296 research outputs found
Ramanujan sums for signal processing of low frequency noise
An aperiodic (low frequency) spectrum may originate from the error term in
the mean value of an arithmetical function such as M\"obius function or
Mangoldt function, which are coding sequences for prime numbers. In the
discrete Fourier transform the analyzing wave is periodic and not well suited
to represent the low frequency regime. In place we introduce a new signal
processing tool based on the Ramanujan sums c_q(n), well adapted to the
analysis of arithmetical sequences with many resonances p/q. The sums are
quasi-periodic versus the time n of the resonance and aperiodic versus the
order q of the resonance. New results arise from the use of this
Ramanujan-Fourier transform (RFT) in the context of arithmetical and
experimental signalsComment: 11 pages in IOP style, 14 figures, 2 tables, 16 reference
Pauli graphs when the Hilbert space dimension contains a square: why the Dedekind psi function ?
We study the commutation relations within the Pauli groups built on all
decompositions of a given Hilbert space dimension , containing a square,
into its factors. Illustrative low dimensional examples are the quartit ()
and two-qubit () systems, the octit (), qubit/quartit () and three-qubit () systems, and so on. In the single qudit case,
e.g. , one defines a bijection between the maximal
commuting sets [with the sum of divisors of ] of Pauli
observables and the maximal submodules of the modular ring ,
that arrange into the projective line and a independent set
of size [with the Dedekind psi function]. In the
multiple qudit case, e.g. , the Pauli graphs rely on
symplectic polar spaces such as the generalized quadrangles GQ(2,2) (if
) and GQ(3,3) (if ). More precisely, in dimension ( a
prime) of the Hilbert space, the observables of the Pauli group (modulo the
center) are seen as the elements of the -dimensional vector space over the
field . In this space, one makes use of the commutator to define
a symplectic polar space of cardinality , that
encodes the maximal commuting sets of the Pauli group by its totally isotropic
subspaces. Building blocks of are punctured polar spaces (i.e. a
observable and all maximum cliques passing to it are removed) of size given by
the Dedekind psi function . For multiple qudit mixtures (e.g.
qubit/quartit, qubit/octit and so on), one finds multiple copies of polar
spaces, ponctured polar spaces, hypercube geometries and other intricate
structures. Such structures play a role in the science of quantum information.Comment: 18 pages, version submiited to J. Phys. A: Math. Theo
A SU(2) recipe for mutually unbiased bases
A simple recipe for generating a complete set of mutually unbiased bases in
dimension (2j+1)**e, with 2j + 1 prime and e positive integer, is developed
from a single matrix acting on a space of constant angular momentum j and
defined in terms of the irreducible characters of the cyclic group C(2j+1). As
two pending results, this matrix is used in the derivation of a polar
decomposition of SU(2) and of a FFZ algebra.Comment: v2: abstract enlarged, a corollary added, acknowledgments added, one
reference added, presentation improved; v3: two misprints correcte
Qudits of composite dimension, mutually unbiased bases and projective ring geometry
The Pauli operators attached to a composite qudit in dimension may
be mapped to the vectors of the symplectic module
( the modular ring). As a result, perpendicular vectors
correspond to commuting operators, a free cyclic submodule to a maximal
commuting set, and disjoint such sets to mutually unbiased bases. For
dimensions , and 18, the fine structure and the incidence
between maximal commuting sets is found to reproduce the projective line over
the rings , , ,
and ,
respectively.Comment: 10 pages (Fast Track communication). Journal of Physics A
Mathematical and Theoretical (2008) accepte
Mutually unbiased phase states, phase uncertainties, and Gauss sums
Mutually unbiased bases (MUBs), which are such that the inner product between
two vectors in different orthogonal bases is a constant equal to 1/sqrt{d),
with d the dimension of the finite Hilbert space, are becoming more and more
studied for applications such as quantum tomography and cryptography, and in
relation to entangled states and to the Heisenberg-Weil group of quantum
optics. Complete sets of MUBs of cardinality d+1 have been derived for prime
