Journal IssueDOI
Lower bounds in communication complexity based on factorization norms
Nati Linial,Adi Shraibman +1 more
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TLDR
It follows from the results that this bound on the saving in communication is tight almost always, and shed some light on the question how much communication can be saved by using entanglement.Abstract:
We introduce a new method to derive lower bounds on randomized and quantum communication complexity. Our method is based on factorization norms, a notion from Banach Space theory. This approach gives us access to several powerful tools from this area such as normed spaces duality and Grothendiek's inequality. This extends the arsenal of methods for deriving lower bounds in communication complexity.
As we show, our method subsumes most of the previously known general approaches to lower bounds on communication complexity. Moreover, we extend all (but one) of these lower bounds to the realm of quantum communication complexity with entanglement.
Our results also shed some light on the question how much communication can be saved by using entanglement. It is known that entanglement can save one of every two qubits, and examples for which this is tight are also known. It follows from our results that this bound on the saving in communication is tight almost always. © 2008 Wiley Periodicals, Inc. Random Struct. Alg., 2009read more
Citations
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Journal ArticleDOI
Grothendieck's Theorem, past and present
TL;DR: The Grothendieck constant of a graph has been introduced in graph theory and in computer science as discussed by the authors, where it is invoked to replace certain NP hard problems by others that can be treated by semidefinite programming and hence solved in polynomial time.
Journal ArticleDOI
The Pattern Matrix Method
TL;DR: The pattern matrix method gives a new and simple proof of Razborov's breakthrough quantum lower bounds for disjointness and other symmetric predicates, and characterize the discrepancy, approximate rank, and approximate trace norm of A_f in terms of well-studied analytic properties of f.
Proceedings ArticleDOI
Quantum Query Complexity of State Conversion
TL;DR: It is obtained that the general adversary bound characterizes the quantum query complexity of any function whatsoever, implying that discrete and continuous-time query models are equivalent in the bounded-error setting, even for the general state-conversion problem.
Journal ArticleDOI
Large Violation of Bell Inequalities with Low Entanglement
Marius Junge,Carlos Palazuelos +1 more
TL;DR: In this article, it was shown that, even though entanglement is necessary to obtain violation of Bell inequalities, the entropy of the underlying state is essentially irrelevant in obtaining large violation.
Book
Lower Bounds in Communication Complexity
Troy Lee,Adi Shraibman +1 more
TL;DR: Lower Bounds in Communication Complexity focuses on showing lower bounds on the communication complexity of explicit functions, and treats different variants of communication complexity, including randomized, quantum, and multiparty models.
References
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Journal ArticleDOI
Communication via One- and Two-Particle Operators on Einstein-Podolsky-Rosen States
TL;DR: The set of states accessible from an initial EPR state by one-particle operations are characterized and it is shown that in a sense they allow two bits to be encoded reliably in one spin-1/2 particle.
Journal ArticleDOI
On “bent” functions
TL;DR: The polynomial degree of a bent function P ( x ) is studied, as are the properties of the Fourier transform of (−1) P(x) , and a connection with Hadamard matrices.
Proceedings ArticleDOI
Quantum circuit complexity
TL;DR: It is shown that any function computable in polynomial time by a quantum Turing machine has aPolynomial-size quantum circuit, and this result enables us to construct a universal quantum computer which can simulate a broader class of quantum machines than that considered by E. Bernstein and U. Vazirani (1993), thus answering an open question raised by them.
Proceedings ArticleDOI
Complexity classes in communication complexity theory
TL;DR: The main objective is to exploit the analogy between Turing machine (TM) and communication complexity (CC) classes to provide a more amicable environment for the study of questions analogous to the most notorious problems in TM complexity.
Proceedings ArticleDOI
Lower bounds by probabilistic arguments
TL;DR: It is proved that, to compute the majority function of n Boolean variables, the size of any depth-3 monotone circuit must be greater than 2nε, and thesize of any width-2 branching program must have super-polynomial growth.