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 aspects

We develop a new approach of the quantum phase in an Hilbert space of finite dimension which is based on the relation between the physical concept of phase locking and mathematical concepts such as cyclotomy and the Ramanujan sums. As a result phase variability looks quite similar to its classical counterpart, having peaks at dimensions equal to a power of a prime number. Squeezing of that noise is allowed for specific quantum states. The concept of phase entanglement for pairs of phase-locked states is introduced.

We develop a new approach of the quantum phase in an Hilbert space of finite dimension which is based on the relation between the physical concept of phase locking and mathematical concepts such as cyclotomy and the Ramanujan sums. As a result, phase variability looks quite similar to its classical counterpart, having peaks at dimensions equal to a power of a prime number. Squeezing of the phase noise is allowed for specific quantum states. The concept of phase entanglement for Kloosterman pairs of phase-locked states is introduced.

It is conjectured that the question of the existence of projective planes whose order is not a power of prime is intimately linked with the problem whether there exists a set of d+1 mutually unbiased bases in a d-dimensional Hilbert space if d differs from a power of prime.

I list from the literature some Schroedinger hamiltonians related to prime numbers adding a few stimulating comments on each case. Introduction The problem of the nontrivial zeros of Riemann’s zeta function, i.e., whether they all lie on the line z = 1 2 + it in the complex plane or not, is a famous unsolved mathematical problem in which physics, especially quantum mechanics and chaos theory, could have a substantial and rewarding contribution. There is a lot of online information. My goal here is to provide a short survey of the hamiltonians that so far have been proposed to give hints for a spectral (Hilbert-Polya) solution of the location of the zeros on the critical line. Bhaduri, Khare, Low (1995) [1] BKL showed that the density of zeros of Riemann’s ζ function is determined by its phase exp[2iθ(t)] = exp(−it lnπ) Γ (

Laboratoire de Physique et M´etrologie des Oscillateurs du CNRS, 25044 Besanc¸onCedex, FranceE-mail: planat@lpmo.eduDated: August 2003; File: RP.texProc. ICSSUR-8, Puebla, Mexico, 9-13 June 2003Eds. H. Moya-Cessa, R. J´auregui, S. Hacyan, O. Castan˜osRinton Press, ISBN 1-58949-040-1, pp. 366-372 (Nov. 2003)

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 entanglement.

If stationary, the spectrum of vacuum field noise (VFN) is an important ingredient to get information about the curvature invariants of classical worldlines (relativistic classical trajectories). For scalar quantum field vacua there are six stationary cases as shown by Letaw some time ago, these are reviewed here. However, the non-stationary vacuum noises are not out of reach and can be processed by a few mathematical methods which I briefly comment on. Since the information about the kinematical curvature invariants of the worldlines is of radiometric origin, hints are given on a more useful application to radiation and beam radiometric standards at relativistic energies

The connection of unbroken SUSY quantum mechanics in its strictly isospectral form with the nonlinear Riccati superposition principle is pointed out

Short remarks on the problem of assigning frequency spectra to Casimir, sonoluminescence, Hawking, Unruh, and quantum optical squeezing effects are presented.

The Wheeler-DeWitt equation for empty FRW minisuperspace universes of Hartle-Hawking factor ordering parameter Q=0 is mapped onto the dynamics of a unit mass classical oscillator. The latter is studied by the classical Ermakov invariant method. Angle quantities are presented in the same context

For the one-dimensional Helmholtz equation we write the corresponding time-dependent Helmholtz Hamiltonian in order to study it as an Ermakov problem and derive geometrical angles and phases in this context.

This is a Physics World features paper on Chen-Tajima proposal to detect Unruh radiation by means of high-intensity lasers (PRL 83, 256 (1999)).

We present the supersymmetric Witten and double Darboux (strictly isospectral) constructions as applied to the diffusion of thermal neutrons from an infinitely long line source. While the Witten construction is just a mathematical scheme, the double Darboux method introduces a one-parameter family of diffusion solutions which are strictly isospectral to the stationary solution. They correspond to a Darboux-transformed diffusion length which is flux dependent

If at least some Wheeler-DeWitt solutions can be interpreted as zero-energy resonances, then the total s-wave cross-section of the corresponding quantum universes is infinite.

We apply the strictly isospectral technique of standard supersymmetric quantum mechanics to the Q=0 factor ordered Wheeler-DeWitt equation for the Friedmann-Robertson-Walker (FRW) minisuperspace model. The resulting strictly isospectral one-parameter families of both FRW cosmological potentials and "wavefunctions of the universe" are exhibited with relevant plots

The connection between the strictly isospectral construction in supersymmetric quantum mechanics and the general zero-mode solutions of the Schroedinger equation is explained by introducing slightly generalized first-order intertwining operators. We also present a multiple-parameter generalization of the strictly isospectral construction in the same perspective.