Graph-theoretical Bounds on the Entangled Value of Non-local Games

We introduce a novel technique to give bounds to the entangled value of non-local games. The technique is based on a class of graphs used by Cabello, Severini and Winter in 2010. The upper bound uses the famous Lov\'asz theta number and is efficiently computable; the lower one is based on the quantum independence number, which is a quantity used in the study of entanglement-assisted channel capacities and graph homomorphism games.Comment: 10 pages, submission to the 9th Conference on the Theory of Quantum Computation, Communication, and Cryptography (TQC 2014

On preparing ground states of gapped Hamiltonians: An efficient Quantum Lovász Local Lemma

A frustration-free local Hamiltonian has the property that its ground state minimises the energy of all local terms simultaneously. In general, even deciding whether a Hamiltonian is frustration-free is a hard task, as it is closely related to the QMA1-complete quantum satisfiability problem (QSAT) -- the quantum analogue of SAT, which is the archetypal NP-complete problem in classical computer science. This connection shows that the frustration-free property is not only relevant to physics but also to computer science. The Quantum Lovász Local Lemma (QLLL) provides a sufficient condition for frustration-freeness. A natural question is whether there is an efficient way to prepare a frustration-free state under the conditions of the QLLL. Previous results showed that the answer is positive if all local terms commute. In this work we improve on the previous constructive results by designing an algorithm that works efficiently for non-commuting terms as well, assuming that the system is "uniformly" gapped, by which we mean that the system and all its subsystems have an inverse polynomial energy gap. Also, ou

Graph-theoretic approach to dimension witnessing

A fundamental problem in quantum computation and quantum information is finding the minimum quantum dimension needed for a task. For tasks involving state preparation and measurements, this problem can be addressed using only the input-output correlations. This has been applied to Bell, prepare-and-measure, and Kochen-Specker contextuality scenarios. Here, we introduce a novel approach to quantum dimension witnessing for scenarios with one preparation and several measurements, which uses the graphs of mutual exclusivity between sets of measurement events. We present the concepts and tools needed for graph-theoretic quantum dimension witnessing and illustrate their use by identifying novel quantum dimension witnesses, including a family that can certify arbitrarily high quantum dimensions with few events.Comment: 20 pages. V2 corrected minor typos and improved presentatio

Graph parameters from symplectic group invariants

In this paper we introduce, and characterize, a class of graph parameters obtained from tensor invariants of the symplectic group. These parameters are similar to partition functions of vertex models, as introduced by de la Harpe and Jones, [P. de la Harpe, V.F.R. Jones, Graph invariants related to statistical mechanical models: examples and problems, Journal of Combinatorial Theory, Series B 57 (1993) 207-227]. Yet they give a completely different class of graph invariants. We moreover show that certain evaluations of the cycle partition polynomial, as defined by Martin [P. Martin, Enum\'erations eul\'eriennes dans les multigraphes et invariants de Tutte-Grothendieck, Diss. Institut National Polytechnique de Grenoble-INPG; Universit\'e Joseph-Fourier-Grenoble I, 1977], give examples of graph parameters that can be obtained this way.Comment: Some corrections have been made on the basis of referee comments. 21 pages, 1 figure. Accepted in JCT