Simple proof of the impossibility of bit-commitment in generalised probabilistic theories using cone programming

TL;DR

This paper proves that perfect bit-commitment cannot be achieved in generalised probabilistic theories by employing cone programming techniques to establish a fundamental trade-off in cheating probabilities, extending to integer-commitment.

Contribution

It introduces a cone programming framework to demonstrate the impossibility of bit-commitment in generalised probabilistic theories, generalizing previous quantum results.

Findings

Impossibility of ideal bit-commitment in generalised probabilistic theories.

A quantitative trade-off between Alice's and Bob's cheating probabilities.

The method extends to the more general task of integer-commitment.

Abstract

Bit-commitment is a fundamental cryptographic task, in which Alice commits a bit to Bob such that she cannot later change the value of the bit, while, simultaneously, the bit is hidden from Bob. It is known that ideal bit-commitment is impossible within quantum theory. In this work, we show that it is also impossible in generalised probabilistic theories (under a small set of assumptions) by presenting a quantitative trade-off between Alice's and Bob's cheating probabilities. Our proof relies crucially on a formulation of cheating strategies as cone programs, a natural generalisation of semidefinite programs. In fact, using the generality of this technique, we prove that this result holds for the more general task of integer-commitment.