Cores of Cooperative Games in Information Theory

Cores of cooperative games are ubiquitous in information theory, and arise most frequently in the characterization of fundamental limits in various scenarios involving multiple users. Examples include classical settings in network information theory such as Slepian-Wolf source coding and multiple access channels, classical settings in statistics such as robust hypothesis testing, and new settings at the intersection of networking and statistics such as distributed estimation problems for sensor networks. Cooperative game theory allows one to understand aspects of all of these problems from a fresh and unifying perspective that treats users as players in a game, sometimes leading to new insights. At the heart of these analyses are fundamental dualities that have been long studied in the context of cooperative games; for information theoretic purposes, these are dualities between information inequalities on the one hand and properties of rate, capacity or other resource allocation regions on the other.Comment: 12 pages, published at http://www.hindawi.com/GetArticle.aspx?doi=10.1155/2008/318704 in EURASIP Journal on Wireless Communications and Networking, Special Issue on "Theory and Applications in Multiuser/Multiterminal Communications", April 200

Quantum de Finetti Theorems under Local Measurements with Applications

Quantum de Finetti theorems are a useful tool in the study of correlations in quantum multipartite states. In this paper we prove two new quantum de Finetti theorems, both showing that under tests formed by local measurements one can get a much improved error dependence on the dimension of the subsystems. We also obtain similar results for non-signaling probability distributions. We give the following applications of the results: We prove the optimality of the Chen-Drucker protocol for 3-SAT, under the exponential time hypothesis. We show that the maximum winning probability of free games can be estimated in polynomial time by linear programming. We also show that 3-SAT with m variables can be reduced to obtaining a constant error approximation of the maximum winning probability under entangled strategies of O(m^{1/2})-player one-round non-local games, in which the players communicate O(m^{1/2}) bits all together. We show that the optimization of certain polynomials over the hypersphere can be performed in quasipolynomial time in the number of variables n by considering O(log(n)) rounds of the Sum-of-Squares (Parrilo/Lasserre) hierarchy of semidefinite programs. As an application to entanglement theory, we find a quasipolynomial-time algorithm for deciding multipartite separability. We consider a result due to Aaronson -- showing that given an unknown n qubit state one can perform tomography that works well for most observables by measuring only O(n) independent and identically distributed (i.i.d.) copies of the state -- and relax the assumption of having i.i.d copies of the state to merely the ability to select subsystems at random from a quantum multipartite state. The proofs of the new quantum de Finetti theorems are based on information theory, in particular on the chain rule of mutual information.Comment: 39 pages, no figure. v2: changes to references and other minor improvements. v3: added some explanations, mostly about Theorem 1 and Conjecture 5. STOC version. v4, v5. small improvements and fixe

Quantum Proofs

Quantum information and computation provide a fascinating twist on the notion of proofs in computational complexity theory. For instance, one may consider a quantum computational analogue of the complexity class \class{NP}, known as QMA, in which a quantum state plays the role of a proof (also called a certificate or witness), and is checked by a polynomial-time quantum computation. For some problems, the fact that a quantum proof state could be a superposition over exponentially many classical states appears to offer computational advantages over classical proof strings. In the interactive proof system setting, one may consider a verifier and one or more provers that exchange and process quantum information rather than classical information during an interaction for a given input string, giving rise to quantum complexity classes such as QIP, QSZK, and QMIP* that represent natural quantum analogues of IP, SZK, and MIP. While quantum interactive proof systems inherit some properties from their classical counterparts, they also possess distinct and uniquely quantum features that lead to an interesting landscape of complexity classes based on variants of this model. In this survey we provide an overview of many of the known results concerning quantum proofs, computational models based on this concept, and properties of the complexity classes they define. In particular, we discuss non-interactive proofs and the complexity class QMA, single-prover quantum interactive proof systems and the complexity class QIP, statistical zero-knowledge quantum interactive proof systems and the complexity class \class{QSZK}, and multiprover interactive proof systems and the complexity classes QMIP, QMIP*, and MIP*.Comment: Survey published by NOW publisher

Conditional Hardness for Approximate Coloring

We study the coloring problem: Given a graph G, decide whether $c(G) \leq q$ or $c(G) \ge Q$, where c(G) is the chromatic number of G. We derive conditional hardness for this problem for any constant $3 \le q < Q$. For $q\ge 4$, our result is based on Khot's 2-to-1 conjecture [Khot'02]. For $q=3$, we base our hardness result on a certain `fish shaped' variant of his conjecture. We also prove that the problem almost coloring is hard for any constant \eps>0, assuming Khot's Unique Games conjecture. This is the problem of deciding for a given graph, between the case where one can 3-color all but a \eps fraction of the vertices without monochromatic edges, and the case where the graph contains no independent set of relative size at least \eps. Our result is based on bounding various generalized noise-stability quantities using the invariance principle of Mossel et al [MOO'05]

Measuring Intergenerational Mobility and Equality of Opportunity

This paper explores the link between the measurement of intergenerational mobility and the notion of equality of opportunity. We show how recently proposed theories of equality of opportunity can be meaningfully adapted to the intergenerational context. This throws a new light on the interpretation of existing mobility measures: these may be interesting to measure mobility as movement, but they are inadequate to capture the notion of equality of opportunity. We propose some new mobility measures, which start from the idea that the intergenerational transition matrix gives useful information about the opportunity sets of the children of different social classes. These measures are used in an empirical illustration to evaluate the degree of inequality of opportunity in the US, Great Britain and Italy.