1,090 research outputs found
Low entropy output states for products of random unitary channels
In this paper, we study the behaviour of the output of pure entangled states
after being transformed by a product of conjugate random unitary channels. This
study is motivated by the counterexamples by Hastings and Hayden-Winter to the
additivity problems. In particular, we study in depth the difference of
behaviour between random unitary channels and generic random channels. In the
case where the number of unitary operators is fixed, we compute the limiting
eigenvalues of the output states. In the case where the number of unitary
operators grows linearly with the dimension of the input space, we show that
the eigenvalue distribution converges to a limiting shape that we characterize
with free probability tools. In order to perform the required computations, we
need a systematic way of dealing with moment problems for random matrices whose
blocks are i.i.d. Haar distributed unitary operators. This is achieved by
extending the graphical Weingarten calculus introduced in Collins and Nechita
(2010)
Gaussianization and eigenvalue statistics for random quantum channels (III)
In this paper, we present applications of the calculus developed in Collins
and Nechita [Comm. Math. Phys. 297 (2010) 345-370] and obtain an exact formula
for the moments of random quantum channels whose input is a pure state thanks
to Gaussianization methods. Our main application is an in-depth study of the
random matrix model introduced by Hayden and Winter [Comm. Math. Phys. 284
(2008) 263-280] and used recently by Brandao and Horodecki [Open Syst. Inf.
Dyn. 17 (2010) 31-52] and Fukuda and King [J. Math. Phys. 51 (2010) 042201] to
refine the Hastings counterexample to the additivity conjecture in quantum
information theory. This model is exotic from the point of view of random
matrix theory as its eigenvalues obey two different scalings simultaneously. We
study its asymptotic behavior and obtain an asymptotic expansion for its von
Neumann entropy.Comment: Published in at http://dx.doi.org/10.1214/10-AAP722 the Annals of
Applied Probability (http://www.imstat.org/aap/) by the Institute of
Mathematical Statistics (http://www.imstat.org
On Courant's nodal domain property for linear combinations of eigenfunctions, Part I
According to Courant's theorem, an eigenfunction as\-sociated with the -th
eigenvalue has at most nodal domains. A footnote in the book
of Courant and Hilbert, states that the same assertion is true for any linear
combination of eigenfunctions associated with eigenvalues less than or equal to
. We call this assertion the \emph{Extended Courant
Property}.\smallskipIn this paper, we propose simple and explicit examples for
which the extended Courant property is false: convex domains in
(hypercube and equilateral triangle), domains with cracks in , on
the round sphere , and on a flat torus .Comment: To appear in Documenta Mathematica.Modifications with respect to
version 4: Introduction rewritten. To shorten the paper two sections (Section
7, Numerical simulations and Section 8, Conjectures) have been removed and
will be published elsewhere. Related to the paper arXiv:1803.00449v2. Small
overlap with arXiv:1803.00449v1 which will be modified accordingl
Eigenvalue and Entropy Statistics for Products of Conjugate Random Quantum Channels
Using the graphical calculus and integration techniques introduced by the
authors, we study the statistical properties of outputs of products of random
quantum channels for entangled inputs. In particular, we revisit and generalize
models of relevance for the recent counterexamples to the minimum output
entropy additivity problems. Our main result is a classification of regimes for
which the von Neumann entropy is lower on average than the elementary bounds
that can be obtained with linear algebra techniques
The tame-wild principle for discriminant relations for number fields
Consider tuples of separable algebras over a common local or global number
field, related to each other by specified resolvent constructions. Under the
assumption that all ramification is tame, simple group-theoretic calculations
give best possible divisibility relations among the discriminants. We show that
for many resolvent constructions, these divisibility relations continue to hold
even in the presence of wild ramification.Comment: 31 pages, 11 figures. Version 2 fixes a normalization error: |G| is
corrected to n in Section 7.5. Version 3 fixes an off-by-one error in Section
6.
Vertices of Gelfand-Tsetlin Polytopes
This paper is a study of the polyhedral geometry of Gelfand-Tsetlin patterns
arising in the representation theory \mathfrak{gl}_n \C and algebraic
combinatorics. We present a combinatorial characterization of the vertices and
a method to calculate the dimension of the lowest-dimensional face containing a
given Gelfand-Tsetlin pattern.
As an application, we disprove a conjecture of Berenstein and Kirillov about
the integrality of all vertices of the Gelfand-Tsetlin polytopes. We can
construct for each a counterexample, with arbitrarily increasing
denominators as grows, of a non-integral vertex. This is the first infinite
family of non-integral polyhedra for which the Ehrhart counting function is
still a polynomial. We also derive a bound on the denominators for the
non-integral vertices when is fixed.Comment: 14 pages, 3 figures, fixed attribution
Random matrix techniques in quantum information theory
The purpose of this review article is to present some of the latest
developments using random techniques, and in particular, random matrix
techniques in quantum information theory. Our review is a blend of a rather
exhaustive review, combined with more detailed examples -- coming from research
projects in which the authors were involved. We focus on two main topics,
random quantum states and random quantum channels. We present results related
to entropic quantities, entanglement of typical states, entanglement
thresholds, the output set of quantum channels, and violations of the minimum
output entropy of random channels
- …