Contextuality with Pauli observables in cycle scenarios

TL;DR

This paper systematically studies Pauli contextuality in cycle scenarios, establishing bounds on realizability, demonstrating that only 4-cycle scenarios can witness contextuality, and analyzing the role of cycles in general Pauli scenarios.

Contribution

It provides the first detailed bounds on cycle realizability with multi-qubit Paulis and characterizes when Pauli scenarios can exhibit contextuality, especially highlighting the special role of 4-cycles.

Findings

Maximum cycle size realizable by m-qubit Paulis is 3m.

All 4-cycle Pauli realizations exhibit contextuality.

No n-cycle (n>4) Pauli realization can witness contextuality.

Abstract

Contextuality is a fundamental marker of quantum non-classicality, which has emerged as a key resource for quantum computational advantage in multiple settings. Many such results hinge specifically on contextuality witnessed through Pauli measurements. In this work, we initiate a systematic study of (state-dependent) Pauli contextuality, focusing on cycle measurement scenarios, the simplest scenarios capable of exhibiting contextual behaviour. First, we study realizability of cycle scenarios with multi-qubit Pauli observables: we show that the maximum size of a cycle faithfully realizable by $m$-qubit Paulis is upper bounded by $3m$, while we construct explicit realizations of cycles of size $2m-1$ or $2m$, depending on whether $m \not\equiv 1 \pmod{3}$ or $m \equiv 1 \pmod{3}$. Then, we investigate the presence of contextuality: we prove that no $n$-cycle Pauli realization for $n > 4$ can witness contextuality (on any quantum state), whereas for $n = 4$ every Pauli realization exhibits contextuality, attaining the quantum bound for all noncontextuality inequalities on some pure state. Finally, we discuss arbitrary Pauli scenarios in light of Vorob'ev's theorem, and show that, contrary to what our cycle characterization might suggest, the presence of $4$-cycles is not necessary for witnessing contextuality in general Pauli scenarios.