429 research outputs found
Tensor Norms and the Classical Communication Complexity of Nonlocal Quantum Measurement
We initiate the study of quantifying nonlocalness of a bipartite measurement
by the minimum amount of classical communication required to simulate the
measurement. We derive general upper bounds, which are expressed in terms of
certain tensor norms of the measurement operator. As applications, we show that
(a) If the amount of communication is constant, quantum and classical
communication protocols with unlimited amount of shared entanglement or shared
randomness compute the same set of functions; (b) A local hidden variable model
needs only a constant amount of communication to create, within an arbitrarily
small statistical distance, a distribution resulted from local measurements of
an entangled quantum state, as long as the number of measurement outcomes is
constant.Comment: A preliminary version of this paper appears as part of an article in
Proceedings of the the 37th ACM Symposium on Theory of Computing (STOC 2005),
460--467, 200
Local tests of global entanglement and a counterexample to the generalized area law
We introduce a technique for applying quantum expanders in a distributed
fashion, and use it to solve two basic questions: testing whether a bipartite
quantum state shared by two parties is the maximally entangled state and
disproving a generalized area law. In the process these two questions which
appear completely unrelated turn out to be two sides of the same coin.
Strikingly in both cases a constant amount of resources are used to verify a
global property.Comment: 21 pages, to appear FOCS 201
Explicit lower and upper bounds on the entangled value of multiplayer XOR games
XOR games are the simplest model in which the nonlocal properties of
entanglement manifest themselves. When there are two players, it is well known
that the bias --- the maximum advantage over random play --- of entangled
players can be at most a constant times greater than that of classical players.
Recently, P\'{e}rez-Garc\'{i}a et al. [Comm. Math. Phys. 279 (2), 2008] showed
that no such bound holds when there are three or more players: the advantage of
entangled players over classical players can become unbounded, and scale with
the number of questions in the game. Their proof relies on non-trivial results
from operator space theory, and gives a non-explicit existence proof, leading
to a game with a very large number of questions and only a loose control over
the local dimension of the players' shared entanglement.
We give a new, simple and explicit (though still probabilistic) construction
of a family of three-player XOR games which achieve a large quantum-classical
gap (QC-gap). This QC-gap is exponentially larger than the one given by
P\'{e}rez-Garc\'{i}a et. al. in terms of the size of the game, achieving a
QC-gap of order with questions per player. In terms of the
dimension of the entangled state required, we achieve the same (optimal) QC-gap
of for a state of local dimension per player. Moreover, the
optimal entangled strategy is very simple, involving observables defined by
tensor products of the Pauli matrices.
Additionally, we give the first upper bound on the maximal QC-gap in terms of
the number of questions per player, showing that our construction is only
quadratically off in that respect. Our results rely on probabilistic estimates
on the norm of random matrices and higher-order tensors which may be of
independent interest.Comment: Major improvements in presentation; results identica
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
Bounding quantum-classical separations for classes of nonlocal games
We bound separations between the entangled and classical values for several classes of nonlocal t-player games. Our motivating question is whether there is a family of t-player XOR games for which the entangled bias is 1 but for which the classical bias goes down to 0, for fixed t. Answering this question would have important consequences in the study of multi-party communication complexity, as a positive answer would imply an unbounded separation between randomized communication complexity with and without entanglement. Our contribution to answering the question is identifying several general classes of games for which the classical bias can not go to zero when the entangled bias stays above a constant threshold. This rules out the possibility of using these games to answer our motivating question. A previously studied set of XOR games, known not to give a positive answer to the question, are those for which there is a quantum strategy that attains value 1 using a so-called Schmidt state. We generalize this class to mod-m games and show that their classical value is always at least 1/m + (m-1)/m t^{1-t}. Secondly, for free XOR games, in which the input distribution is of product form, we show beta(G) >= beta^*(G)^{2^t} where beta(G) and beta^*(G) are the classical and entangled biases of the game respectively. We also introduce so-called line games, an example of which is a slight modification of the Magic Square game, and show that they can not give a positive answer to the question either. Finally we look at two-player unique games and show that if the entangled value is 1-epsilon then the classical value is at least 1-O(sqrt{epsilon log k}) where k is the number of outputs in the game. Our proofs use semidefinite-programming techniques, the Gowers inverse theorem and hypergraph norms
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 (up
to a logarithmic factor) 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
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
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