# On the impossibility of coin-flipping in generalized probabilistic theories via discretizations of semi-infinite programs

## TL;DR

This paper proves that ideal coin-flipping is impossible in generalized probabilistic theories under the Generalized No-Restriction Hypothesis, using a novel semi-infinite programming approach to model cheating strategies.

## Contribution

It introduces a new formalism of semi-infinite programs for analyzing cheating strategies in cryptographic tasks within generalized probabilistic theories.

## Key findings

- Coin-flipping impossible in classical, quantum, and generalized probabilistic theories.
- Semi-infinite programs effectively model cheating strategies.
- New formalism may benefit future cryptographic and quantum information research.

## Abstract

Coin-flipping is a fundamental cryptographic task where a spatially separated Alice and Bob wish to generate a fair coin-flip over a communication channel. It is known that ideal coin-flipping is impossible in both classical and quantum theory. In this work, we give a short proof that it is also impossible in generalized probabilistic theories under the Generalized No-Restriction Hypothesis. Our proof relies crucially on a formulation of cheating strategies as semi-infinite programs, i.e., cone programs with infinitely many constraints. This introduces a new formalism which may be of independent interest to the quantum community.