20,108 research outputs found
Entanglement Wedge Cross Sections Require Tripartite Entanglement
We argue that holographic CFT states require a large amount of tripartite
entanglement, in contrast to the conjecture that their entanglement is mostly
bipartite. Our evidence is that this mostly-bipartite conjecture is in sharp
conflict with two well-supported conjectures about the entanglement wedge cross
section surface . If is related to either the CFT's reflected
entropy or its entanglement of purification, then those quantities can differ
from the mutual information at . We prove that this
implies holographic CFT states must have amounts
of tripartite entanglement. This proof involves a new Fannes-type inequality
for the reflected entropy, which itself has many interesting applications.Comment: 20 pages, 5 figures, comments added in v
Relaxation-Based Coarsening for Multilevel Hypergraph Partitioning
Multilevel partitioning methods that are inspired by principles of
multiscaling are the most powerful practical hypergraph partitioning solvers.
Hypergraph partitioning has many applications in disciplines ranging from
scientific computing to data science. In this paper we introduce the concept of
algebraic distance on hypergraphs and demonstrate its use as an algorithmic
component in the coarsening stage of multilevel hypergraph partitioning
solvers. The algebraic distance is a vertex distance measure that extends
hyperedge weights for capturing the local connectivity of vertices which is
critical for hypergraph coarsening schemes. The practical effectiveness of the
proposed measure and corresponding coarsening scheme is demonstrated through
extensive computational experiments on a diverse set of problems. Finally, we
propose a benchmark of hypergraph partitioning problems to compare the quality
of other solvers
Eigendecompositions of Transfer Operators in Reproducing Kernel Hilbert Spaces
Transfer operators such as the Perron--Frobenius or Koopman operator play an
important role in the global analysis of complex dynamical systems. The
eigenfunctions of these operators can be used to detect metastable sets, to
project the dynamics onto the dominant slow processes, or to separate
superimposed signals. We extend transfer operator theory to reproducing kernel
Hilbert spaces and show that these operators are related to Hilbert space
representations of conditional distributions, known as conditional mean
embeddings in the machine learning community. Moreover, numerical methods to
compute empirical estimates of these embeddings are akin to data-driven methods
for the approximation of transfer operators such as extended dynamic mode
decomposition and its variants. One main benefit of the presented kernel-based
approaches is that these methods can be applied to any domain where a
similarity measure given by a kernel is available. We illustrate the results
with the aid of guiding examples and highlight potential applications in
molecular dynamics as well as video and text data analysis
Recurrence for discrete time unitary evolutions
We consider quantum dynamical systems specified by a unitary operator U and
an initial state vector \phi. In each step the unitary is followed by a
projective measurement checking whether the system has returned to the initial
state. We call the system recurrent if this eventually happens with probability
one. We show that recurrence is equivalent to the absence of an absolutely
continuous part from the spectral measure of U with respect to \phi. We also
show that in the recurrent case the expected first return time is an integer or
infinite, for which we give a topological interpretation. A key role in our
theory is played by the first arrival amplitudes, which turn out to be the
(complex conjugated) Taylor coefficients of the Schur function of the spectral
measure. On the one hand, this provides a direct dynamical interpretation of
these coefficients; on the other hand it links our definition of first return
times to a large body of mathematical literature.Comment: 27 pages, 5 figures, typos correcte
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