Infinitely Many Information Inequalities

Matúš, František

doi:10.1109/isit.2007.4557201

Cited by 133 publications

(99 citation statements)

References 14 publications

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“…It has been a long-standing open question to determine whether there are any further universal entropy inequalities apart from the classical ones [11][12][13]. For probability distributions, this question has been answered to the affirmative in the breakthrough works [14,15]: for n ≥ 4 random variables, there is an infinite number of independent entropy inequalities satisfied by the Shannon entropy. In network coding theory, they give rise to tighter capacity bounds [16].…”

Section: Jhep09(2015)130mentioning

confidence: 99%

“…For these, not only are there extreme rays for n ≥ 3 that can only be attained approximately, so that the corresponding entropy cones are not closed, but there are in fact linear inequalities that constrain some of the lower-dimensional faces when n ≥ 4 [12,17,18]. It is moreover known for the Shannon entropy (and likewise conjectured for the von Neumann entropy) that the cones are not polyhedral for n ≥ 4 [15].…”

Section: Jhep09(2015)130mentioning

confidence: 99%

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The holographic entropy cone

Bao

Nezami

Ooguri

et al. 2015

J. High Energ. Phys.

173

510

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We initiate a systematic enumeration and classification of entropy inequalities satisfied by the Ryu-Takayanagi formula for conformal field theory states with smooth holographic dual geometries. For 2, 3, and 4 regions, we prove that the strong subadditivity and the monogamy of mutual information give the complete set of inequalities. This is in contrast to the situation for generic quantum systems, where a complete set of entropy inequalities is not known for 4 or more regions. We also find an infinite new family of inequalities applicable to 5 or more regions. The set of all holographic entropy inequalities bounds the phase space of Ryu-Takayanagi entropies, defining the holographic entropy cone. We characterize this entropy cone by reducing geometries to minimal graph models that encode the possible cutting and gluing relations of minimal surfaces. We find that, for a fixed number of regions, there are only finitely many independent entropy inequalities. To establish new holographic entropy inequalities, we introduce a combinatorial proof technique that may also be of independent interest in Riemannian geometry and graph theory.

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Section: Jhep09(2015)130mentioning

confidence: 99%

Section: Jhep09(2015)130mentioning

confidence: 99%

The holographic entropy cone

Bao

Nezami

Ooguri

et al. 2015

J. High Energ. Phys.

173

510

View full text Add to dashboard Cite

show abstract

“…Using the same technique proposed in [1], more and more linear information inequalities have been discovered [8,[13][14][15]. Later in [9], Matúš obtained a countable infinite set of linear information inequalities for a set of four random variables. Using the same set of inequalities, Matúš further proved thatΓ * (N 4 ) is not a polyhedral.…”

Section: Non-polyhedral Propertymentioning

confidence: 99%

“…This particular approach for construction has been the main ingredient in every non-Shannon inequality that has been subsequently discovered. Using this approach, new inequalities can be found mechanically [8] and there are in fact infinitely many such independent inequalities even when there are only four random variables involved [9]. Despite this progress, a complete characterisation is still missing however.…”

Section: Introductionmentioning

confidence: 99%

Recent Progresses in Characterising Information Inequalities

Chan

2011

Entropy

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In this paper, we present a revision on some of the recent progresses made in characterising and understanding information inequalities, which are the fundamental physical laws in communications and compression. We will begin with the introduction of a geometric framework for information inequalities, followed by the first non-Shannon inequality proved by Zhang et al. in 1998 [1]. The discovery of this non-Shannon inequality is a breakthrough in the area and has led to the subsequent discovery of many more non-Shannon inequalities. We will also review the close relations between information inequalities and other research areas such as Kolmogorov complexity, determinantal inequalities, and group-theoretic inequalities. These relations have led to non-traditional techniques in proving information inequalities and at the same time made impacts back on those related areas by the introduction of information-theoretic tools.

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“…The non-Shannon inequalities provide outer bounds for the entropy region and it has been shown that no finite number of these inequalities can characterize the entropy region completely [13]. Innerbounds are less often studied in the literature and the most well-known inner region is defined through the Ingleton inequality which was first obtained for the ranks of vectors spaces [14].…”

Section: A What Is Known About the Entropy Region?mentioning

confidence: 99%

On the entropy region of discrete and continuous random variables and network information theory

Shadbakht

Hassibi

2008

2008 42nd Asilomar Conference on Signals, Systems and Computers

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Abstract-We show that a large class of network information theory problems can be cast as convex optimization over the convex space of entropy vectors. A vector in 2 n − 1 dimensional space is called entropic if each of its entries can be regarded as the joint entropy of a particular subset of n random variables (note that any set of size n has 2 n − 1 nonempty subsets.) While an explicit characterization of the space of entropy vectors is wellknown for n = 2, 3 random variables, it is unknown for n > 3 (which is why most network information theory problems are open.) We will construct inner bounds to the space of entropic vectors using tools such as quasi-uniform distributions, lattices, and Cayley's hyperdeterminant.

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Infinitely Many Information Inequalities

Cited by 133 publications

References 14 publications

The holographic entropy cone

The holographic entropy cone

Recent Progresses in Characterising Information Inequalities

On the entropy region of discrete and continuous random variables and network information theory

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