Table of contents

The International Conference on 'Quantum Control, Exact or Perturbative, Linear or Nonlinear', took place in Mexico City on 22-24 October 2014. It was held with the aim of celebrating the first fifty years of scientific career of Bogdan Mielnik, an outstanding scientist whose professional trajectory spans over Poland and Mexico and who is currently Professor Emeritus in the Physics Department of Centro de Investigación y de Estudios Avanzados del IPN (Cinvestav) in Mexico.

Bogdan Mielnik was born on May 6th, 1936 in Warsaw, Poland. He studied elementary and high school until 1953. In the autumn of 1953 he started the studies in the Faculty of Mathematics and Physics at the University of Warsaw, and at the end of 1957 he did his master work under the direction of Professor Jerzy Plebański. In 1962 he was invited to the newly opened Research Center of IPN (Cinvestav), in Mexico, as an assistant and PhD student of Jerzy Plebański. On October 22nd, 1964, he submitted to Cinvestav his PhD Thesis entitled ''Analytic functions of the displacement operator'', marking the offcial beginning of his scientific career. It is worth mentioning that Bogdan Mielnik is the first PhD graduate of the Physics Department of Cinvestav, so with this Conference our Department was also celebrating an important date on its calendar. A more detailed information can be found in the website http://www.fis.cinvestav.mx/mielnik50/.

It was our great pleasure to see that many collaborators and former students of Bogdan Mielnik attended this Conference. The articles collected in this volume are the written contributions of the majority of talks presented at the conference. They have been organized according to the research subjects that Bogdan Mielnik has been involved in. Thus, the articles of JG Hirsch, L Hughston, G Morales-Luna, O Rosas-Ortiz and G Torres-Vega deal with Fundamental Problems in Quantum Mechanics. On the other hand, the papers by F Delgado, H Hernández-Coronado, G Herrera Corral, F Rojas, KB Wolf and M Znojil belong to the subject of Quantum Control and Dynamical Manipulation, while the articles of D Bermudez, A Contreras-Astorga, E Díaz-Bautista, JC González, V Hussin and VS Morales-Salgado are related with Factorization Method, Supersymmetric Quantum Mechanics and Coherent States. Finally, the papers of S Cruz y Cruz, M Enríquez, A Jaimes-Nájera and R Kerner address some Interdisciplinary Problems in Quantum Mechanics.

We would like to conclude by thanking for the support of the Physics Department of Cinvestav, the Academic Affairs Offce of Cinvestav, and the Mexican National Council of Science and Technology (Conacyt, projects 152574 and 166581). Without their support, neither the Conference would have been held nor this Conference Volume would have been published.

The organizers of the conference would like to thank for the support from the Physics Department and the Offce of Academic Affairs of the Centro de Investigación y de Estudios Avanzados and from the Consejo Nacional de Ciencia y Tecnología (Conacyt), projects 152574 and 166581.

All papers published in this volume of Journal of Physics: Conference Series have been peer reviewed through processes administered by the proceedings Editors. Reviews were conducted by expert referees to the professional and scientific standards expected of a proceedings journal published by IOP Publishing.

Randomness plays a central role in the quantum mechanical description of our interactions. We review the relationship between the violation of Bell inequalities, non signaling and randomness. We discuss the challenge in defining a random string, and show that algorithmic information theory provides a necessary condition for randomness using Borel normality. We close with a view on incomputablity and its implications in physics.

We introduce a family of operations in quantum mechanics that one can regard as "universal quantum measurements" (UQMs). These measurements are applicable to all finite dimensional quantum systems and entail the specification of only a minimal amount of structure. The first class of UQM that we consider involves the specification of the initial state of the system—no further structure is brought into play. We call operations of this type "tomographic measurements", since given the statistics of the outcomes one can deduce the original state of the system. Next, we construct a disentangling operation, the outcome of which, when the procedure is applied to a general mixed state of an entangled composite system, is a disentangled product of pure constituent states. This operation exists whenever the dimension of the Hilbert space is not a prime, and can be used to model the decay of a composite system. As another example, we show how one can make a measurement of the direction along which the spin of a particle of spin s is oriented (s = 1/2, 1,...). The required additional structure in this case involves the embedding of CP¹ as a rational curve of degree 2s in CP^2s.

We recall several cryptographic protocols based on entanglement alone and also on entanglement swapping. We make an exposition in terms of the geometrical aspects of the involved Hilbert spaces, and we concentrate on the formal nature of the used transformations.

The time-evolution of the quantum correlations between two qubits that are coupled to a pair of photon baths is studied. We show that conditioned transitions occurring in the entire system have influence on the time-evolution of the subsystems. Then, we show that the study of the population inversion of each of the qubits is a measure of the correlations between them that is in agreement with the notion of concurrence.

