Norm Bounds on Sums of Positive Operators with Applications to Monogamy Games and Quantum Key Distribution

Abstract

Bounding the acceptance probabilities of quantum measurements is a recurring task in quantum information theory. An acceptance probability averaged over a random measurement setting can be bounded uniformly over the input state by the operator norm of an average of positive semidefinite operators. We prove a general inequality for such norms, valid for arbitrary finite families of positive semidefinite operators and averaging distributions. We give two applications of our operator-norm inequality. First, we determine the sharp robustness threshold for the parallel BB84 monogamy-of-entanglement game. For error thresholds $β,γ\in[0,1/2]$, when the two players may make at most $βn$ and $γn$ errors, respectively, the optimal winning probability decays exponentially if $(1-2β)^2+(1-2γ)^2>1$, tends to one exponentially fast under the reverse strict inequality, and remains bounded away from zero on the boundary. Second, we establish the achievability of every asymptotic key rate $0<R<1- 2h_2(γ)$ for one-sided device-independent BB84 quantum key distribution with tolerated bit-error rate $γ$ and asymptotically optimal reconciliation, where $h_2(\cdot)$ is the binary entropy function. Alice&#39;s measurements are trusted, while Bob may perform an arbitrary joint measurement depending on the entire basis string, subject to the protocol&#39;s receipt and isolation requirements. This recovers the usual BB84 rate in this model.