State-space Manifold and Rotating Black Holes

We study a class of fluctuating higher dimensional black hole configurations obtained in string theory/ $M$-theory compactifications. We explore the intrinsic Riemannian geometric nature of Gaussian fluctuations arising from the Hessian of the coarse graining entropy, defined over an ensemble of brane microstates. It has been shown that the state-space geometry spanned by the set of invariant parameters is non-degenerate, regular and has a negative scalar curvature for the rotating Myers-Perry black holes, Kaluza-Klein black holes, supersymmetric $AdS_5$ black holes, $D_1$-$D_5$ configurations and the associated BMPV black holes. Interestingly, these solutions demonstrate that the principal components of the state-space metric tensor admit a positive definite form, while the off diagonal components do not. Furthermore, the ratio of diagonal components weakens relatively faster than the off diagonal components, and thus they swiftly come into an equilibrium statistical configuration. Novel aspects of the scaling property suggest that the brane-brane statistical pair correlation functions divulge an asymmetric nature, in comparison with the others. This approach indicates that all above configurations are effectively attractive and stable, on an arbitrary hyper-surface of the state-space manifolds. It is nevertheless noticed that there exists an intriguing relationship between non-ideal inter-brane statistical interactions and phase transitions. The ramifications thus described are consistent with the existing picture of the microscopic CFTs. We conclude with an extended discussion of the implications of this work for the physics of black holes in string theory.Comment: 44 pages, Keywords: Rotating Black Holes; State-space Geometry; Statistical Configurations, String Theory, M-Theory. PACS numbers: 04.70.-s Physics of black holes; 04.70.Bw Classical black holes; 04.70.Dy Quantum aspects of black holes, evaporation, thermodynamics; 04.50.Gh Higher-dimensional black holes, black strings, and related objects. Edited the bibliograph

Incomplete descriptions and relevant entropies

Statistical mechanics relies on the complete though probabilistic description of a system in terms of all the microscopic variables. Its object is to derive therefrom static and dynamic properties involving some reduced set of variables. The elimination of the irrelevant variables is guided by the maximum entropy criterion, which produces the probability law carrying the least amount of information compatible with the relevant variables. This defines relevant entropies which measure the missing information (the disorder) associated with the sole variables retained in an incomplete description. Relevant entropies depend not only on the state of the system but also on the coarseness of its reduced description. Their use sheds light on questions such as the Second Law, both in equilibrium an in irreversible thermodynamics, the projection method of statistical mechanics, Boltzmann's \textit{H}-theorem or spin-echo experiment.Comment: flatex relevant_entropies.tex, 1 file Submitted to: Am. J. Phy

Granger causality and the inverse Ising problem

We study Ising models for describing data and show that autoregressive methods may be used to learn their connections, also in the case of asymmetric connections and for multi-spin interactions. For each link the linear Granger causality is two times the corresponding transfer entropy (i.e. the information flow on that link) in the weak coupling limit. For sparse connections and a low number of samples, the L1 regularized least squares method is used to detect the interacting pairs of spins. Nonlinear Granger causality is related to multispin interactions.Comment: 6 pages and 8 figures. Revised version in press on Physica

Local versus non-local information in quantum information theory: formalism and phenomena

In spite of many results in quantum information theory, the complex nature of compound systems is far from being clear. In general the information is a mixture of local, and non-local ("quantum") information. To make this point more clear, we develop and investigate the quantum information processing paradigm in which parties sharing a multipartite state distill local information. The amount of information which is lost because the parties must use a classical communication channel is the deficit. This scheme can be viewed as complementary to the notion of distilling entanglement. After reviewing the paradigm, we show that the upper bound for the deficit is given by the relative entropy distance to so-called psuedo-classically correlated states; the lower bound is the relative entropy of entanglement. This implies, in particular, that any entangled state is informationally nonlocal i.e. has nonzero deficit. We also apply the paradigm to defining the thermodynamical cost of erasing entanglement. We show the cost is bounded from below by relative entropy of entanglement. We demonstrate the existence of several other non-local phenomena. For example,we prove the existence of a form of non-locality without entanglement and with distinguishability. We analyze the deficit for several classes of multipartite pure states and obtain that in contrast to the GHZ state, the Aharonov state is extremely nonlocal (and in fact can be thought of as quasi-nonlocalisable). We also show that there do not exist states, for which the deficit is strictly equal to the whole informational content (bound local information). We then discuss complementary features of information in distributed quantum systems. Finally we discuss the physical and theoretical meaning of the results and pose many open questions.Comment: 35 pages in two column, 4 figure