Complexity measures from interaction structures

Kahle, Thomas; Olbrich, Eckehard; Jost, Juergen; Ay, Nihat

doi:10.1103/physreve.79.026201

Cited by 34 publications

(64 citation statements)

References 23 publications

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“…A geometric lower bound is given by the weighted distances to the entropy maximisers with same k-marginals, [16,17,34,[41][42][43]. The interest in complexity measures was spurred by the association with enhanced neuronal activity, evaluating the functionality of equally correlated neural networks.…”

Section: S) Total Correlations Are Superadditive It Is Given a Coarmentioning

confidence: 99%

Quantifying Genuine Multipartite Correlations and their Pattern Complexity

2017

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We propose an information-theoretic framework to quantify multipartite correlations in classical and quantum systems, answering questions such as: what is the amount of seven-partite correlations in a given state of ten particles? We identify measures of genuine multipartite correlations, i.e. statistical dependencies which cannot be ascribed to bipartite correlations, satisfying a set of desirable properties. Inspired by ideas developed in complexity science, we then introduce the concept of weaving to classify states which display different correlation patterns, but cannot be distinguished by correlation measures. The weaving of a state is defined as the weighted sum of correlations of every order. Weaving measures are good descriptors of the complexity of correlation structures in multipartite systems. Correlations describe global properties which cannot be inferred from the features of the system parts, e.g. phases of many-body systems [5]. They are also resources. Entanglement, a kind of quantum correlation, enables speed-up in quantum information processing [6]. Yet, the very notion of genuine multipartite correlations still generates discussion [7]. There is no consistent way to quantify dependencies which do not manifest bipartite correlations, encoding joint properties of k > 2 particles instead, while witnesses of multipartite entanglement of at least order k have been proposed [8][9][10][11][12][13][14][15]. A further problem is that computing correlations is not always sufficient to fully describe multipartite correlation patterns. Equally correlated networks of multivariate variables can display different structures and properties [16,17]. Also, quantum states can be correlated in inherently inequivalent ways [18][19][20]. Here we propose a framework to describe genuine multipartite correlations in classical and quantum systems. We identify distance-based measures which satisfy a set of desirable properties when parts of the systems are added or discarded, and local operations are performed. We show that adopting the relative entropy allows for simplifying computations and meeting even stronger constraints. We then introduce the notion of weaving to classify multipartite states by studying how correlations scale with their order. The weaving of a state is given by the weighted sum of genuine multipartite correlations of any order, inheriting the properties of correlation measures. We compute the weaving of correlated classical and quantum states. In such cases, states which have equal total correlations or highest order correlations, but display a different correlation pattern, take different weaving values.

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Section: S) Total Correlations Are Superadditive It Is Given a Coarmentioning

confidence: 99%

Quantifying Genuine Multipartite Correlations and their Pattern Complexity

2017

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“…This decomposition was introduced in [9] and studied for several examples in [10] or [17] with the single terms called connected information or interaction complexities, respectively. The idea that synergy should capture everything beyond pair interactions motivates us to define:…”

Section: Interaction Spacesmentioning

confidence: 99%

Information Decomposition and Synergy

Olbrich

Bertschinger

Rauh

2015

Entropy

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Abstract:Recently, a series of papers addressed the problem of decomposing the information of two random variables into shared information, unique information and synergistic information. Several measures were proposed, although still no consensus has been reached. Here, we compare these proposals with an older approach to define synergistic information based on the projections on exponential families containing only up to k-th order interactions. We show that these measures are not compatible with a decomposition into unique, shared and synergistic information if one requires that all terms are always non-negative (local positivity). We illustrate the difference between the two measures for multivariate Gaussians.

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“…It should be also possible to adapt the spacewise updating algorithm to classify automatically traveling gliders and to investigate what sort of gliders are capable of surviving repeated collisions in the lattice and, thus, may be used as carriers of useful information across extended spatial domains. The present algorithm combined with spacewise search is also expected to help finding exhaustive answers concerning the spatio-temporal organization of cellular automaton models of interesting applications such as, for example, computer networks [14][15][16][17], secure schemes to share encrypted color images [18], modeling of biological patterns and processes like, e.g., tumor growth [19,20], efficient means of simulating mixing and segregation of granular media [21] as well as in a number of fundamental open questions in physics [22][23][24][25][26][27][28][29][30][31].…”

Section: Discussionmentioning

confidence: 99%