Entanglement Entropy in Extended Quantum Systems

After a brief introduction to the concept of entanglement in quantum systems, I apply these ideas to many-body systems and show that the von Neumann entropy is an effective way of characterising the entanglement between the degrees of freedom in different regions of space. Close to a quantum phase transition it has universal features which serve as a diagnostic of such phenomena. In the second part I consider the unitary time evolution of such systems following a `quantum quench' in which a parameter in the hamiltonian is suddenly changed, and argue that finite regions should effectively thermalise at late times, after interesting transient effects.Comment: 6 pages. Plenary talk delivered at Statphys 23, Genoa, July 200

Inference algorithms for gene networks: a statistical mechanics analysis

The inference of gene regulatory networks from high throughput gene expression data is one of the major challenges in systems biology. This paper aims at analysing and comparing two different algorithmic approaches. The first approach uses pairwise correlations between regulated and regulating genes; the second one uses message-passing techniques for inferring activating and inhibiting regulatory interactions. The performance of these two algorithms can be analysed theoretically on well-defined test sets, using tools from the statistical physics of disordered systems like the replica method. We find that the second algorithm outperforms the first one since it takes into account collective effects of multiple regulators

Unifying approach to the quantification of bipartite correlations by Bures distance

The notion of distance defined on the set of states of a composite quantum system can be used to quantify total, quantum and classical correlations in a unifying way. We provide new closed formulae for classical and total correlations of two-qubit Bell-diagonal states by considering the Bures distance. Complementing the known corresponding expressions for entanglement and more general quantum correlations, we thus complete the quantitative hierarchy of Bures correlations for Bell-diagonal states. We then explicitly calculate Bures correlations for two relevant families of states: Werner states and rank-2 Bell-diagonal states, highlighting the subadditivity which holds for total correlations with respect to the sum of classical and quantum ones when using Bures distance. Finally, we analyse a dynamical model of two independent qubits locally exposed to non-dissipative decoherence channels, where both quantum and classical correlations measured by Bures distance exhibit freezing phenomena, in analogy with other known quantifiers of correlations.Comment: 18 pages, 4 figures; published versio

The entanglement of few-particle systems when using the local-density approximation

In this chapter we discuss methods to calculate the entanglement of a system using density-functional theory. We firstly introduce density-functional theory and the local-density approximation (LDA). We then discuss the concept of the `interacting LDA system'. This is characterised by an interacting many-body Hamiltonian which reproduces, uniquely and exactly, the ground state density obtained from the single-particle Kohn-Sham equations of density-functional theory when the local-density approximation is used. We motivate why this idea can be useful for appraising the local-density approximation in many-body physics particularly with regards to entanglement and related quantum information applications. Using an iterative scheme, we find the Hamiltonian characterising the interacting LDA system in relation to the test systems of Hooke's atom and helium-like atoms. The interacting LDA system ground state wavefunction is then used to calculate the spatial entanglement and the results are compared and contrasted with the exact entanglement for the two test systems. For Hooke's atom we also compare the entanglement to our previous estimates of an LDA entanglement. These were obtained using a combination of evolutionary algorithm and gradient descent, and using an LDA-based perturbative approach. We finally discuss if the position-space information entropy of the density---which can be obtained directly from the system density and hence easily from density-functional theory methods---can be considered as a proxy measure for the spatial entanglement for the test systems.Comment: 12 pages and 5 figures

Langevin processes, agent models and socio-economic systems

We review some approaches to the understanding of fluctuations in some models used to describe socio and economic systems. Our approach builds on the development of a simple Langevin equation that characterises stochastic processes. This provides a unifying approach that allows first a straightforward description of the early approaches of Bachelier. We generalise the approach to stochastic equations that model interacting agents. Using a simple change of variable, we show that the peer pressure model of Marsilli and the wealth dynamics model of Solomon are closely related. The methods are further shown to be consistent with a global free energy functional that invokes an entropy term based on the Boltzmann formula. A more recent approach by Michael and Johnson maximised a Tsallis entropy function subject to simple constraints. We show how this approach can be developed from an agent model where the simple Langevin process is now conditioned by local rather than global noise. The approach yields a BBGKY type hierarchy of equations for the system correlation functions. Of especial interest is that the results can be obtained from a new free energy functional similar to that mentioned above except that a Tsallis like entropy term replaces the Boltzmann entropy term. A mean field approximation yields the results of Michael and Johnson. We show how personal income data for Brazil, the US, Germany and the UK, analysed recently by Borgas can be qualitatively understood by this approach.Comment: 1 figur