Quantum replication at the Heisenberg limit

No process in nature can perfectly clone an arbitrary quantum state. But is it possible to engineer processes that replicate quantum information with vanishingly small error? Here we demonstrate the possibility of probabilistic super-replication phenomena where N equally prepared quantum clocks are transformed into a much larger number of M nearly perfect replicas, with an error that rapidly vanishes whenever M is small compared to the square of N. The quadratic replication rate is the ultimate limit imposed by Quantum Mechanics to the proliferation of information and is fundamentally linked with the Heisenberg limit of quantum metrology.Comment: 9 + 16 pages, 2 figures, published versio

Reduction of Markov Chains using a Value-of-Information-Based Approach

In this paper, we propose an approach to obtain reduced-order models of Markov chains. Our approach is composed of two information-theoretic processes. The first is a means of comparing pairs of stationary chains on different state spaces, which is done via the negative Kullback-Leibler divergence defined on a model joint space. Model reduction is achieved by solving a value-of-information criterion with respect to this divergence. Optimizing the criterion leads to a probabilistic partitioning of the states in the high-order Markov chain. A single free parameter that emerges through the optimization process dictates both the partition uncertainty and the number of state groups. We provide a data-driven means of choosing the `optimal' value of this free parameter, which sidesteps needing to a priori know the number of state groups in an arbitrary chain.Comment: Submitted to Entrop

Fuzzy Extractors: How to Generate Strong Keys from Biometrics and Other Noisy Data

We provide formal definitions and efficient secure techniques for - turning noisy information into keys usable for any cryptographic application, and, in particular, - reliably and securely authenticating biometric data. Our techniques apply not just to biometric information, but to any keying material that, unlike traditional cryptographic keys, is (1) not reproducible precisely and (2) not distributed uniformly. We propose two primitives: a "fuzzy extractor" reliably extracts nearly uniform randomness R from its input; the extraction is error-tolerant in the sense that R will be the same even if the input changes, as long as it remains reasonably close to the original. Thus, R can be used as a key in a cryptographic application. A "secure sketch" produces public information about its input w that does not reveal w, and yet allows exact recovery of w given another value that is close to w. Thus, it can be used to reliably reproduce error-prone biometric inputs without incurring the security risk inherent in storing them. We define the primitives to be both formally secure and versatile, generalizing much prior work. In addition, we provide nearly optimal constructions of both primitives for various measures of ``closeness'' of input data, such as Hamming distance, edit distance, and set difference.Comment: 47 pp., 3 figures. Prelim. version in Eurocrypt 2004, Springer LNCS 3027, pp. 523-540. Differences from version 3: minor edits for grammar, clarity, and typo

Local Exchangeability

Exchangeability---in which the distribution of an infinite sequence is invariant to reorderings of its elements---implies the existence of a simple conditional independence structure that may be leveraged in the design of probabilistic models, efficient inference algorithms, and randomization-based testing procedures. In practice, however, this assumption is too strong an idealization; the distribution typically fails to be exactly invariant to permutations and de Finetti's representation theory does not apply. Thus there is the need for a distributional assumption that is both weak enough to hold in practice, and strong enough to guarantee a useful underlying representation. We introduce a relaxed notion of local exchangeability---where swapping data associated with nearby covariates causes a bounded change in the distribution. We prove that locally exchangeable processes correspond to independent observations from an underlying measure-valued stochastic process. We thereby show that de Finetti's theorem is robust to perturbation and provide further justification for the Bayesian modelling approach. Using this probabilistic result, we develop three novel statistical procedures for (1) estimating the underlying process via local empirical measures, (2) testing via local randomization, and (3) estimating the canonical premetric of local exchangeability. These three procedures extend the applicability of previous exchangeability-based methods without sacrificing rigorous statistical guarantees. The paper concludes with examples of popular statistical models that exhibit local exchangeability

Probabilistic Spectral Sparsification In Sublinear Time

In this paper, we introduce a variant of spectral sparsification, called probabilistic $(\varepsilon,\delta)$-spectral sparsification. Roughly speaking, it preserves the cut value of any cut $(S,S^{c})$ with an $1\pm\varepsilon$ multiplicative error and a $\delta\left|S\right|$ additive error. We show how to produce a probabilistic $(\varepsilon,\delta)$-spectral sparsifier with $O(n\log n/\varepsilon^{2})$ edges in time $\tilde{O}(n/\varepsilon^{2}\delta)$ time for unweighted undirected graph. This gives fastest known sub-linear time algorithms for different cut problems on unweighted undirected graph such as - An $\tilde{O}(n/OPT+n^{3/2+t})$ time $O(\sqrt{\log n/t})$-approximation algorithm for the sparsest cut problem and the balanced separator problem. - A $n^{1+o(1)}/\varepsilon^{4}$ time approximation minimum s-t cut algorithm with an $\varepsilon n$ additive error

Short seed extractors against quantum storage

Some, but not all, extractors resist adversaries with limited quantum storage. In this paper we show that Trevisan's extractor has this property, thereby showing an extractor against quantum storage with logarithmic seed length

Improved Quantum Communication Complexity Bounds for Disjointness and Equality

We prove new bounds on the quantum communication complexity of the disjointness and equality problems. For the case of exact and non-deterministic protocols we show that these complexities are all equal to n+1, the previous best lower bound being n/2. We show this by improving a general bound for non-deterministic protocols of de Wolf. We also give an O(sqrt{n}c^{log^* n})-qubit bounded-error protocol for disjointness, modifying and improving the earlier O(sqrt{n}log n) protocol of Buhrman, Cleve, and Wigderson, and prove an Omega(sqrt{n}) lower bound for a large class of protocols that includes the BCW-protocol as well as our new protocol.Comment: 11 pages LaTe