Contextuality with Pauli observables in cycle scenarios

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.