Causal Boxes: Quantum Information-Processing Systems Closed under Composition

Complex information-processing systems, for example quantum circuits, cryptographic protocols, or multi-player games, are naturally described as networks composed of more basic information-processing systems. A modular analysis of such systems requires a mathematical model of systems that is closed under composition, i.e., a network of these objects is again an object of the same type. We propose such a model and call the corresponding systems causal boxes. Causal boxes capture superpositions of causal structures, e.g., messages sent by a causal box A can be in a superposition of different orders or in a superposition of being sent to box B and box C. Furthermore, causal boxes can model systems whose behavior depends on time. By instantiating the Abstract Cryptography framework with causal boxes, we obtain the first composable security framework that can handle arbitrary quantum protocols and relativistic protocols.Comment: 44+24 pages, 16 figures. v3: minor edits based on referee comments, matches published version up to layout. v2: definition of causality weakened, new reference

Unextendible Product Bases, Uncompletable Product Bases and Bound Entanglement

We report new results and generalizations of our work on unextendible product bases (UPB), uncompletable product bases and bound entanglement. We present a new construction for bound entangled states based on product bases which are only completable in a locally extended Hilbert space. We introduce a very useful representation of a product basis, an orthogonality graph. Using this representation we give a complete characterization of unextendible product bases for two qutrits. We present several generalizations of UPBs to arbitrary high dimensions and multipartite systems. We present a sufficient condition for sets of orthogonal product states to be distinguishable by separable superoperators. We prove that bound entangled states cannot help increase the distillable entanglement of a state beyond its regularized entanglement of formation assisted by bound entanglement.Comment: 24 pages RevTex, 15 figures; appendix removed, several small corrections, to appear in Comm. Math. Phy

Clock and Category; IS QUANTUM GRAVITY ALGEBRAIC

We investigate the possibility that the quantum theory of gravity could be constructed discretely using algebraic methods. The algebraic tools are similar to ones used in constructing topological quantum field theories.The algebraic tools are related to ideas about the reinterpretation of quantum mechanics in a general relativistic context.Comment: To appear in special issue of JMP. Latex documen

The Hilbert Zonotope and a Polynomial Time Algorithm for Universal Grobner Bases

We provide a polynomial time algorithm for computing the universal Gr\"obner basis of any polynomial ideal having a finite set of common zeros in fixed number of variables. One ingredient of our algorithm is an effective construction of the state polyhedron of any member of the Hilbert scheme Hilb^d_n of n-long d-variate ideals, enabled by introducing the Hilbert zonotope H^d_n and showing that it simultaneously refines all state polyhedra of ideals on Hilb^d_n

(3+1)-dimensional topological phases and self-dual quantum geometries encoded on Heegard surfaces

We apply the recently suggested strategy to lift state spaces and operators for (2+1)-dimensional topological quantum field theories to state spaces and operators for a (3+1)-dimensional TQFT with defects. We start from the (2+1)-dimensional Turaev-Viro theory and obtain a state space, consistent with the state space expected from the Crane-Yetter model with line defects. This work has important applications for quantum gravity as well as the theory of topological phases in (3+1) dimensions. It provides a self-dual quantum geometry realization based on a vacuum state peaked on a homogeneously curved geometry. The state spaces and operators we construct here provide also an improved version of the Walker-Wang model, and simplify its analysis considerably. We in particular show that the fusion bases of the (2+1)-dimensional theory lead to a rich set of bases for the (3+1)-dimensional theory. This includes a quantum deformed spin network basis, which in a loop quantum gravity context diagonalizes spatial geometry operators. We also obtain a dual curvature basis, that diagonalizes the Walker-Wang Hamiltonian. Furthermore, the construction presented here can be generalized to provide state spaces for the recently introduced dichromatic four-dimensional manifold invariants.Comment: 27 pages, many figures, v2: minor correction