power dimensions d=p^m using the tools of abstract algebra. Presumably, for non
prime dimensions the cardinality is much less. Here we reinterpret MUBs as
quantum phase states, i.e. as eigenvectors of Hermitean phase operators
generalizing those introduced by Pegg & Barnett in 1989. We relate MUB states
to additive characters of Galois fields (in odd characteristic p) and to Galois
rings (in characteristic 2). Quantum Fourier transforms of the components in
vectors of the bases define a more general class of MUBs with multiplicative
characters and additive ones altogether. We investigate the complementary
properties of the above phase operator with respect to the number operator. We
also study the phase probability distribution and variance for general pure
quantum electromagnetic states and find them to be related to the Gauss sums,
which are sums over all elements of the field (or of the ring) of the product
of multiplicative and additive characters. Finally, we relate the concepts of
mutual unbiasedness and maximal entanglement. This allows to use well studied
algebraic concepts as efficient tools in the study of entanglement and its
information aspectsComment: 16 pages, a few typos corrected, some references updated, note
acknowledging I. Shparlinski adde
Projective Ring Line of an Arbitrary Single Qudit
As a continuation of our previous work (arXiv:0708.4333) an algebraic
geometrical study of a single -dimensional qudit is made, with being
{\it any} positive integer. The study is based on an intricate relation between
the symplectic module of the generalized Pauli group of the qudit and the fine
structure of the projective line over the (modular) ring \bZ_{d}. Explicit
formulae are given for both the number of generalized Pauli operators commuting
with a given one and the number of points of the projective line containing the
corresponding vector of \bZ^{2}_{d}. We find, remarkably, that a perp-set is
not a set-theoretic union of the corresponding points of the associated
projective line unless is a product of distinct primes. The operators are
also seen to be structured into disjoint `layers' according to the degree of
their representing vectors. A brief comparison with some multiple-qudit cases
is made
MUBs: From finite projective geometry to quantum phase enciphering
This short note highlights the most prominent mathematical problems and
physical questions associated with the existence of the maximum sets of
mutually unbiased bases (MUBs) in the Hilbert space of a given dimensionComment: 5 pages, accepted for AIP Conf Book, QCMC 2004, Strathclyde, Glasgow,
minor correction
Pauli graphs, Riemann hypothesis, Goldbach pairs
Let consider the Pauli group with unitary quantum
generators (shift) and (clock) acting on the vectors of the
-dimensional Hilbert space via and , with
. It has been found that the number of maximal mutually
commuting sets within is controlled by the Dedekind psi
function (with a prime)
\cite{Planat2011} and that there exists a specific inequality , involving the Euler constant , that is only satisfied at specific low dimensions . The set is closely related to
the set of integers that are totally Goldbach, i.e.
that consist of all primes ) is equivalent to Riemann hypothesis.
Introducing the Hardy-Littlewood function (with the twin prime constant),
that is used for estimating the number of
Goldbach pairs, one shows that the new inequality is also equivalent to Riemann hypothesis. In this paper,
these number theoretical properties are discusssed in the context of the qudit
commutation structure.Comment: 11 page
Multi-Line Geometry of Qubit-Qutrit and Higher-Order Pauli Operators
The commutation relations of the generalized Pauli operators of a
qubit-qutrit system are discussed in the newly established graph-theoretic and
finite-geometrical settings. The dual of the Pauli graph of this system is
found to be isomorphic to the projective line over the product ring Z2xZ3. A
"peculiar" feature in comparison with two-qubits is that two distinct
points/operators can be joined by more than one line. The multi-line property
is shown to be also present in the graphs/geometries characterizing two-qutrit
and three-qubit Pauli operators' space and surmised to be exhibited by any
other higher-level quantum system.Comment: 8 pages, 6 figures. International Journal of Theoretical Physics
(2007) accept\'
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