Algebraic quantum structures for reference frame-independent quantum teleportation and pseudo-telepathy - Verdon, D (2019)
Part I: We present two new schemes for teleportation of a quantum state between parties whose local reference frames are misaligned by the action of a compact Lie group G. These schemes require no prior alignment of reference frames and are unaffected by arbitrary changes in reference frame alignment during execution, suiting them to situations of rapid reference frame drift. Our tight scheme yields improved purity compared to standard teleportation, in some cases substantially ---including the case of qubit teleportation under arbitrary SU(2) reference frame uncertainty--- while communicating no information about either party's reference frame alignment at any time. Our perfect scheme performs perfect teleportation, but does communicate some reference frame information. The mathematical foundation of these schemes is a unitary error basis permuted up to a phase by the conjugation action of a finite subgroup of G; we completely classify such unitary error bases for qubits, exhibit constructions in higher dimension, and provide a method for proving nonexistence in some cases. Part II: Quantum pseudo-telepathy is a phenomenon whereby two non-communicating parties can use the non-signalling correlations from shared quantum entanglement to perform a task that would ordinarily be impossible without communication. Here we consider the graph isomorphism game: an instance of the game is defined by a pair of graphs, and a perfect deterministic classical strategy precisely corresponds to an isomorphism of those graphs. We call a pair of graphs for which a perfect strategy using shared entanglement (a quantum strategy) exists pseudo-telepathic, and call a quantum strategy a quantum isomorphism between the graphs. Using a deep connection between noncommutative mathematics and nonlocal games, we classify all graphs quantum isomorphic to a given graph in terms of the quantum automorphism group of the graph. This implies a construction of quantum isomorphisms from central type subgroups of the ordinary automorphism group; we give an explicit description of the resulting quantum strategy and the new graph. We consider generation of new examples of quantum contextuality from central type groups. For abelian central type groups we show that this reduces to construction of quantum solutions of a binary linear constraint system from the abelianisation of its homogeneous solution group.