4,099 research outputs found
Grothendieck inequalities for semidefinite programs with rank constraint
Grothendieck inequalities are fundamental inequalities which are frequently
used in many areas of mathematics and computer science. They can be interpreted
as upper bounds for the integrality gap between two optimization problems: a
difficult semidefinite program with rank-1 constraint and its easy semidefinite
relaxation where the rank constrained is dropped. For instance, the integrality
gap of the Goemans-Williamson approximation algorithm for MAX CUT can be seen
as a Grothendieck inequality. In this paper we consider Grothendieck
inequalities for ranks greater than 1 and we give two applications:
approximating ground states in the n-vector model in statistical mechanics and
XOR games in quantum information theory.Comment: 22 page
Grothendieck ring of semialgebraic formulas and motivic real Milnor fibres
We define a Grothendieck ring for basic real semialgebraic formulas, that is
for systems of real algebraic equations and inequalities. In this ring the
class of a formula takes into consideration the algebraic nature of the set of
points satisfying this formula and contains as a ring the usual Grothendieck
ring of real algebraic formulas. We give a realization of our ring that allows
to express a class as a Z[1/2]- linear combination of classes of real algebraic
formulas, so this realization gives rise to a notion of virtual Poincar\'e
polynomial for basic semialgebraic formulas. We then define zeta functions with
coefficients in our ring, built on semialgebraic formulas in arc spaces. We
show that they are rational and relate them to the topology of real Milnor
fibres.Comment: 30 pages, 1 figur
Quantum XOR Games
We introduce quantum XOR games, a model of two-player one-round games that
extends the model of XOR games by allowing the referee's questions to the
players to be quantum states. We give examples showing that quantum XOR games
exhibit a wide range of behaviors that are known not to exist for standard XOR
games, such as cases in which the use of entanglement leads to an arbitrarily
large advantage over the use of no entanglement. By invoking two deep
extensions of Grothendieck's inequality, we present an efficient algorithm that
gives a constant-factor approximation to the best performance players can
obtain in a given game, both in case they have no shared entanglement and in
case they share unlimited entanglement. As a byproduct of the algorithm we
prove some additional interesting properties of quantum XOR games, such as the
fact that sharing a maximally entangled state of arbitrary dimension gives only
a small advantage over having no entanglement at all.Comment: 43 page
The little Grothendieck theorem and Khintchine inequalities for symmetric spaces of measurable operators
We prove the little Grothendieck theorem for any 2-convex noncommutative
symmetric space. Let \M be a von Neumann algebra equipped with a normal
faithful semifinite trace \t, and let be an r.i. space on (0, \8). Let
E(\M) be the associated symmetric space of measurable operators. Then to any
bounded linear map from E(\M) into a Hilbert space
corresponds a positive norm one functional f\in E_{(2)}(\M)^* such that
\forall x\in E(\M)\quad \|T(x)\|^2\le K^2 \|T\|^2 f(x^*x+xx^*), where
denotes the 2-concavification of and is a universal constant.
As a consequence we obtain the noncommutative Khintchine inequalities for
E(\M) when is either 2-concave or 2-convex and -concave for some
q<\8. We apply these results to the study of Schur multipliers from a
2-convex unitary ideal into a 2-concave one.Comment: 14 pages. To appear in J. Funct. Ana
Qutrit witness from the Grothendieck constant of order four
In this paper, we prove that , where denotes the
Grothendieck constant of order . To this end, we use a branch-and-bound
algorithm commonly used in the solution of NP-hard problems. It has recently
been proven that . Here we prove that ,
which has implications for device-independent witnessing dimensions greater
than two. Furthermore, the algorithm with some modifications may find
applications in various black-box quantum information tasks with large number
of inputs and outputs.Comment: 13 pages, 2 figure
The positive semidefinite Grothendieck problem with rank constraint
Given a positive integer n and a positive semidefinite matrix A = (A_{ij}) of
size m x m, the positive semidefinite Grothendieck problem with
rank-n-constraint (SDP_n) is
maximize \sum_{i=1}^m \sum_{j=1}^m A_{ij} x_i \cdot x_j, where x_1, ..., x_m
\in S^{n-1}.
In this paper we design a polynomial time approximation algorithm for SDP_n
achieving an approximation ratio of
\gamma(n) = \frac{2}{n}(\frac{\Gamma((n+1)/2)}{\Gamma(n/2)})^2 = 1 -
\Theta(1/n).
We show that under the assumption of the unique games conjecture the achieved
approximation ratio is optimal: There is no polynomial time algorithm which
approximates SDP_n with a ratio greater than \gamma(n). We improve the
approximation ratio of the best known polynomial time algorithm for SDP_1 from
2/\pi to 2/(\pi\gamma(m)) = 2/\pi + \Theta(1/m), and we show a tighter
approximation ratio for SDP_n when A is the Laplacian matrix of a graph with
nonnegative edge weights.Comment: (v3) to appear in Proceedings of the 37th International Colloquium on
Automata, Languages and Programming, 12 page
A generalized Grothendieck inequality and entanglement in XOR games
Suppose Alice and Bob make local two-outcome measurements on a shared
entangled state. For any d, we show that there are correlations that can only
be reproduced if the local dimension is at least d. This resolves a conjecture
of Brunner et al. Phys. Rev. Lett. 100, 210503 (2008) and establishes that the
amount of entanglement required to maximally violate a Bell inequality must
depend on the number of measurement settings, not just the number of
measurement outcomes. We prove this result by establishing the first lower
bounds on a new generalization of Grothendieck's constant.Comment: Version submitted to QIP on 10-20-08. See also arxiv:0812.1572 for
related results, obtained independentl
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