Entanglement and nonclassical properties of hypergraph states

Hypergraph states are multi-qubit states that form a subset of the locally maximally entangleable states and a generalization of the well--established notion of graph states. Mathematically, they can conveniently be described by a hypergraph that indicates a possible generation procedure of these states; alternatively, they can also be phrased in terms of a non-local stabilizer formalism. In this paper, we explore the entanglement properties and nonclassical features of hypergraph states. First, we identify the equivalence classes under local unitary transformations for up to four qubits, as well as important classes of five- and six-qubit states, and determine various entanglement properties of these classes. Second, we present general conditions under which the local unitary equivalence of hypergraph states can simply be decided by considering a finite set of transformations with a clear graph-theoretical interpretation. Finally, we consider the question whether hypergraph states and their correlations can be used to reveal contradictions with classical hidden variable theories. We demonstrate that various noncontextuality inequalities and Bell inequalities can be derived for hypergraph states.Comment: 29 pages, 5 figures, final versio

Tensor network representations from the geometry of entangled states

Tensor network states provide successful descriptions of strongly correlated quantum systems with applications ranging from condensed matter physics to cosmology. Any family of tensor network states possesses an underlying entanglement structure given by a graph of maximally entangled states along the edges that identify the indices of the tensors to be contracted. Recently, more general tensor networks have been considered, where the maximally entangled states on edges are replaced by multipartite entangled states on plaquettes. Both the structure of the underlying graph and the dimensionality of the entangled states influence the computational cost of contracting these networks. Using the geometrical properties of entangled states, we provide a method to construct tensor network representations with smaller effective bond dimension. We illustrate our method with the resonating valence bond state on the kagome lattice.Comment: 35 pages, 9 figure

Towards Ryser\u27s Conjecture: Bounds on the Cardinality of Partitioned Intersecting Hypergraphs

This work is motivated by the open conjecture concerning the size of a minimum vertex cover in a partitioned hypergraph. In an r-uniform r-partite hypergraph, the size of the minimum vertex cover C is conjectured to be related to the size of its maximum matching M by the relation (|C|\u3c= (r-1)|M|). In fact it is not known whether this conjecture holds when |M| = 1. We consider r-partite hypergraphs with maximal matching size |M| = 1, and pose a novel algorithmic approach to finding a vertex cover of size (r - 1) in this case. We define a reactive hypergraph to be a back-and-forth algorithm for a hypergraph which chooses new edges in response to a choice of vertex cover, and prove that this algorithm terminates for all hypergraphs of orders r = 3 and 4. We introduce the idea of optimizing the size of the reactive hypergraph and find that the reactive hypergraph terminates for r = 5...20. We then consider the case where the intersection of any two edges is exactly 1. We prove bounds on the size of this 1-intersecting hypergraph and relate the 1-intersecting hypergraph maximization problem to mutually orthogonal Latin squares. We propose a generative algorithm for 1-intersecting hypergraphs of maximal size for prime powers r-1 = pd under the constraint pd+1 is also a prime power of the same form, and therefore pose a new generating algorithm for MOLS based upon intersecting hypergraphs. We prove this algorithm generates a valid set of mutually orthogonal Latin squares and prove the construction guarantees certain symmetric properties. We conclude that a conjecture by Lovasz, that the inequality in Ryser\u27s Conjecture cannot be improved when (r-1) is a prime power, is correct for the 1-intersecting hypergraph of prime power orders

Distinguishing partitions of complete multipartite graphs

A \textit{distinguishing partition} of a group $X$ with automorphism group ${aut}(X)$ is a partition of $X$ that is fixed by no nontrivial element of ${aut}(X)$. In the event that $X$ is a complete multipartite graph with its automorphism group, the existence of a distinguishing partition is equivalent to the existence of an asymmetric hypergraph with prescribed edge sizes. An asymptotic result is proven on the existence of a distinguishing partition when $X$ is a complete multipartite graph with $m_1$ parts of size $n_1$ and $m_2$ parts of size $n_2$ for small $n_1$, $m_2$ and large $m_1$, $n_2$. A key tool in making the estimate is counting the number of trees of particular classes

Finite-Function-Encoding Quantum States

We investigate the encoding of higher-dimensional logic into quantum states. To that end we introduce finite-function-encoding (FFE) states which encode arbitrary $d$-valued logic functions and investigate their structure as an algebra over the ring of integers modulo $d$. We point out that the polynomiality of the function is the deciding property for associating hypergraphs to states. Given a polynomial, we map it to a tensor-edge hypergraph, where each edge of the hypergraph is associated with a tensor. We observe how these states generalize the previously defined qudit hypergraph states, especially through the study of a group of finite-function-encoding Pauli stabilizers. Finally, we investigate the structure of FFE states under local unitary operations, with a focus on the bipartite scenario and its connections to the theory of complex Hadamard matrices.Comment: Comments welcom

Sparse Hypergraphs and Pebble Game Algorithms

A hypergraph G=(V,E) is (k,ℓ)-sparse if no subset V′⊂V spans more than k|V′|−ℓ hyperedges. We characterize (k,ℓ)-sparse hypergraphs in terms of graph theoretic, matroidal and algorithmic properties. We extend several well-known theorems of Haas, Lovász, Nash-Williams, Tutte, and White and Whiteley, linking arboricity of graphs to certain counts on the number of edges. We also address the problem of finding lower-dimensional representations of sparse hypergraphs, and identify a critical behavior in terms of the sparsity parameters k and ℓ. Our constructions extend the pebble games of Lee and Streinu [A. Lee, I. Streinu, Pebble game algorithms and sparse graphs, Discrete Math. 308 (8) (2008) 1425–1437] from graphs to hypergraphs