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IEEE International Symposium on Information Theory (ISIT)

Non-entropic inequalities from information constraints

Siu‐Wai Ho, Terence Chan, Alex J. Grant

Published Jul 1, 2012• 10 citations
Publisher

Abstract

This paper investigates a new method in proving converses in secure communication problems. The method gives a converse result in terms of the logarithm of support size instead of entropy. The results are connected to constrained information inequalities involving three random variables. A new constrained non-Shannon type inequality is shown.

DOI
10.1109/isit.2012.6282169

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This paper investigates a new method in proving converses in secure communication problems. The method gives a converse result in terms of the logarithm of support size instead of entropy. The results are connected to constrained information inequalities involving three random variables. A new constrained non-Shannon type inequality is shown.
The entropic region is formed by the collection of the Shannon entropies of all subvectors of finitely many jointly distributed discrete random variables. For four or more variables, the structure of the entropic region is mostly unknown. We utilize a variant of the Maximum Entropy Method to obtain five-variable non-Shannon entropy inequalities, which delimit the five-variable entropy region. This method adds copies of some of the random variables in generations. A significant reduction in computational complexity, achieved through theoretical considerations and by harnessing the inherent symmetries, allowed us to calculate all five-variable non-Shannon inequalities provided by the first nine generations. Based on the results, we define two infinite collections of such inequalities and prove them to be entropy inequalities. We investigate downward-closed subsets of non-negative lattice points that parameterize these collections, and based on this, we develop an algorithm to enumerate all extremal inequalities. The discovered set of entropy inequalities is conjectured to characterize the applied method completely.
A causal structure is a relationship between observed variables that in general restricts the possible correlations between them. This relationship can be mediated by unobserved systems, modelled by random variables in the classical case or joint quantum systems in the quantum case. One way to differentiate between the correlations realisable by two different causal structures is to use entropy vectors, i.e., vectors whose components correspond to the entropies of each subset of the observed variables. To date, the starting point for deriving entropic constraints within causal structures are the so-called Shannon inequalities (positivity of entropy, conditional entropy and conditional mutual information). In the present work we investigate what happens when non-Shannon entropic inequalities are included as well. We show that in general these lead to tighter outer approximations of the set of realisable entropy vectors and hence enable a sharper distinction of different causal structures. Since non-Shannon inequalities can only be applied amongst classical variables, it might be expected that their use enables an entropic distinction between classical and quantum causal structures. However, this remains an open question. We also introduce techniques for deriving inner approximations to the allowed sets of entropy vectors for a given causal structure. These are useful for proving tightness of outer approximations or for finding interesting regions of entropy space. We illustrate these techniques in several scenarios, including the triangle causal structure.
A causal structure is a relationship between observed variables that in general restricts the possible correlations between them. This relationship can be mediated by unobserved systems, modelled by random variables in the classical case or joint quantum systems in the quantum case. One way to differentiate between the correlations realisable by two different causal structures is to use entropy vectors, i.e., vectors whose components correspond to the entropies of each subset of the observed variables. To date, the starting point for deriving entropic constraints within causal structures are the so-called Shannon inequalities (positivity of entropy, conditional entropy and conditional mutual information). In the present work we investigate what happens when non-Shannon entropic inequalities are included as well. We show that in general these lead to tighter outer approximations of the set of realisable entropy vectors and hence enable a sharper distinction of different causal structures. Since non-Shannon inequalities can only be applied amongst classical variables, it might be expected that their use enables an entropic distinction between classical and quantum causal structures. However, this remains an open question. We also introduce techniques for deriving inner approximations to the allowed sets of entropy vectors for a given causal structure. These are useful for proving tightness of outer approximations or for finding interesting regions of entropy space. We illustrate these techniques in several scenarios, including the triangle causal structure.
The known secret-sharing schemes for most access structures are not efficient; even for a one-bit secret the length of the shares in the schemes is 2O(n), wherenis the number of participants in the access structure. It is a long standing open problem to improve these schemes or prove that they cannot be improved. The best known lower bound is by Csirmaz, who proved that there exist access structures withnparticipants such that the size of the share of at least one party isn/logntimes the secret size. Csirmaz's proof uses Shannon information inequalities, which were the only information inequalities known when Csirmaz published his result. On the negative side, Csirmaz proved that by only using Shannon information inequalities one cannot prove a lower bound of ω(n) on the share size. In the last decade, a sequence of non-Shannon information inequalities were discovered. In fact, it was proved that there are infinity many independent information inequalities even in four variables. This raises the hope that these inequalities can help in improving the lower bounds beyondn. However, we show that any information inequality with four or five variables cannot prove a lower bound of ω(n) on the share size. In addition, we show that the same negative result holds for all information inequalities with more than five variables that are known to date.
Information inequalities govern the ultimate limitations in information theory and as such play a pivotal role in characterizing what values the entropy of multipartite states can take. Proving an information inequality, however, quickly becomes arduous when the number of involved parties increases. For classical systems, [Yeung, IEEE Trans. Inf. Theory (1997)] proposed a framework to prove Shannon-type inequalities via linear programming. Here, we derive an analogous framework for quantum systems, based on the strong sub-additivity and weak monotonicity inequalities for the von-Neumann entropy. Importantly, this also allows us to handle constrained inequalities, which - in the classical case - served as a crucial tool in proving the existence of non-standard, so-called non-Shannon-type inequalities [Zhang & Yeung, IEEE Trans. Inf. Theory (1998)]. Our main contribution is the Python package qITIP, for which we present the theory and demonstrate its capabilities with several illustrative examples.