222 research outputs found
Monotone graph limits and quasimonotone graphs
The recent theory of graph limits gives a powerful framework for
understanding the properties of suitable (convergent) sequences of
graphs in terms of a limiting object which may be represented by a symmetric
function on , i.e., a kernel or graphon. In this context it is
natural to wish to relate specific properties of the sequence to specific
properties of the kernel. Here we show that the kernel is monotone (i.e.,
increasing in both variables) if and only if the sequence satisfies a
`quasi-monotonicity' property defined by a certain functional tending to zero.
As a tool we prove an inequality relating the cut and norms of kernels of
the form with and monotone that may be of interest in its
own right; no such inequality holds for general kernels.Comment: 38 page
Explicit lower and upper bounds on the entangled value of multiplayer XOR games
XOR games are the simplest model in which the nonlocal properties of
entanglement manifest themselves. When there are two players, it is well known
that the bias --- the maximum advantage over random play --- of entangled
players can be at most a constant times greater than that of classical players.
Recently, P\'{e}rez-Garc\'{i}a et al. [Comm. Math. Phys. 279 (2), 2008] showed
that no such bound holds when there are three or more players: the advantage of
entangled players over classical players can become unbounded, and scale with
the number of questions in the game. Their proof relies on non-trivial results
from operator space theory, and gives a non-explicit existence proof, leading
to a game with a very large number of questions and only a loose control over
the local dimension of the players' shared entanglement.
We give a new, simple and explicit (though still probabilistic) construction
of a family of three-player XOR games which achieve a large quantum-classical
gap (QC-gap). This QC-gap is exponentially larger than the one given by
P\'{e}rez-Garc\'{i}a et. al. in terms of the size of the game, achieving a
QC-gap of order with questions per player. In terms of the
dimension of the entangled state required, we achieve the same (optimal) QC-gap
of for a state of local dimension per player. Moreover, the
optimal entangled strategy is very simple, involving observables defined by
tensor products of the Pauli matrices.
Additionally, we give the first upper bound on the maximal QC-gap in terms of
the number of questions per player, showing that our construction is only
quadratically off in that respect. Our results rely on probabilistic estimates
on the norm of random matrices and higher-order tensors which may be of
independent interest.Comment: Major improvements in presentation; results identica
A new look at C*-simplicity and the unique trace property of a group
We characterize when the reduced C*-algebra of a group has unique tracial
state, respectively, is simple, in terms of Dixmier-type properties of the
group C*-algebra. We also give a simple proof of the recent result by
Breuillard, Kalantar, Kennedy and Ozawa that the reduced C*-algebra of a group
has unique tracial state if and only if the amenable radical of the group is
trivial.Comment: 8 page
Mutually unbiased bases in dimension six: The four most distant bases
We consider the average distance between four bases in dimension six. The
distance between two orthonormal bases vanishes when the bases are the same,
and the distance reaches its maximal value of unity when the bases are
unbiased. We perform a numerical search for the maximum average distance and
find it to be strictly smaller than unity. This is strong evidence that no four
mutually unbiased bases exist in dimension six. We also provide a two-parameter
family of three bases which, together with the canonical basis, reach the
numerically-found maximum of the average distance, and we conduct a detailed
study of the structure of the extremal set of bases.Comment: 10 pages, 2 figures, 1 tabl
Association analysis identifies TLR7 and TLR8 as novel risk genes in asthma and related disorders
Constructing Mutually Unbiased Bases in Dimension Six
The density matrix of a qudit may be reconstructed with optimal efficiency if
the expectation values of a specific set of observables are known. In dimension
six, the required observables only exist if it is possible to identify six
mutually unbiased complex 6x6 Hadamard matrices. Prescribing a first Hadamard
matrix, we construct all others mutually unbiased to it, using algebraic
computations performed by a computer program. We repeat this calculation many
times, sampling all known complex Hadamard matrices, and we never find more
than two that are mutually unbiased. This result adds considerable support to
the conjecture that no seven mutually unbiased bases exist in dimension six.Comment: As published version. Added discussion of the impact of numerical
approximations and corrected the number of triples existing for non-affine
families (cf Table 3
A Characterization of right coideals of quotient type and its application to classification of Poisson boundaries
Let be a co-amenable compact quantum group. We show that a right coideal
of is of quotient type if and only if it is the range of a conditional
expectation preserving the Haar state and is globally invariant under the left
action of the dual discrete quantum group. We apply this result to theory of
Poisson boundaries introduced by Izumi for discrete quantum groups and
generalize a work of Izumi-Neshveyev-Tuset on for co-amenable compact
quantum groups with the commutative fusion rules. More precisely, we prove that
the Poisson integral is an isomorphism between the Poisson boundary and the
right coideal of quotient type by maximal quantum subgroup of Kac type. In
particular, the Poisson boundary and the quantum flag manifold are isomorphic
for any q-deformed classical compact Lie group.Comment: 28 pages, Remark 4.9 adde
Spectral density of generalized Wishart matrices and free multiplicative convolution
We investigate the level density for several ensembles of positive random
matrices of a Wishart--like structure, , where stands for a
nonhermitian random matrix. In particular, making use of the Cauchy transform,
we study free multiplicative powers of the Marchenko-Pastur (MP) distribution,
, which for an integer yield Fuss-Catalan
distributions corresponding to a product of independent square random
matrices, . New formulae for the level densities are derived
for and . Moreover, the level density corresponding to the
generalized Bures distribution, given by the free convolution of arcsine and MP
distributions is obtained. We also explain the reason of such a curious
convolution. The technique proposed here allows for the derivation of the level
densities for several other cases.Comment: 10 latex pages including 4 figures, Ver 4, minor improvements and
references updat
Exotic complex Hadamard matrices, and their equivalence
In this paper we use a design theoretical approach to construct new,
previously unknown complex Hadamard matrices. Our methods generalize and extend
the earlier results of de la Harpe--Jones and Munemasa--Watatani and offer a
theoretical explanation for the existence of some sporadic examples of complex
Hadamard matrices in the existing literature. As it is increasingly difficult
to distinguish inequivalent matrices from each other, we propose a new
invariant, the fingerprint of complex Hadamard matrices. As a side result, we
refute a conjecture of Koukouvinos et al. on (n-8)x(n-8) minors of real
Hadamard matrices.Comment: 10 pages. To appear in Cryptography and Communications: Discrete
Structures, Boolean Functions and Sequence
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