186 research outputs found
The Collatz-Wielandt quotient for pairs of nonnegative operators
In this paper we consider two versions of the Collatz-Wielandt quotient for a
pair of nonnegative operators A,B that map a given pointed generating cone in
the first space into a given pointed generating cone in the second space. If
the two spaces and two cones are identical, and B is the identity operator then
one version of this quotient is the spectral radius of A. In some applications,
as commodity pricing, power control in wireless networks and quantum
information theory, one needs to deal with the Collatz-Wielandt quotient for
two nonnegative operators. In this paper we treat the two important cases: a
pair of rectangular nonnegative matrices and a pair completely positive
operators. We give a characterization of minimal optimal solutions and
polynomially computable bounds on the Collatz-Wielandt quotient.Comment: 24 pages. To appear in Applications of Mathematics, ISSN 0862-794
Mathematical open problems in Projected Entangled Pair States
Projected Entangled Pair States (PEPS) are used in practice as an efficient
parametrization of the set of ground states of quantum many body systems. The
aim of this paper is to present, for a broad mathematical audience, some
mathematical questions about PEPS.Comment: Notes associated to the Santal\'o Lecture 2017, Universidad
Complutense de Madrid (UCM), minor typos correcte
Inequalities for selected eigenvalues of the product of matrices
The product of a Hermitian matrix and a positive semidefinite matrix has only
real eigenvalues. We present bounds for sums of eigenvalues of such a product.Comment: to appear in AMS Proceeding
Antieigenvalues and antisingularvalues of a matrix and applications to problems in statistics
Let A be p × p positive definite matrix. A p-vector x such that Ax =
x is called an eigenvector with the associated with eigenvalue . Equivalent
characterizations are:
(i) cos = 1, where is the angle between x and Ax.
(ii) (x0Ax)−1 = xA−1x.
(iii) cos = 1, where is the angle between A1/2x and A−1/2x.
We ask the question what is x such that cos as defined in (i) is a minimum
or the angle of separation between x and Ax is a maximum. Such a vector
is called an anti-eigenvector and cos an anti-eigenvalue of A. This is the
basis of operator trigonometry developed by K. Gustafson and P.D.K.M. Rao
(1997), Numerical Range: The Field of Values of Linear Operators and Matrices,
Springer. We may define a measure of departure from condition (ii) as
min[(x0Ax)(x0A−1x)]−1 which gives the same anti-eigenvalue. The same result
holds if the maximum of the angle between A1/2x and A−1/2x as in condition
(iii) is sought. We define a hierarchical series of anti-eigenvalues, and also consider
optimization problems associated with measures of separation between an
r(< p) dimensional subspace S and its transform AS.
Similar problems are considered for a general matrix A and its singular
values leading to anti-singular values.
Other possible definitions of anti-eigen and anti-singular values, and applications
to problems in statistics will be presented
The quantum communication complexity of sampling
Sampling is an important primitive in probabilistic and quantum algorithms. In the spirit of communication complexity, given a function f : X × Y → {0, 1} and a probability distribution D over X × Y , we define the sampling complexity of (f,D) as the minimum number of bits that Alice and Bob must communicate for Alice to pick x ∈ X and Bob to pick y ∈ Y as well as a value z such that the resulting distribution of (x, y, z) is close to the distribution (D, f(D)).
In this paper we initiate the study of sampling complexity, in both the classical and quantum models. We give several variants of a definition. We completely characterize some of these variants and give upper and lower bounds on others. In particular, this allows us to establish an exponential gap between quantum and classical sampling complexity for the set-disjointness function
- …