No-signaling Quantum Key Distribution: Solution by Linear Programming

TL;DR

This paper presents a linear programming method to determine secret key rates in no-signaling quantum key distribution, simplifying calculations and enabling potential generalizations beyond existing approaches.

Contribution

It introduces a straightforward linear programming approach to bound Eve's guessing probability, improving upon prior methods for calculating secret key rates in no-signaling QKD.

Findings

Achieved positive key rates consistent with known results

Provided a simpler, more generalizable calculation method

Demonstrated the approach with binary Eve outcomes

Abstract

We outline a straightforward approach for obtaining a secret key rate using only no-signaling constraints and linear programming. Assuming an individual attack, we consider all possible joint probabilities. Initially, we study only the case where Eve has binary outcomes, and we impose constraints due to the no-signaling principle and given measurement outcomes. Within the remaining space of joint probabilities, by using linear programming, we get bound on the probability of Eve correctly guessing Bob's bit. We then make use of an inequality that relates this guessing probability to the mutual information between Bob and a more general Eve, who is not binary-restricted. Putting our computed bound together with the Csisz\'ar-K\"orner formula, we obtain a positive key generation rate. The optimal value of this rate agrees with known results, but was calculated in a more straightforward way, offering the potential of generalization to different scenarios.