Irreducible magic sets for $n$-qubit systems

TL;DR

This paper identifies new irreducible magic sets for 3 to 6 qubits, expanding understanding of quantum advantage structures beyond known two- and three-qubit configurations, and provides algorithms for their analysis.

Contribution

It demonstrates the existence of irreducible magic sets requiring more than three qubits and generalizes Arkhipov's theorem with an efficient decision algorithm.

Findings

Existence of irreducible magic sets for 3 to 6 qubits.

Generalized Arkhipov's theorem with an algorithm for hypergraph compatibility.

Determined tight bounds for noncontextuality inequalities.

Abstract

Magic sets of observables are minimal structures that capture quantum state-independent advantage for systems of $n\ge 2$ qubits and are, therefore, fundamental tools for investigating the interface between classical and quantum physics. A theorem by Arkhipov (arXiv:1209.3819) states that $n$-qubit magic sets in which each observable is in exactly two subsets of compatible observables can be reduced either to the two-qubit magic square or the three-qubit magic pentagram [N. D. Mermin, Phys. Rev. Lett. 65, 3373 (1990)]. An open question is whether there are magic sets that cannot be reduced to the square or the pentagram. If they exist, a second key question is whether they require $n >3$ qubits, since, if this is the case, these magic sets would capture minimal state independent quantum advantage that is specific for $n$-qubit systems with specific values of $n$. Here, we answer both questions affirmatively. We identify magic sets which cannot be reduced to the square or the pentagram and require $n=3,4,5$, or $6$ qubits. In addition, we prove a generalized version of Arkhipov's theorem providing an efficient algorithm for, given a hypergraph, deciding whether or not it can accommodate a magic set, and solve another open problem, namely, given a magic set, obtaining the tight bound of its associated noncontextuality inequality.