195 research outputs found
Multiscale Transforms for Signals on Simplicial Complexes
Our previous multiscale graph basis dictionaries/graph signal transforms --
Generalized Haar-Walsh Transform (GHWT); Hierarchical Graph Laplacian Eigen
Transform (HGLET); Natural Graph Wavelet Packets (NGWPs); and their relatives
-- were developed for analyzing data recorded on nodes of a given graph. In
this article, we propose their generalization for analyzing data recorded on
edges, faces (i.e., triangles), or more generally -dimensional
simplices of a simplicial complex (e.g., a triangle mesh of a manifold). The
key idea is to use the Hodge Laplacians and their variants for hierarchical
partitioning of a set of -dimensional simplices in a given simplicial
complex, and then build localized basis functions on these partitioned subsets.
We demonstrate their usefulness for data representation on both illustrative
synthetic examples and real-world simplicial complexes generated from a
co-authorship/citation dataset and an ocean current/flow dataset.Comment: 19 Pages, Comments welcom
Probing the anomalous dynamical phase in long-range quantum spin chains through Fisher-zero lines
Using the framework of infinite Matrix Product States, the existence of an
\textit{anomalous} dynamical phase for the transverse-field Ising chain with
sufficiently long-range interactions was first reported in [J.~C.~Halimeh and
V.~Zauner-Stauber, arXiv:1610:02019], where it was shown that
\textit{anomalous} cusps arise in the Loschmidt-echo return rate for
sufficiently small quenches within the ferromagnetic phase. In this work we
further probe the nature of the anomalous phase through calculating the
corresponding Fisher-zero lines in the complex time plane. We find that these
Fisher-zero lines exhibit a qualitative difference in their behavior, where,
unlike in the case of the regular phase, some of them terminate before
intersecting the imaginary axis, indicating the existence of smooth peaks in
the return rate preceding the cusps. Additionally, we discuss in detail the
infinite Matrix Product State time-evolution method used to calculate Fisher
zeros and the Loschmidt-echo return rate using the Matrix Product State
transfer matrix. Our work sheds further light on the nature of the anomalous
phase in the long-range transverse-field Ising chain, while the numerical
treatment presented can be applied to more general quantum spin chains.Comment: Journal article. 9 pages and 6 figures. Includes in part what used to
be supplemental material in arXiv:1610:0201
Genuinely multipartite entangled states and orthogonal arrays
A pure quantum state of N subsystems with d levels each is called
k-multipartite maximally entangled state, written k-uniform, if all its
reductions to k qudits are maximally mixed. These states form a natural
generalization of N-qudits GHZ states which belong to the class 1-uniform
states. We establish a link between the combinatorial notion of orthogonal
arrays and k-uniform states and prove the existence of several new classes of
such states for N-qudit systems. In particular, known Hadamard matrices allow
us to explicitly construct 2-uniform states for an arbitrary number of N>5
qubits. We show that finding a different class of 2-uniform states would imply
the Hadamard conjecture, so the full classification of 2-uniform states seems
to be currently out of reach. Additionally, single vectors of another class of
2-uniform states are one-to-one related to maximal sets of mutually unbiased
bases. Furthermore, we establish links between existence of k-uniform states,
classical and quantum error correction codes and provide a novel graph
representation for such states.Comment: 24 pages, 7 figures. Comments are very welcome
Entanglement of formation of mixed many-body quantum states via Tree Tensor Operators
We present a numerical strategy to efficiently estimate bipartite
entanglement measures, and in particular the Entanglement of Formation, for
many-body quantum systems on a lattice. Our approach exploits the Tree Tensor
Operator tensor network ansatz, a positive loopless representation for density
matrices which, as we demonstrate, efficiently encodes information on bipartite
entanglement, enabling the up-scaling of entanglement estimation. Employing
this technique, we observe a finite-size scaling law for the entanglement of
formation in 1D critical lattice models at finite temperature for up to 128
spins, extending to mixed states the scaling law for the entanglement entropy
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