Bounds on the power of proofs and advice in general physical theories

TL;DR

This paper explores how physical principles in various theories constrain the computational and communicative power of systems, revealing bounds and conditions for powerful computation beyond quantum theory.

Contribution

It establishes bounds on computation with advice and interactive proof systems in general physical theories, highlighting the role of non-trivial dynamics and measurement capabilities.

Findings

Non-trivial dynamics imply bounds on advice-based computation.

A theory with trivial dynamics can have unbounded advice computation.

Interactive proof systems are bounded in theories with local tomography.

Abstract

Quantum theory presents us with the tools for computational and communication advantages over classical theory. One approach to uncovering the source of these advantages is to determine how computation and communication power vary as quantum theory is replaced by other operationally-defined theories from a broad framework of such theories. Such investigations may reveal some of the key physical features required for powerful computation and communication. In this paper we investigate how simple physical principles bound the power of two different computational paradigms which combine computation and communication in a non-trivial fashion: computation with advice and interactive proof systems. We show that the existence of non-trivial dynamics in a theory implies a bound on the power of computation with advice. Moreover, we provide an explicit example of a theory with no non-trivial dynamics in which the power of computation with advice is unbounded. Finally we show that the power of simple interactive proof systems in theories where local measurements suffice for tomography is non-trivially bounded. This result provides a proof that QMA is contained in PP which does not make use of any uniquely quantum structure - such as the fact that observables correspond to self-adjoint operators - and thus may be of independent interest.