1,605 research outputs found
Eigenstate entanglement between quantum chaotic subsystems: universal transitions and power laws in the entanglement spectrum
We derive universal entanglement entropy and Schmidt eigenvalue behaviors for
the eigenstates of two quantum chaotic systems coupled with a weak interaction.
The progression from a lack of entanglement in the noninteracting limit to the
entanglement expected of fully randomized states in the opposite limit is
governed by the single scaling transition parameter, . The behaviors
apply equally well to few- and many-body systems, e.g.\ interacting particles
in quantum dots, spin chains, coupled quantum maps, and Floquet systems as long
as their subsystems are quantum chaotic, and not localized in some manner. To
calculate the generalized moments of the Schmidt eigenvalues in the
perturbative regime, a regularized theory is applied, whose leading order
behaviors depend on . The marginal case of the moment,
which is related to the distance to closest maximally entangled state, is an
exception having a leading order and a logarithmic
dependence on subsystem size. A recursive embedding of the regularized
perturbation theory gives a simple exponential behavior for the von Neumann
entropy and the Havrda-Charv{\' a}t-Tsallis entropies for increasing
interaction strength, demonstrating a universal transition to nearly maximal
entanglement. Moreover, the full probability densities of the Schmidt
eigenvalues, i.e.\ the entanglement spectrum, show a transition from power laws
and L\'evy distribution in the weakly interacting regime to random matrix
results for the strongly interacting regime. The predicted behaviors are tested
on a pair of weakly interacting kicked rotors, which follow the universal
behaviors extremely well
Random pure states: quantifying bipartite entanglement beyond the linear statistics
We analyze the properties of entangled random pure states of a quantum system
partitioned into two smaller subsystems of dimensions and . Framing the
problem in terms of random matrices with a fixed-trace constraint, we
establish, for arbitrary , a general relation between the -point
densities and the cross-moments of the eigenvalues of the reduced density
matrix, i.e. the so-called Schmidt eigenvalues, and the analogous functionals
of the eigenvalues of the Wishart-Laguerre ensemble of the random matrix
theory. This allows us to derive explicit expressions for two-level densities,
and also an exact expression for the variance of von Neumann entropy at finite
. Then we focus on the moments of the Schmidt number
, the reciprocal of the purity. This is a random variable supported on
, which quantifies the number of degrees of freedom effectively
contributing to the entanglement. We derive a wealth of analytical results for
for and and arbitrary , and also for
square systems by spotting for the latter a connection with the
probability that the smallest eigenvalue
of a matrix belonging to the Gaussian Unitary
Ensemble is larger than . As a byproduct, we present an exact
asymptotic expansion for for finite as
. Our results are corroborated by numerical simulations
whenever possible, with excellent agreement.Comment: 22 pages, 8 figures. Minor changes, typos fixed. Accepted for
publication in PR
Thermodynamics of network model fitting with spectral entropies
An information theoretic approach inspired by quantum statistical mechanics
was recently proposed as a means to optimize network models and to assess their
likelihood against synthetic and real-world networks. Importantly, this method
does not rely on specific topological features or network descriptors, but
leverages entropy-based measures of network distance. Entertaining the analogy
with thermodynamics, we provide a physical interpretation of model
hyperparameters and propose analytical procedures for their estimate. These
results enable the practical application of this novel and powerful framework
to network model inference. We demonstrate this method in synthetic networks
endowed with a modular structure, and in real-world brain connectivity
networks.Comment: 11 pages, 3 figure
From RNA folding to inverse folding: a computational study: Folding and design of RNA molecules
Since the discovery of the structure of DNA in the early 1953s and its double-chained complement of information hinting at its means of replication, biologists have recognized the strong connection between molecular structure and function. In the past two decades, there has been a surge of research on an ever-growing class of RNA molecules that are non-coding but whose various folded structures allow a diverse array of vital functions. From the well-known splicing and modification of ribosomal RNA, non-coding RNAs (ncRNAs) are now known to be intimately involved in possibly every stage of DNA translation and protein transcription, as well as RNA signalling and gene regulation processes.
Despite the rapid development and declining cost of modern molecular methods, they typically can only describe ncRNA's structural conformations in vitro, which differ from their in vivo counterparts. Moreover, it is estimated that only a tiny fraction of known ncRNAs has been documented experimentally, often at a high cost. There is thus a growing realization that computational methods must play a central role in the analysis of ncRNAs. Not only do computational approaches hold the promise of rapidly characterizing many ncRNAs yet to be described, but there is also the hope that by understanding the rules that determine their structure, we will gain better insight into their function and design. Many studies revealed that the ncRNA functions are performed by high-level structures that often depend on their low-level structures, such as the secondary structure. This thesis studies the computational folding mechanism and inverse folding of ncRNAs at the secondary level.
In this thesis, we describe the development of two bioinformatic tools that have the potential to improve our understanding of RNA secondary structure. These tools are as follows: (1) RAFFT for efficient prediction of pseudoknot-free RNA folding pathways using the fast Fourier transform (FFT)}; (2) aRNAque, an evolutionary algorithm inspired by Lévy flights for RNA inverse folding with or without pseudoknot (A secondary structure that often poses difficulties for bio-computational detection).
The first tool, RAFFT, implements a novel heuristic to predict RNA secondary structure formation pathways that has two components: (i) a folding algorithm and (ii) a kinetic ansatz. When considering the best prediction in the ensemble of 50 secondary structures predicted by RAFFT, its performance matches the recent deep-learning-based structure prediction methods. RAFFT also acts as a folding kinetic ansatz, which we tested on two RNAs: the CFSE and a classic bi-stable sequence. In both test cases, fewer structures were required to reproduce the full kinetics, whereas known methods (such as Treekin) required a sample of 20,000 structures and more.
The second tool, aRNAque, implements an evolutionary algorithm (EA) inspired by the Lévy flight, allowing both local global search and which supports pseudoknotted target structures. The number of point mutations at every step of aRNAque's EA is drawn from a Zipf distribution. Therefore, our proposed method increases the diversity of designed RNA sequences and reduces the average number of evaluations of the evolutionary algorithm. The overall performance showed improved empirical results compared to existing tools through intensive benchmarks on both pseudoknotted and pseudoknot-free datasets.
In conclusion, we highlight some promising extensions of the versatile RAFFT method to RNA-RNA interaction studies. We also provide an outlook on both tools' implications in studying evolutionary dynamics
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