are only approximately idempotent, quantified by ||Φ^2 − Φ||_cb ≤ η. A natural idea is to approximate Φ by an exactly idempotent map using functional calculus, e.g., Θ = θ(2Φ − 1). Although Θ is idempotent and cb-close to Φ, it need not be completely positive, as shown by an explicit 2×2 example.

This raises the central question of whether one can instead approximate any η-idempotent UCP map by a genuinely idempotent UCP map with an accuracy that depends only on η and not on the system dimension or other parameters. The text notes that for the exhibited example there exists an idempotent UCP map that is O(√η)-close, suggesting a potential universal dependence such as O(√η).

Resolving this would give a direct route to representing approximately idempotent channels by genuine idempotent channels with dimension-independent control, strengthening the structural results on the associated approximate C*-algebras and their factorization properties.