74 research outputs found
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
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 refereeâs 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 Grothendieckâs inequality, we present an efficient algorithm that gives a constant-factor approximation to the best performance that players can obtain in a given game, both in the case that they have no shared entanglement and that 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
On the Bohnenblust-Hille inequality and a variant of Littlewood's 4/3 inequality
The search for sharp constants for inequalities of the type Littlewood's 4/3
and Bohnenblust-Hille, besides its pure mathematical interest, has shown
unexpected applications in many different fields, such as Analytic Number
Theory, Quantum Information Theory, or (for instance) in deep results on the
-dimensional Bohr radius. The recent estimates obtained for the multilinear
Bohnenblust-Hille inequality (in the case of real scalars) have been recently
used, as a crucial step, by A. Montanaro in order to solve problems in the
theory of quantum XOR games. Here, among other results, we obtain new upper
bounds for the Bohnenblust-Hille constants in the case of complex scalars. For
bilinear forms, we obtain the optimal constants of variants of Littlewood's 4/3
inequality (in the case of real scalars) when the exponent 4/3 is replaced by
any As a consequence of our estimates we show that the optimal
constants for the real case are always strictly greater than the constants for
the complex case
Noise in Quantum and Classical Computation & Non-locality
Quantum computers seem to have capabilities which go beyond those of classical computers. A particular example which is important for cryptography is that quantum computers are able to factor numbers much faster than what seems possible on classical machines.
In order to actually build quantum computers it is necessary to build sufficiently accurate hardware, which is a big challenge.
In part 1 of this thesis we prove lower bounds on the accuracy of the hardware needed to do quantum computation.
We also present a similar result for classical computers.
One resource that quantum computers have but classical computers do not have is entanglement. In Part 2 of this thesis we study certain general aspects of entanglement in terms of quantum XOR games and non-locality
On the relation between completely bounded and (1, cb)- summing maps with applications to quantum xor games
In this work we show that, given a linear map from a general operator space into the dual of a Câ -algebra, its completely bounded norm is upper bounded by a universal constant times its (1, cb)-summing norm. This problem is motivated by the study of quantum XOR games in the field of quantum information theory. In particular, our results imply that for such games entangled strategies cannot be arbitrarily better than those strategies using one-way classical communication
- âŠ