10,059 research outputs found
Invariant and polynomial identities for higher rank matrices
We exhibit explicit expressions, in terms of components, of discriminants,
determinants, characteristic polynomials and polynomial identities for matrices
of higher rank. We define permutation tensors and in term of them we construct
discriminants and the determinant as the discriminant of order , where
is the dimension of the matrix. The characteristic polynomials and the
Cayley--Hamilton theorem for higher rank matrices are obtained there from
Nonparametric maximum likelihood estimation of probability densities by penalty function methods
When it is known a priori exactly to which finite dimensional manifold the probability density function gives rise to a set of samples, the parametric maximum likelihood estimation procedure leads to poor estimates and is unstable; while the nonparametric maximum likelihood procedure is undefined. A very general theory of maximum penalized likelihood estimation which should avoid many of these difficulties is presented. It is demonstrated that each reproducing kernel Hilbert space leads, in a very natural way, to a maximum penalized likelihood estimator and that a well-known class of reproducing kernel Hilbert spaces gives polynomial splines as the nonparametric maximum penalized likelihood estimates
A random number generator for continuous random variables
A FORTRAN 4 routine is given which may be used to generate random observations of a continuous real valued random variable. Normal distribution of F(x), X, E(akimas), and E(linear) is presented in tabular form
Optimal discrimination of quantum states with a fixed rate of inconclusive outcomes
In this paper we present the solution to the problem of optimally
discriminating among quantum states, i.e., identifying the states with maximum
probability of success when a certain fixed rate of inconclusive answers is
allowed. By varying the inconclusive rate, the scheme optimally interpolates
between Unambiguous and Minimum Error discrimination, the two standard
approaches to quantum state discrimination. We introduce a very general method
that enables us to obtain the solution in a wide range of cases and give a
complete characterization of the minimum discrimination error as a function of
the rate of inconclusive answers. A critical value of this rate is identified
that coincides with the minimum failure probability in the cases where
unambiguous discrimination is possible and provides a natural generalization of
it when states cannot be unambiguously discriminated. The method is illustrated
on two explicit examples: discrimination of two pure states with arbitrary
prior probabilities and discrimination of trine states
Beating noise with abstention in state estimation
We address the problem of estimating pure qubit states with non-ideal (noisy)
measurements in the multiple-copy scenario, where the data consists of a number
N of identically prepared qubits. We show that the average fidelity of the
estimates can increase significantly if the estimation protocol allows for
inconclusive answers, or abstentions. We present the optimal such protocol and
compute its fidelity for a given probability of abstention. The improvement
over standard estimation, without abstention, can be viewed as an effective
noise reduction. These and other results are exemplified for small values of N.
For asymptotically large N, we derive analytical expressions of the fidelity
and the probability of abstention, and show that for a fixed fidelity gain the
latter decreases with N at an exponential rate given by a Kulback-Leibler
(relative) entropy. As a byproduct, we obtain an asymptotic expression in terms
of this very entropy of the probability that a system of N qubits, all prepared
in the same state, has a given total angular momentum. We also discuss an
extreme situation where noise increases with N and where estimation with
abstention provides a most significant improvement as compared to the standard
approach
Universal field equations for metric-affine theories of gravity
We show that almost all metric--affine theories of gravity yield Einstein
equations with a non--null cosmological constant . Under certain
circumstances and for any dimension, it is also possible to incorporate a Weyl
vector field and therefore the presence of an anisotropy. The viability
of these field equations is discussed in view of recent astrophysical
observations.Comment: 13 pages. This is a copy of the published paper. We are posting it
here because of the increasing interest in f(R) theories of gravit
Phase-Covariant Quantum Benchmarks
We give a quantum benchmark for teleportation and quantum storage experiments
suited for pure and mixed test states. The benchmark is based on the average
fidelity over a family of phase-covariant states and certifies that an
experiment can not be emulated by a classical setup, i.e., by a
measure-and-prepare scheme. We give an analytical solution for qubits, which
shows important differences with standard state estimation approach, and
compute the value of the benchmark for coherent and squeezed states, both pure
and mixed.Comment: 4 pages, 2 figure
Cooper pairs as bosons
Although BCS pairs of fermions are known not to obey Bose-Einstein (BE)
commutation relations nor BE statistics, we show how Cooper pairs (CPs),
whether the simple original ones or the CPs recently generalized in a many-body
Bethe-Salpeter approach, being clearly distinct from BCS pairs at least obey BE
statistics. Hence, contrary to widespread popular belief, CPs can undergo BE
condensation to account for superconductivity if charged, as well as for
neutral-atom fermion superfluidity where CPs, but uncharged, are also expected
to form.Comment: 8 pages, 2 figures, full biblio info adde
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