On the Ingleton-Violating Finite Groups

Given n discrete random variables, its entropy vector is the 2^n - 1-dimensional vector obtained from the joint entropies of all non-empty subsets of the random variables. It is well known that there is a close relation between such an entropy vector and a certain group-characterizable vector obtained from a finite group and n of its subgroups; indeed, roughly speaking, knowing the region of all such group-characterizable vectors is equivalent to knowing the region of all entropy vectors. This correspondence may be useful for characterizing the space of entropic vectors and for designing network codes. If one restricts attention to abelian groups then not all entropy vectors can be obtained. This is an explanation for the fact shown by Dougherty et al. that linear network codes cannot achieve capacity in general network coding problems (since linear network codes come from abelian groups). All abelian group-characterizable vectors, and by fiat all entropy vectors generated by linear network codes, satisfy a linear inequality called the Ingleton inequality. General entropy vectors, however, do not necessarily have this property. It is, therefore, of interest to identify groups that violate the Ingleton inequality. In this paper, we study the problem of finding nonabelian finite groups that yield characterizable vectors, which violate the Ingleton inequality. Using a refined computer search, we find the symmetric group S_5 to be the smallest group that violates the Ingleton inequality. Careful study of the structure of this group, and its subgroups, reveals that it belongs to the Ingleton-violating family PGL(2,q) with a prime power q ≥ 5 , i.e., the projective group of 2×2 nonsingular matrices with entries in F_q . We further interpret this family of groups, and their subgroups, using the theory of group actions and identify the subgroups as certain stabilizers. We also extend the construction to more general groups such as PGL(n,q) and GL(n,q) . The families of groups identified here are therefore good candidates for constructing network codes more powerful than linear network codes, and we discuss some considerations for constructing such group network codes

Non-Shannon inequalities in the entropy vector approach to causal structures

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.Comment: 23 pages + appendix; v2: minor changes to Section IV A; v3: paper has been significantly shortened, an expanded version of the removed review section can be found in arXiv:1709.08988; v4: version to be published, supplementary information available as ancillary file

Entropy Region and Convolution

The entropy region is constructed from vectors of random variables by collecting Shannon entropies of all subvectors. Its shape is studied here by means of polymatroidal constructions, notably by convolution. The closure of the region is decomposed into the direct sum of tight and modular parts, reducing the study to the tight part. The relative interior of the reduction belongs to the entropy region. Behavior of the decomposition under self-adhesivity is clarified. Results are specialized and extended to the region constructed from four tuples of random variables. This and computer experiments help to visualize approximations of a symmetrized part of the entropy region. The four-atom conjecture on the minimal Ingleton score is refuted. © 2016 IEEE

Information-Theoretic Studies and Capacity Bounds: Group Network Codes and Energy Harvesting Communication Systems

Network information theory and channels with memory are two important but difficult frontiers of information theory. In this two-parted dissertation, we study these two areas, each comprising one part. For the first area we study the so-called entropy vectors via finite group theory, and the network codes constructed from finite groups. In particular, we identify the smallest finite group that violates the Ingleton inequality, an inequality respected by all linear network codes, but not satisfied by all entropy vectors. Based on the analysis of this group we generalize it to several families of Ingleton-violating groups, which may be used to design good network codes. Regarding that aspect, we study the network codes constructed with finite groups, and especially show that linear network codes are embedded in the group network codes constructed with these Ingleton-violating families. Furthermore, such codes are strictly more powerful than linear network codes, as they are able to violate the Ingleton inequality while linear network codes cannot. For the second area, we study the impact of memory to the channel capacity through a novel communication system: the energy harvesting channel. Different from traditional communication systems, the transmitter of an energy harvesting channel is powered by an exogenous energy harvesting device and a finite-sized battery. As a consequence, each time the system can only transmit a symbol whose energy consumption is no more than the energy currently available. This new type of power supply introduces an unprecedented input constraint for the channel, which is random, instantaneous, and has memory. Furthermore, naturally, the energy harvesting process is observed causally at the transmitter, but no such information is provided to the receiver. Both of these features pose great challenges for the analysis of the channel capacity. In this work we use techniques from channels with side information, and finite state channels, to obtain lower and upper bounds of the energy harvesting channel. In particular, we study the stationarity and ergodicity conditions of a surrogate channel to compute and optimize the achievable rates for the original channel. In addition, for practical code design of the system we study the pairwise error probabilities of the input sequences

Analysing causal structures with entropy

A central question for causal inference is to decide whether a set of correlations fit a given causal structure. In general, this decision problem is computationally infeasible and hence several approaches have emerged that look for certificates of compatibility. Here we review several such approaches based on entropy. We bring together the key aspects of these entropic techniques with unified terminology, filling several gaps and establishing new connections regarding their relation, all illustrated with examples. We consider cases where unobserved causes are classical, quantum and post-quantum and discuss what entropic analyses tell us about the difference. This has applications to quantum cryptography, where it can be crucial to eliminate the possibility of classical causes. We discuss the achievements and limitations of the entropic approach in comparison to other techniques and point out the main open problems.Comment: 19 (+3) pages, 5 (+1) figures. A few minor updates and corrections. There is a small error in the published version of this manuscript: the claim in the last sentence of Section 2(a)(ii) should be restricted to four variables. This is correct in the arXiv versio

Violence Exposure in Human Rights Defenders\u27 Work

The work carried out by Human Rights Defenders (HRDs) is fundamental for a more just and democratic society. Therefore, the acts of aggression against them, whether committed by individuals, the State, armed groups or corporative interests, constitute an indirect attack against the rights of the whole population. By restricting access to information and limiting political participation, these aggressions impede society from actively engaging in public affairs, something that ultimately takes the power away from the people. From a perspective empathetic to HRDs, this Independent Practitioner Inquiry Capstone paper (IPIC) explores possible causes of violence exposure in HRDs’ work and its consequences, some of the actions that are being developed to confront this violence and also, potential pathways of how to better address risk in rough contexts. The work is the result of working with the Protection and Defense Program of Article 19 in Mexico City during the summer of 2017 and with the joint project Protection and Welfare at the Time of Reporting implemented by the Plurinational Legislative Assembly (PLA) of Bolivia and the Journalists Association of La Paz (JALP) during the spring of 2018 in La Paz. Keywords: Violence, human rights, journalism, activism, risk, security