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 $\mathcal{G}$ of a subgroup $H$ of the two-generator free group $G=\left\langle x,y\right\rangle$. One defines geometric contextuality from the discrepancy between the commutativity of cosets on $\mathcal{G}$ and that of quantum observables.It is shown in this paper that Kleinian subgroups $K=\left\langle f,g\right\rangle$ that are non-compact, arithmetic, and generated by two elliptic isometries $f$ and $g$ (the Martin-Maclachlan classification), are appropriate contextuality filters. Standard contextual geometries such as some thin generalized polygons (starting with Mermin's $3 \times 3$ grid) belong to this frame. The Bianchi groups $PSL(2,O\_d)$, $d \in \{1,3\}$ defined over the imaginary quadratic field $O\_d=\mathbb{Q}(\sqrt{-d})$ 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

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 C$\pm$n . 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

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

Entangling gates in even Euclidean lattices such as the Leech lattice

The group of automorphisms of Euclidean (embedded in $\mathbb{R}^n$) dense lattices such as the root lattices $D_4$ and $E_8$, the Barnes-Wall lattice $BW_{16}$, the unimodular lattice $D_{12}^+$ and the Leech lattice $\Lambda_{24}$ 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 $D_4$ and $BW_{16}$, respectively, and the three-qubit real Clifford group is maximal in the Weyl group $W(E_8)$. Technically, the automorphism group $Aut(\Lambda)$ of the lattice $\Lambda$ is the set of orthogonal matrices $B$ such that, following the conjugation action by the generating matrix of the lattice, the output matrix is unimodular (of determinant $\pm 1$, with integer entries). When the degree $n$ is equal to the number of basis elements of $\Lambda$, then $Aut(\Lambda)$ also acts on basis vectors and is generated with matrices $B$ such that the sum of squared entries in a row is one, i.e. $B$ 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 $E_8$ (the two- and three- tangles have equal magnitude 1/4) and a GHZ type entanglement in BW$_{16}$. In this paper, we also investigate the entangled gates from $D_{12}^+$ and $\Lambda_{24}$, 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

Two-letter words and a fundamental homomorphism ruling geometric contextuality

It has recently been recognized by the author that the quantum contextuality paradigm may be formulated in terms of the properties of some subgroups of the two-letter free group $G$ and their corresponding point-line incidence geometry $\mathcal{G}$. I introduce a fundamental homomorphism $f$ mapping the (infinitely many) words of G to the permutations ruling the symmetries of $\mathcal{G}$. The substructure of $f$ is revealing the essence of geometric contextuality in a straightforward way.Comment: 18 pages, 11 figures, 2 tables to appear in "Symmetry: Culture and Science