On the duality of additivity and tensorization

Beigi, Salman; Gohari, Amin

doi:10.1109/isit.2015.7282882

Cited by 19 publications

(16 citation statements)

References 29 publications

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“…In fact, hypercontractivity has an information theoretic characterization (see [8], [136]), which also suggests a duality between additivity and tensorization (see [19]). An information theoretic characterization of the Brascamp-Lieb inequality, which includes the hypercontractivity bound as a special case, has been known earlier in the context of functional analysis [46]; for a recent treatment of the Brascamp-Lieb inequality in the context of information theory, see [20], [114].…”

Section: B Maximal Correlation and Hypercontractivitymentioning

confidence: 99%

“…To see (18) and (19), for transcript U r of protocol π 1 and for every 1 ≤ i ≤ r, by the monotonicity of correlation we get…”

Section: In Order To Derive the Single-letter Characterization Againmentioning

confidence: 99%

“…As was pointed-out in [10], both ρ m (X, Y ) and s * (X, Y ) satisfy the conditions required of Γ in Theorem V.6. We note that a more general class of measures of correlation satisfying these conditions is available [24] (see, also, [19]). As a corollary, we have the following.…”

Section: Simulation Of Correlated Random Variablesmentioning

confidence: 99%

See 2 more Smart Citations

Communication for Generating Correlation: A Unifying Survey

Sudan

Tyagi

Watanabe

2020

IEEE Trans. Inform. Theory

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The task of manipulating correlated random variables in a distributed setting has received attention in the fields of both Information Theory and Computer Science. Often shared correlations can be converted, using a little amount of communication, into perfectly shared uniform random variables. Such perfect shared randomness, in turn, enables the solutions of many tasks. Even the reverse conversion of perfectly shared uniform randomness into variables with a desired form of correlation turns out to be insightful and technically useful. In this article, we describe progress-to-date on such problems and lay out pertinent measures, achievability results, limits of performance, and point to new directions.M. Sudan is with the 13 When X1, Z, and X2 form Markov chain in this order, the SK capacity is 0. 14 The benefit of feedback in the context of the wiretap channel was pointed out in [119].

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Section: B Maximal Correlation and Hypercontractivitymentioning

confidence: 99%

“…To see (18) and (19), for transcript U r of protocol π 1 and for every 1 ≤ i ≤ r, by the monotonicity of correlation we get…”

Section: In Order To Derive the Single-letter Characterization Againmentioning

confidence: 99%

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Communication for Generating Correlation: A Unifying Survey

Sudan

Tyagi

Watanabe

2020

IEEE Trans. Inform. Theory

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“…To address the question of finding a universal bound on the distance from the quantum set to extremal nosignaling boxes, one approach is to use the norm-based bounds developed in [33,34], a more general approach is to utilize the method of proof of the Theorem 1 and lower bound the minimum eigenvalue of the matrix A(P) TÃ (P) associated with the vertex P. Concerning non-locality distillation, recently in [36] a systematic method to construct sets of non-local boxes that are closed under wirings has been introduced. In particular, a measure of correlations called the maximal correlation has been shown to be monotonically decreasing under wirings.…”

Section: Lemmamentioning

confidence: 99%

No Quantum Realization of Extremal No-Signaling Boxes

et al. 2016

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The study of quantum correlations is important for fundamental reasons as well as for quantum communication and information processing tasks. On the one hand, it is of tremendous interest to derive the correlations produced by measurements on separated composite quantum systems from within the set of all correlations obeying the no-signaling principle of relativity, by means of information-theoretic principles [1][2][3][4][5]. On the other hand, cryptographic protocols based on quantum non-local correlations have been proposed for the generation of secure keys [6] and the amplification and expansion of randomness [7,8] against general no-signaling adversaries. In both these research programs, a fundamental question arises : can any measurements on quantum states realize the correlations present in pure extremal no-signaling boxes? Here, we answer this question in full generality showing that no non-trivial (not local realistic) extremal boxes of general no-signaling theories can be realized in quantum theory. We then explore some important consequences of this fact.It is well-known that quantum mechanics allows for non-local correlations between spatially separated systems, i.e., correlations that cannot be explained by any local hidden variable theory as shown by the violation of Bell inequalities. The correlations described by quantum theory form a convex set which is sandwiched between the sets of classical correlations and general nosignaling correlations. By how much and why the quantum set is inside the no-signaling set has been the subject of intense research [1][2][3][4][5]9].The classical and no-signaling sets are known to form convex polytopes, the quantum set while being convex is not a polytope in general. Each point in the set describes a box of correlations, that is a set of conditional probability distributions for outputs given inputs. The extremal boxes (vertices) of the no-signaling polytope are the pure states of the no-signaling theory and their purity implies that they are completely uncorrelated with the environment. Consequently, access to an extremal non-local box would be of great advantage in cryptographic tasks, providing intrinsic certified randomness and key secure against any eavesdropper even when the latter is limited only by the no-signaling principle. Extremal boxes are also of crucial importance in the program of deriving the set of quantum correlations from purely information-theoretic principles, being essential test-beds for checking the validity of such principles [1][2][3][4][5]. Furthermore, the extremal boxes are the ones that maximally violate Bell inequalities in no-signaling theories; for some Bell inequalities there exists a single unique box that gives maximal violation. For instance, for the celebrated CHSH inequality [10], there exists a unique extremal box known as the Popescu-Rohrlich (PR) box [11] which maximally violates the inequality. From these considerations, one arrives at the fundamental question: does quantum mechanics allow for the realization of extre...

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“…Equivalent generalizations for multiuser channels using pairwise setups are of interest. Another observation of independent interest is that recently, the Hypercontractivity (HC) ribbon, a tensorizing measure of correlation [14], was derived as a dual of the GW region [15]. Both the HC ribbon and ARI region behave mono-tonically under local stochastic evolution and are measures of nonlocal correlation.…”

Section: ) For Any P Q|ŷavt ∈ Pŷmentioning

confidence: 99%

Multipartite monotones for secure sampling by public discussion from noisy correlations

Banerjee

2015

2015 Iran Workshop on Communication and Information Theory (IWCIT)

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We address the problem of quantifying the cryptographic content of probability distributions, in relation to an application to secure multi-party sampling against a passive tadversary. We generalize a recently introduced notion of assisted common information of a pair of correlated sources to that of K sources and define a family of monotone rate regions indexed by K. This allows for a simple characterization of all t-private distributions that can be statistically securely sampled without any auxiliary setup of pre-shared noisy correlations. We also give a new monotone called the residual total correlation that admits a simple operational interpretation. Interestingly, for sampling with non-trivial setups (K > 2) in the public discussion model, our definition of a monotone region differs from the one by Prabhakaran and Prabhakaran (ITW 2012).

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On the duality of additivity and tensorization

Cited by 19 publications

References 29 publications

Communication for Generating Correlation: A Unifying Survey

Communication for Generating Correlation: A Unifying Survey

No Quantum Realization of Extremal No-Signaling Boxes

Multipartite monotones for secure sampling by public discussion from noisy correlations

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