Optimal Measurement Structures for Contextuality Applications

TL;DR

This paper introduces optimal measurement structures called gadgets for quantum contextuality, enabling advancements in quantum communication, theory separation, and randomness generation, while also exploring higher-order gadgets for foundational proofs.

Contribution

It presents the design of optimal gadgets and higher-order gadgets for contextuality applications, advancing the understanding and construction of Kochen-Specker proofs.

Findings

Gadgets enable entanglement-assisted zero-error communication.

Large separations between quantum and probabilistic theories identified.

Optimal tests for contextuality-based randomness generation developed.

Abstract

The Kochen-Specker (KS) theorem is a corner-stone result in the foundations of quantum mechanics describing the fundamental difference between quantum theory and classical non-contextual theories. Recently specific substructures termed $01$-gadgets were shown to exist within KS proofs that capture the essential contradiction of the theorem. Here, we show these gadgets and their generalizations provide an optimal toolbox for contextuality applications including (i) constructing classical channels exhibiting entanglement-assisted advantage in zero-error communication, (ii) identifying large separations between quantum theory and binary generalised probabilistic theories, and (iii) finding optimal tests for contextuality-based semi-device-independent randomness generation. Furthermore, we introduce and study a generalisation to definite prediction sets for more general logical propositions, that we term higher-order gadgets. We pinpoint the role these higher-order gadgets play in KS proofs by identifying these as induced subgraphs within KS graphs and showing how to construct proofs of state-independent contextuality using higher-order gadgets as building blocks. The constructions developed here may help in solving some of the remaining open problems regarding minimal proofs of the Kochen-Specker theorem.