3,086 research outputs found
On the cone eigenvalue complementarity problem for higher-order tensors
In this paper, we consider the tensor generalized eigenvalue complementarity
problem (TGEiCP), which is an interesting generalization of matrix eigenvalue
complementarity problem (EiCP). First, we given an affirmative result showing
that TGEiCP is solvable and has at least one solution under some reasonable
assumptions. Then, we introduce two optimization reformulations of TGEiCP,
thereby beneficially establishing an upper bound of cone eigenvalues of
tensors. Moreover, some new results concerning the bounds of number of
eigenvalues of TGEiCP further enrich the theory of TGEiCP. Last but not least,
an implementable projection algorithm for solving TGEiCP is also developed for
the problem under consideration. As an illustration of our theoretical results,
preliminary computational results are reported.Comment: 26 pages, 2 figures, 3 table
A Semismooth Newton Method for Tensor Eigenvalue Complementarity Problem
In this paper, we consider the tensor eigenvalue complementarity problem
which is closely related to the optimality conditions for polynomial
optimization, as well as a class of differential inclusions with nonconvex
processes. By introducing an NCP-function, we reformulate the tensor eigenvalue
complementarity problem as a system of nonlinear equations. We show that this
function is strongly semismooth but not differentiable, in which case the
classical smoothing methods cannot apply. Furthermore, we propose a damped
semismooth Newton method for tensor eigenvalue complementarity problem. A new
procedure to evaluate an element of the generalized Jocobian is given, which
turns out to be an element of the B-subdifferential under mild assumptions. As
a result, the convergence of the damped semismooth Newton method is guaranteed
by existing results. The numerical experiments also show that our method is
efficient and promising
Complementarity of Entanglement and Interference
A complementarity relation is shown between the visibility of interference
and bipartite entanglement in a two qubit interferometric system when the
parameters of the quantum operation change for a given input state. The
entanglement measure is a decreasing function of the visibility of
interference. The implications for quantum computation are briefly discussed.Comment: Final version, to appear on IJMPC; minor revision
Galois Unitaries, Mutually Unbiased Bases, and MUB-balanced states
A Galois unitary is a generalization of the notion of anti-unitary operators.
They act only on those vectors in Hilbert space whose entries belong to some
chosen number field. For Mutually Unbiased Bases the relevant number field is a
cyclotomic field. By including Galois unitaries we are able to remove a
mismatch between the finite projective group acting on the bases on the one
hand, and the set of those permutations of the bases that can be implemented as
transformations in Hilbert space on the other hand. In particular we show that
there exist transformations that cycle through all the bases in every dimension
which is an odd power of an odd prime. (For even primes unitary MUB-cyclers
exist.) These transformations have eigenvectors, which are MUB-balanced states
(i.e. rotationally symmetric states in the original terminology of Wootters and
Sussman) if and only if d = 3 modulo 4. We conjecture that this construction
yields all such states in odd prime power dimension.Comment: 32 pages, 2 figures, AMS Latex. Version 2: minor improvements plus a
few additional reference
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