9 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
Strengths and Weaknesses of Quantum Fingerprinting
We study the power of quantum fingerprints in the simultaneous message
passing (SMP) setting of communication complexity. Yao recently showed how to
simulate, with exponential overhead, classical shared-randomness SMP protocols
by means of quantum SMP protocols without shared randomness
(-protocols). Our first result is to extend Yao's simulation to
the strongest possible model: every many-round quantum protocol with unlimited
shared entanglement can be simulated, with exponential overhead, by
-protocols. We apply our technique to obtain an efficient
-protocol for a function which cannot be efficiently solved
through more restricted simulations. Second, we tightly characterize the power
of the quantum fingerprinting technique by making a connection to arrangements
of homogeneous halfspaces with maximal margin. These arrangements have been
well studied in computational learning theory, and we use some strong results
obtained in this area to exhibit weaknesses of quantum fingerprinting. In
particular, this implies that for almost all functions, quantum fingerprinting
protocols are exponentially worse than classical deterministic SMP protocols.Comment: 13 pages, no figures, to appear in CCC'0
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
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
The communication complexity of non-signaling distributions
We study a model of communication complexity that encompasses many
well-studied problems, including classical and quantum communication
complexity, the complexity of simulating distributions arising from bipartite
measurements of shared quantum states, and XOR games. In this model, Alice gets
an input x, Bob gets an input y, and their goal is to each produce an output
a,b distributed according to some pre-specified joint distribution p(a,b|x,y).
We introduce a new technique based on affine combinations of lower-complexity
distributions. Specifically, we introduce two complexity measures, one which
gives lower bounds on classical communication, and one for quantum
communication. These measures can be expressed as convex optimization problems.
We show that the dual formulations have a striking interpretation, since they
coincide with maximum violations of Bell and Tsirelson inequalities. The dual
expressions are closely related to the winning probability of XOR games. These
lower bounds subsume many known communication complexity lower bound methods,
most notably the recent lower bounds of Linial and Shraibman for the special
case of Boolean functions.
We show that the gap between the quantum and classical lower bounds is at
most linear in the size of the support of the distribution, and does not depend
on the size of the inputs. This translates into a bound on the gap between
maximal Bell and Tsirelson inequality violations, which was previously known
only for the case of distributions with Boolean outcomes and uniform marginals.
Finally, we give an exponential upper bound on quantum and classical
communication complexity in the simultaneous messages model, for any
non-signaling distribution. One consequence is a simple proof that any quantum
distribution can be approximated with a constant number of bits of
communication.Comment: 23 pages. V2: major modifications, extensions and additions compared
to V1. V3 (21 pages): proofs have been updated and simplified, particularly
Theorem 10 and Theorem 22. V4 (23 pages): Section 3.1 has been rewritten (in
particular Lemma 10 and its proof), and various minor modifications have been
made. V5 (24 pages): various modifications in the presentatio