Primitive Nonclassical Structures of the $N$-qubit Pauli Group

Several types of nonclassical structures within the $N$-qubit Pauli group that can be seen as fundamental resources for quantum information processing are presented and discussed. Identity Products (IDs), structures fundamentally related to entanglement, are defined and explored. The Kochen-Specker theorem is proved by particular sets of IDs that we call KS sets. We also present a new theorem that we call the $N$-qubit Kochen-Specker theorem, which is proved by particular sets of IDs that we call NKS sets, and whose utility is that it leads to efficient constructions for KS sets. We define the criticality, or irreducibility, of these structures, and its connection to entanglement. All representative critical IDs for up to $N=4$ qubits are presented, and numerous families of critical IDs for arbitrarily large values of $N$ are discussed. The critical IDs for a given $N$ are a finite set of geometric objects that appear to fully characterize the nonclassicality of the $N$-qubit Pauli group. Methods for constructing critical KS sets and NKS sets from IDs are given, and experimental tests of entanglement, contextuality, and nonlocality are discussed. Possible applications and connections to other work are also discussed