Primitives of Nonclassicality in the N-qubit Pauli Group
The N-qubit Pauli group contains a natural hierarchy of nonclassical structures related to quantum entanglement, contextuality, and nonlocality. Several of these structures will be introduced and built up starting with the simplest, which we call identity products (IDs). Entangled IDs are sets of mutually commuting N-qubit Pauli observables, whose overall product is the N-qubit identity, and which have maximally entangled joint eigenstates. Structures called N-qubit-entanglement Kochen-Specker (NKS) sets composed of one of more IDs prove a simple No-Go theorem for entanglement-free theories, which in some special cases also provide a proof of the Greenberger-Horne-Zeilinger (GHZ) ``all-versus-nothing'' proof of Bell's nonlocality theorem. NKS sets can be generalized into Kochen-Specker (KS) sets in many ways by adding new IDs, where a KS set proves the Kochen Specker contextuality theorem. Using pre- and post-selected states and weak measurements, we can show confined contextuality in some KS sets. IDs are special geometric objects within the N-qubit Pauli group, and we have computationally enumerated all of them up to certain sizes. A specific subset of IDs are error-correcting codes, and some also provide excellent benchmarks for the nonclassical behavior of quantum hardware.