Regularity of the Capacity in Operator Scaling

Abstract: Completely positive operators are fundamental objects in quantum information theory and the capacity of such operators plays a pivotal role in the operator scaling algorithm. Using this algorithm, Garg--Gurvits--Oliveira--Wigderson recently established a certain quantitative continuity result for the capacity map at rational points. We show by different means that operator capacity possesses significantly greater regularity. Our argument gives local H\"{o}lder continuity at all points and rests on a result of Bennett--Bez--Buschenhenke--Cowling--Flock on weighted sums of exponential functions.