Characterization of UCP-complemented operator systems and global UCP cocycles

Characterize which higher-dimensional accessible operator systems are complemented by a unital completely positive projection, and determine necessary and sufficient conditions for the existence of a global unital completely positive cocycle when such projections are unavailable.

Background

For a non-invertible Heisenberg evolution, the dynamically accessible observables at time s form the operator system S_s = Im Λ_s*. The paper proves that Heisenberg CP-divisibility guarantees completely positive reduced propagators on these accessible systems, while Arveson’s extension theorem supplies pairwise unital completely positive extensions to the full matrix algebra. However, independently chosen extensions need not satisfy the composition law required of a propagator cocycle.

The paper establishes a sufficient condition for reconciling complete positivity with composition: each accessible operator system must admit a unital completely positive projection from the full matrix algebra onto that system. This condition is not automatic, as demonstrated by the continuous qutrit counterexample, and the authors explicitly leave open the task of characterizing such complemented systems in higher dimensions and of determining when a globally composable UCP family exists without them.