568 research outputs found
Probabilistic lower bounds on maximal determinants of binary matrices
Let be the maximal determinant for -matrices, and be the ratio of
to the Hadamard upper bound. Using the probabilistic method,
we prove new lower bounds on and in terms of
, where is the order of a Hadamard matrix and is maximal
subject to . For example, if , and if . By a recent result of Livinskyi, as ,
so the second bound is close to for large . Previous
lower bounds tended to zero as with fixed, except in the
cases . For , our bounds are better for all
sufficiently large . If the Hadamard conjecture is true, then , so
the first bound above shows that is bounded below by a positive
constant .Comment: 17 pages, 2 tables, 24 references. Shorter version of
arXiv:1402.6817v4. Typos corrected in v2 and v3, new Lemma 7 in v4, updated
references in v5, added Remark 2.8 and a reference in v6, updated references
in v
The Hadamard Condition for Dirac Fields and Adiabatic States on Robertson-Walker Spacetimes
We characterise the homogeneous and isotropic gauge invariant and quasifree
states for free Dirac quantum fields on Robertson-Walker spacetimes in any even
dimension. Using this characterisation, we construct adiabatic vacuum states of
order corresponding to some Cauchy surface. We then show that any two such
states (of sufficiently high order) are locally quasi-equivalent. We propose a
microlocal version of the Hadamard condition for spinor fields on arbitrary
spacetimes, which is shown to entail the usual short distance behaviour of the
twopoint function. The polarisation set of these twopoint functions is
determined from the Dencker connection of the spinorial Klein-Gordon operator
which we show to equal the (pull-back) of the spin connection. Finally it is
demonstrated that adiabatic states of infinite order are Hadamard, and that
those of order correspond, in some sense, to a truncated Hadamard series
and will therefore allow for a point splitting renormalisation of the expected
stress-energy tensor.Comment: 29 pages, Latex, no figures. v2: corrections in the proof of Thm.
IV.1. v3: published versio
Real eigenvalues of non-Gaussian random matrices and their products
We study the properties of the eigenvalues of real random matrices and their
products. It is known that when the matrix elements are Gaussian-distributed
independent random variables, the fraction of real eigenvalues tends to unity
as the number of matrices in the product increases. Here we present numerical
evidence that this phenomenon is robust with respect to the probability
distribution of matrix elements, and is therefore a general property that
merits detailed investigation. Since the elements of the product matrix are no
longer distributed as those of the single matrix nor they remain independent
random variables, we study the role of these two factors in detail. We study
numerically the properties of the Hadamard (or Schur) product of matrices and
also the product of matrices whose entries are independent but have the same
marginal distribution as that of normal products of matrices, and find that
under repeated multiplication, the probability of all eigenvalues to be real
increases in both cases, but saturates to a constant below unity showing that
the correlations amongst the matrix elements are responsible for the approach
to one. To investigate the role of the non-normal nature of the probability
distributions, we present a thorough analytical treatment of the
single matrix for several standard distributions. Within the class of smooth
distributions with zero mean and finite variance, our results indicate that the
Gaussian distribution has the maximum probability of real eigenvalues, but the
Cauchy distribution characterised by infinite variance is found to have a
larger probability of real eigenvalues than the normal. We also find that for
the two-dimensional single matrices, the probability of real eigenvalues lies
in the range [5/8,7/8].Comment: To appear in J. Phys. A: Math, Theo
Suppression laws for multi-particle interference in Sylvester interferometers
Quantum interference of correlated particles is a fundamental quantum
phenomenon which carries signatures of the statistics properties of the
particles, such as bunching or anti-bunching. In presence of particular
symmetries, interference effects take place with high visibility, one of the
simplest cases being the suppression of coincident detection in the
Hong-Ou-Mandel effect. Tichy et al. recently demonstrated a simple sufficient
criterion for the suppression of output events in the more general case of
Fourier multi-port beam splitters. Here we study the case in which
particles (either bosonic or fermionic) are injected simultaneously in
different ports of a Sylvester interferometer with modes. In
particular, we prove a necessary and sufficient criterion for a significant
fraction of output states to be suppressed, for specific input configurations.
This may find application in assessing the indistinguishability of multiple
single photon sources and in the validation of boson sampling machines
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