Primitives of Nonclassicality in the N-qubit Pauli Group

Throughout the development of quantum mechanics, the striking refusal of nature to obey classical reasoning and intuition has driven both curiosity and confusion. From the apparent inescapably probabilistic nature of the theory, to more subtle issues such as entanglement, nonlocality, and contextuality, it has always been the `nonclassical’ features that present the most interesting puzzle. More recently, it has become apparent that these features are also the primary resource for quantum information processing.
In this talk, we will introduce several types of nonclassical logical structures contained within the N-qubit Pauli group, corresponding in general to preparation and/or measurement schemes for systems of several qubits that demonstrate violation of some notion of classical reasoning. These structures are geometric in nature, and we identify the primitive elements from which more elaborate types are constructed. We will review the key types of structures that are available and explain their hierarchy. Finally we will use them to give simple and transparent proofs of entanglement correlations, quantum contextuality via the Kochen-Specker theorem, and quantum nonlocality via the Bell-GHZ theorem.
These structures have many applications in quantum information processing, but the real purpose of this talk is simply to introduce unfamiliar researchers to the simplest known logical proofs of these nonclassicality theorems, which can be understood using only the algebra of the Pauli spin matrices and simple counting arguments.