Information distance

If the conditional information of a classical probability distribution of three random variables is zero, then it obeys a Markov chain condition. If the conditional information is close to zero, then it is known that the distance (minimum relative entropy) of the distribution to the nearest Markov chain distribution is precisely the conditional information. We prove here that this simple situation does not obtain for quantum conditional information. We show that for tri-partite quantum states the quantum conditional information is always a lower bound for the minimum relative entropy distance to a quantum Markov chain state, but the distance can be much greater; indeed the two quantities can be of different asymptotic order and may even differ by a dimensional factor.

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“…It is also quite amusing to see a quantity appearing that is known as information-distance in other contexts (see e.g. [5]). …”

Section: Example 8 Consider the Class Of Statesmentioning

confidence: 99%

Robustness of Quantum Markov Chains

Ibinson

Linden

Winter

2007

Commun. Math. Phys.

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“…Our basic tool, and a recent development in the field of information theory, is the universal information distance (27,28), which can be applied to any two objects stored on a computer (e.g., networks, genome sequences, or in our case, macrophage system states). This distance uniquely specifies the informational difference between two objects and is defined in terms of the Kolmogorov complexity.…”

Section: Quantifying Information Processing and Flowmentioning

confidence: 99%

Gene expression dynamics in the macrophage exhibit criticality

Nykter

Price

Aldana

et al. 2008

Proc. Natl. Acad. Sci. U.S.A.

195

161

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Cells are dynamical systems of biomolecular interactions that process information from their environment to mount diverse yet specific responses. A key property of many self-organized systems is that of criticality: a state of a system in which, on average, perturbations are neither dampened nor amplified, but are propagated over long temporal or spatial scales. Criticality enables the coordination of complex macroscopic behaviors that strike an optimal balance between stability and adaptability. It has long been hypothesized that biological systems are critical. Here, we address this hypothesis experimentally for system-wide gene expression dynamics in the macrophage. To this end, we have developed a method, based on algorithmic information theory, to assess macrophage criticality, and we have validated the method on networks with known properties. Using global gene expression data from macrophages stimulated with a variety of Toll-like receptor agonists, we found that macrophage dynamics are indeed critical, providing the most compelling evidence to date for this general principle of dynamics in biological systems.complex systems ͉ normalized compression distance ͉ information theory M any complex systems are capable of undergoing a phase transition between a disorganized and an organized state. This phenomenon has been observed in enzyme kinetics (1), growth of bacterial populations (2), foraging in ant colonies (3), brain activity (4), and traffic flow on the Internet (5). A system that is operating near such a phase transition is said to be critical. At equilibrium, this transition will occur at a critical value of a system parameter, such as the Curie temperature in a ferromagnet, below which the system can maintain spontaneous magnetization. Nonequilibrium systems, however, are capable of selforganizing to such a critical state, whereby complex behavior can emerge in a robust manner without fine-tuning the details of the system (6, 7).A hallmark of critical behavior is the spontaneous emergence of complex and coordinated macroscopic behavior in the form of long-range spatial or temporal correlations. Such coordination across many scales enables information to propagate over time from one part of the system to another with a high degree of specificity and sensitivity. For example, measurements of human brain oscillations revealed such critical dynamics of neural networks, implying their ability to effectively propagate information and rapidly reorganize (8). Similarly, measurements of computer network traffic indicate that the Internet exhibits critical dynamics, accordingly, suggesting optimal information transfer (9, 10). Many other complex systems, such as financial markets (11), forest fires (12), neuronal networks supporting our senses (13), and biological macroevolution (14) have been shown to self-organize to a critical state.A living cell is a complex dynamical system of interacting biomolecules. While this system exhibits stability even in varying environments, it is also capable of changing state...

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“…These universal codes are optimal for compressing ergodic sources and are still sufficiently computable for use in practice. The information distance and information metric introduced in [1,11] express how similar two objects are. Complementary to independence tests, similar objects have low distance or metric value.…”

Section: Introductionmentioning

confidence: 99%

“…1 One can think of a sum-test as a test for randomness for the case of a semimeasure on a discrete domain. Namely, if d is a P -sum-test, then for every n it easily follows from (1) that the set {x : d(x) ≥ n} has weight ≤ 2 −n under the semimeasure P .…”

Section: Introductionmentioning

confidence: 99%

Notes on Sum-Tests and Independence Tests

Bauwens

Terwijn

2009

Theory Comput Syst

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We study statistical sum-tests and independence tests, in particular for computably enumerable semimeasures on a discrete domain. Among other things, we prove that for universal semimeasures every 0 1 -sum-test is bounded, but unbounded 0 1 -sum-tests exist, and we study to what extent the latter can be universal. For universal semimeasures, in the unary case of sum-test we leave open whether universal 0 1 -sum-tests exist, whereas in the binary case of independence tests we prove that they do not exist.

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Information distance

Cited by 401 publications

References 25 publications

Robustness of Quantum Markov Chains

Robustness of Quantum Markov Chains

Gene expression dynamics in the macrophage exhibit criticality

Notes on Sum-Tests and Independence Tests

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