A Post-Quantum Associative Memory

TL;DR

This paper explores the limitations and capabilities of generalized probabilistic theories (GPTs) in storing and distinguishing large sets of states, demonstrating exponential advantages over classical and quantum models through theoretical proofs and numerical methods.

Contribution

It provides new bounds on the dimension of GPTs needed to store distinguishable states and introduces a numerical approach for analyzing state distinguishability in GPTs.

Findings

GPTs can outperform classical and quantum theories exponentially in certain state discrimination tasks.

The minimal dimension for 2-state storage in GPTs is linear in the number of bits, unlike exponential in classical/quantum.

An efficient numerical method for maximum distinguishability in GPTs is developed.

Abstract

Associative memories are devices storing information that can be fully retrieved given partial disclosure of it. We examine a toy model of associative memory and the ultimate limitations it is subjected to within the framework of general probabilistic theories (GPTs), which represent the most general class of physical theories satisfying some basic operational axioms. We ask ourselves how large the dimension of a GPT should be so that it can accommodate $2^m$ states with the property that any $N$ of them are perfectly distinguishable. Call $d(N,m)$ the minimal such dimension. Invoking an old result by Danzer and Gr\"unbaum, we prove that $d(2,m)=m+1$, to be compared with $O(2^m)$ when the GPT is required to be either classical or quantum. This yields an example of a task where GPTs outperform both classical and quantum theory exponentially. More generally, we resolve the case of fixed $N$ and asymptotically large $m$, proving that $d(N,m) \leq m^{1+o_N(1)}$ (as $m\to\infty$) for every $N\geq 2$, which yields again an exponential improvement over classical and quantum theories. Finally, we develop a numerical approach to the general problem of finding the largest $N$-wise mutually distinguishable set for a given GPT, which can be seen as an instance of the maximum clique problem on $N$-regular hypergraphs.