977 research outputs found
Cyclotomy and Ramanujan sums in quantum phase locking
Phase-locking governs the phase noise in classical clocks through effects
described in precise mathematical terms. We seek here a quantum counterpart of
these effects by working in a finite Hilbert space. We use a coprimality
condition to define phase-locked quantum states and the corresponding
Pegg-Barnett type phase operator. Cyclotomic symmetries in matrix elements are
revealed and related to Ramanujan sums in the theory of prime numbers. The
employed mathematical procedures also emphasize the isomorphism between
algebraic number theory and the theory of quantum entanglementComment: 6 pages, 3 figures, version accepted at Phys. Lett.
On the Cyclotomic Quantum Algebra of Time Perception
I develop the idea that time perception is the quantum counterpart to time
measurement. Phase-locking and prime number theory were proposed as the
unifying concepts for understanding the optimal synchronization of clocks and
their 1/f frequency noise. Time perception is shown to depend on the
thermodynamics of a quantum algebra of number and phase operators already
proposed for quantum computational tasks, and to evolve according to a
Hamiltonian mimicking Fechner's law. The mathematics is Bost and Connes quantum
model for prime numbers. The picture that emerges is a unique perception state
above a critical temperature and plenty of them allowed below, which are
parametrized by the symmetry group for the primitive roots of unity. Squeezing
of phase fluctuations close to the phase transition temperature may play a role
in memory encoding and conscious activity
Geometric contextuality from the Maclachlan-Martin Kleinian groups
There are contextual sets of multiple qubits whose commutation is
parametrized thanks to the coset geometry of a subgroup of
the two-generator free group . One defines
geometric contextuality from the discrepancy between the commutativity of
cosets on and that of quantum observables.It is shown in this
paper that Kleinian subgroups that are
non-compact, arithmetic, and generated by two elliptic isometries and
(the Martin-Maclachlan classification), are appropriate contextuality filters.
Standard contextual geometries such as some thin generalized polygons (starting
with Mermin's grid) belong to this frame. The Bianchi groups
, defined over the imaginary quadratic field
play a special role
Invitation to the "Spooky" Quantum Phase-Locking Effect and its Link to 1/F Fluctuations
An overview of the concept of phase-locking at the non linear, geometric and
quantum level is attempted, in relation to finite resolution measurements in a
communication receiver and its 1/f noise. Sine functions, automorphic functions
and cyclotomic arithmetic are respectively used as the relevant trigonometric
tools. The common point of the three topics is found to be the Mangoldt
function of prime number theory as the generator of low frequency noise in the
coupling coefficient, the scattering coefficient and in quantum critical
statistical states. Huyghens coupled pendulums, the Adler equation, the Arnold
map, continued fraction expansions, discrete Mobius transformations, Ford
circles, coherent and squeezed phase states, Ramanujan sums, the Riemann zeta
function and Bost and Connes KMS states are some but a few concepts which are
used synchronously in the paper.Comment: submitted to the journal: Fluctuation and Noise Letters, March 13,
200
A moonshine dialogue in mathematical physics
Phys and Math are two colleagues at the University of Sa{\c c}enbon (Crefan
Kingdom), dialoguing about the remarkable efficiency of mathematics for
physics. They talk about the notches on the Ishango bone, the various uses of
psi in maths and physics, they arrive at dessins d'enfants, moonshine concepts,
Rademacher sums and their significance in the quantum world. You should not
miss their eccentric proposal of relating Bell's theorem to the Baby Monster
group. Their hyperbolic polygons show a considerable singularity/cusp structure
that our modern age of computers is able to capture. Henri Poincar{\'e} would
have been happy to see it.Comment: new version expanded for publication in Mathematics (MDPI), special
issue "Mathematical physics" initial: Trick or Truth: the Mysterious
Connection Between Physics and Mathematics, FQXi essay contest - Spring, 201
Clifford group dipoles and the enactment of Weyl/Coxeter group W(E8) by entangling gates
Peres/Mermin arguments about no-hidden variables in quantum mechanics are
used for displaying a pair (R, S) of entangling Clifford quantum gates, acting
on two qubits. From them, a natural unitary representation of Coxeter/Weyl
groups W(D5) and W(F4) emerges, which is also reflected into the splitting of
the n-qubit Clifford group Cn into dipoles Cn . The union of the
three-qubit real Clifford group C+ 3 and the Toffoli gate ensures a orthogonal
representation of the Weyl/Coxeter group W(E8), and of its relatives. Other
concepts involved are complex reflection groups, BN pairs, unitary group
designs and entangled states of the GHZ family.Comment: version revised according the recommendations of a refere
Entangling gates in even Euclidean lattices such as the Leech lattice
The group of automorphisms of Euclidean (embedded in ) dense
lattices such as the root lattices and , the Barnes-Wall lattice
, the unimodular lattice and the Leech lattice
may be generated by entangled quantum gates of the corresponding
dimension. These (real) gates/lattices are useful for quantum error correction:
for instance, the two and four-qubit real Clifford groups are the automorphism
groups of the lattices and , respectively, and the three-qubit
real Clifford group is maximal in the Weyl group . Technically, the
automorphism group of the lattice is the set of
orthogonal matrices such that, following the conjugation action by the
generating matrix of the lattice, the output matrix is unimodular (of
determinant , with integer entries). When the degree is equal to the
number of basis elements of , then also acts on basis
vectors and is generated with matrices such that the sum of squared entries
in a row is one, i.e. may be seen as a quantum gate. For the dense lattices
listed above, maximal multipartite entanglement arises. In particular, one
finds a balanced tripartite entanglement in (the two- and three- tangles
have equal magnitude 1/4) and a GHZ type entanglement in BW. In this
paper, we also investigate the entangled gates from and
, by seeing them as systems coupling a qutrit to two- and
three-qubits, respectively. Apart from quantum computing, the work may be
related to particle physics in the spirit of \cite{PLS2010}.Comment: 11 pages, second updated versio
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