Ignacio Villanueva

Unbounded Violation of Tripartite Bell Inequalities

We prove that there are tripartite quantum states (constructed from random unitaries) that can lead to arbitrarily large violations of Bell inequalities for dichotomic observables. As a consequence these states can withstand an arbitrary amount of white noise before they admit a description within a local hidden variable model. This is in sharp contrast with the bipartite case, where all violations are bounded by Grothendieck’s constant. We will discuss the possibility of determining the Hilbert space dimension from the obtained violation and comment on implications for communication complexity theory. Moreover, we show that the violation obtained from generalized Greenberger-Horne-Zeilinger (GHZ) states is always bounded so that, in contrast to many other contexts, GHZ states do not lead to extremal quantum correlations in this case. In order to derive all these physical consequences, we will have to obtain new mathematical results in the theories of operator spaces and tensor norms. In particular, we will prove the existence of bounded but not completely bounded trilinear forms from commutative C*-algebras. Finally, we will relate the existence of diagonal states leading to unbounded violations with a long-standing open problem in the context of Banach algebras.

Multiple summing operators on Banach spaces

Abstract In this paper, we improve some previous results about multiple p-summing multilinear operators by showing that every multilinear form from L 1 spaces is multiple p-summing for 1⩽p⩽2. The proof is based on the existence of a predual for the Banach space of multiple p-summing multilinear forms. We also show the failure of the inclusion theorem in this class of operators and improve some results of Y. Melendez and A. Tonge about dominated multilinear operators.

Unbounded Violations of Bipartite Bell Inequalities via Operator Space Theory

In this work we show that bipartite quantum states with local Hilbert space dimension n can violate a Bell inequality by a factor of order

Multiple summing operators onC(K) spaces

{{\rm \Omega} \left(\frac{\sqrt{n}}{\log^2n} \right)}

Extensions of multilinear operators and Banach space properties

when observables with n possible outcomes are used. A central tool in the analysis is a close relation between this problem and operator space theory and, in particular, the very recent noncommutative Lp embedding theory.As a consequence of this result, we obtain better Hilbert space dimension witnesses and quantum violations of Bell inequalities with better resistance to noise.

Completely continuous multilinear operators on C(K) spaces

In this Letter we show that the field of operator space theory provides a general and powerful mathematical framework for arbitrary Bell inequalities, in particular, regarding the scaling of their violation within quantum mechanics. We illustrate the power of this connection by showing that bipartite quantum states with local, Hilbert space dimension n can violate a Bell inequality by a factor of order sqrt[n]/(log{2}n) when observables with n possible outcomes are used. Applications to resistance to noise, Hilbert space dimension estimates, and communication complexity are given.

Unconditionally converging multilinear operators

In this paper, we characterize, for 1≤p<∞, the multiple (p, 1)-summing multilinear operators on the product ofC(K) spaces in terms of their representing polymeasures. As consequences, we obtain a new characterization of (p, 1)-summing linear operators onC(K) in terms of their representing measures and a new multilinear characterization ofL∞ spaces. We also solve a problem stated by M.S. Ramanujan and E. Schock, improve a result of H. P. Rosenthal and S. J. Szarek, and give new results about polymeasures.

Radial continuous rotation invariant valuations on star bodies

Abstract We consider the classes of “Grothendieck-integral” (G-integral) and “Pietsch-integral” (P-integral) linear and multilinear operators (see definitions below), and we prove that a multilinear operator between Banach spaces is G-integral (resp. P-integral) if and only if its linearization is G-integral (resp. P-integral) on the injective tensor product of the spaces, together with some related results concerning certain canonically associated linear operators. As an application we give a new proof of a result on the Radon–Nikodym property of the dual of the injective tensor product of Banach spaces. Moreover, we give a simple proof of a characterization of the G-integral operators on C(K,X) spaces and we also give a partial characterization of P-integral operators on C(K,X) spaces.