Almost-idempotent quantum channels and approximate $C^{\ast }$-algebras

Abstract

Let $\Phi$ be a unital completely positive (UCP) map on the space of operators on some Hilbert space. We assume that $\Phi$ is $\eta$-idempotent, namely, $\Vert {\Phi }^2-\Phi {\Vert }_{\mathrm{cb}}\le \eta$, and construct an associated $\epsilon$-$C^{\ast }$ algebra (of almost-invariant observables) for $\epsilon =O(\eta )$. This type of structure has the axioms of a unital $C^{\ast }$ algebra but the associativity and other axioms involving the multiplication and the unit hold up to $\epsilon$. We prove that any finite-dimensional $\epsilon$-$C^{\ast }$ algebra $A$ is $O(\epsilon )$-isomorphic to some genuine $C^{\ast }$ algebra $B$. These bounds are universal, i.e. do not depend on the dimensionality or other parameters. When $A$ comes from a finite-dimensional $\eta$-idempotent UCP map $\Phi$, the $O(\eta )$-isomorphism and its inverse can be realized by UCP maps. This gives an approximate factorization of the quantum channel ${\Phi }^{\ast }$ into a decoding channel, producing a state on $B$, and an encoding channel.