4 research outputs found
Curious properties of free hypergraph C*-algebras
A finite hypergraph consists of a finite set of vertices and a
collection of subsets which we consider as partition
of unity relations between projection operators. These partition of unity
relations freely generate a universal C*-algebra, which we call the "free
hypergraph C*-algebra" . General free hypergraph C*-algebras were first
studied in the context of quantum contextuality. As special cases, the class of
free hypergraph C*-algebras comprises quantum permutation groups, maximal group
C*-algebras of graph products of finite cyclic groups, and the C*-algebras
associated to quantum graph homomorphism, isomorphism, and colouring.
Here, we conduct the first systematic study of aspects of free hypergraph
C*-algebras. We show that they coincide with the class of finite colimits of
finite-dimensional commutative C*-algebras, and also with the class of
C*-algebras associated to synchronous nonlocal games. We had previously shown
that it is undecidable to determine whether is nonzero for given .
We now show that it is also undecidable to determine whether a given
is residually finite-dimensional, and similarly whether it only has
infinite-dimensional representations, and whether it has a tracial state. It
follows that for each one of these properties, there is such that the
question whether has this property is independent of the ZFC axioms,
assuming that these are consistent. We clarify some of the subtleties
associated with such independence results in an appendix.Comment: 19 pages. v2: minor clarifications. v3: terminology 'free hypergraph
C*-algebra', added Remark 2.2
Quantum Bilinear Optimization
We study optimization programs given by a bilinear form over noncommutative variables subject to linear inequalities. Problems of this form include the entangled value of two-prover games, entanglement-assisted coding for classical channels, and quantum-proof randomness extractors. We introduce an asymptotically converging hierarchy of efficiently computable semidefinite programming (SDP) relaxations for this quantum optimization. This allows us to give upper bounds on the quantum advantage for all of these problems. Compared to previous work of Pironio, Navascués, and AcÃn [SIAM J. Optim., 20 (2010), pp. 2157-2180], our hierarchy has additional constraints. By means of examples, we illustrate the importance of these new constraints both in practice and for analytical properties. Moreover, this allows us to give a hierarchy of SDP outer approximations for the completely positive semidefinite cone introduced by Laurent and Piovesan
Lower bounds on matrix factorization ranks via noncommutative polynomial optimization
We use techniques from (tracial noncommutative) polynomial optimization to formulate hierarchies of semidefinite programming lower bounds on matrix factorization ranks. In particular, we consider the nonnegative rank, the completely positive rank, and their symmetric analogues: the positive semidefinite rank and the completely positive semidefinite rank. We study the convergence properties of our hierarchies, compare them extensively to known lower bounds, and provide some (numerical) examples