The usual method for obtaining the eigenstates of an operator is to solve the corresponding eigenvalue equation. This procedure cannot be applied when the operator of interest is not known at all. We develop a method which generates the eigenstates of an operator, and the operator itself, which will be conjugate to a given known operator. This is particularly useful for the case of the time operator in Quantum Mechanics. We illustrate the method by obtaining time eigenstates for the free particle.

Teleportation is possibly the most representative quantum algorithm in the public domain. Nevertheless, though this quantum procedure transmits only information and not objects, its coverage is still very limited and easily subject to errors. Based on a fine control of quantum resources, particularly those entangled, the research to extend its coverage and flexibility is open, in particular on matter based quantum systems. This work shows how anisotropic Ising interactions could be used as a natural basis for this procedure, based on a sequence of magnetic pulses driving Ising interaction, stating results in specialized quantum gates designed for magnetic systems.

In the separation of rotations from internal motions in the n-body problem, there appear some gauge fields which physically represent Coriolis effects. These fields are also present in the "falling cat" problem: at the kinematical level they map changes in the cat's shape to changes in its orientation whereas at the dynamical level they show up as gauge potentials in the Hamiltonian. Classically, the vanishing angular momentum condition allows for the orientation degrees of freedom to decouple from the internal ones and the cat's re-orientation can be accounted for at the kinematical level, partially. In the quantum case the cat's reorientation description requires to be done on dynamical grounds. In this paper we explore the quantum version of the falling cat modelled as a three body problem.

We report on the status of the construction and installation of a new detector in the forward region of the ALICE experiment at the LHC. This detector will allow the study of processes with gaps at larger rapidity than those presently covered. A setup of two stations called AD (stands for ALICE Diffractive) on the right and on the left of the interaction point enhances significantly the efficiency to study diffractive physics and photon induced processes.

We present a physical example model of how Quantum Control with genetic algorithms is applied to implement the quantum superdense code protocol. We studied a model consisting of two quantum dots with an electron with spin, including spin-orbit interaction. The electron and the spin get hybridized with the site acquiring two degrees of freedom, spin and charge. The system has tunneling and site energies as time dependent control parameters that are optimized by means of genetic algorithms to prepare a hybrid Bell-like state used as a transmission channel. This state is transformed to obtain any state of the four Bell basis as required by superdense protocol to transmit two bits of classical information. The control process protocol is equivalent to implement one of the quantum gates in the charge subsystem. Fidelities larger than 99.5% are achieved for the hybrid entangled state preparation and the superdense operations.

The problem of controlling quantum wavefunctions by means of potential jolts and periods of free evolution was broached by Bogdan Mielnik in 1977. This quantum control became a subject of great interest for the preparation of atomic and particle systems. We point out here that these manipulations are also realized in paraxial geometric and wave optics, with lenses and free spaces, and even more transparently than with matter.

In a way inspired by the brief 2002 note "The challenge of nonhermitian structures in physics" by Ramirez and Mielnik (with the text most easily available via arXiv:quant- ph/0211048) the situation in the theory is briefly summarized here as it looks twelve years later. Our text has three parts. In the first one we briefly mention the pre-history (dating back to the Freeman Dyson's proposal of the non-Hermitian-Hamiltonian method in 1956 and to its subsequent successful "interacting boson model" applications in nuclear physics) and, first of all, the amazing recent progress reached, in the stationary case, using, in essence, an inversion of the Dyson's approach. The impact on the latter idea upon abstract quantum physics is sampled, first of all, by the reference to papers by Bender et al. (who made the non-Hermitian model-building popular under the nickname of parity-times-time-reflection- symmetric alias PT-symmetric quantum mechanics) and by Mostafazadeh (who reinterpreted PT-symmetry as P-pseudo-Hermiticity). In the second part of our review the emphasis is shifted to the newest, non-stationary upgrade of the formalism which we proposed in the year 2009 and which is characterized by the simultaneous participation of a triplet of Hilbert spaces H in the representation of a single quantum system. In the third part of the review we finally emphasize that the majority of applications of our three-Hilbert-space (THS) recipe is still ahead of us because the enhancement of the flexibility is necessarily accompanied by an enhancement of the technical difficulties. An escape out of the technical trap is proposed to be sought in a restriction of attention to quantum models living in finite-dimensional Hilbert spaces H. As long as the use of such spaces is so typical for the quantum-control considerations, we conclude with conjecture that the THS formalism should start searching for implementations in the field of quantum control.

In this article we introduce the relation between supersymmetric quantum mechanics (SUSY QM) and a second-order non-linear differential equation known as Painleve V (PV) equation. To that end, we will first make a swift examination on the SUSY QM treatment of the radial oscillator and we will revisit its relation with the polynomial Heisenberg algebras (PHA). After that, we will formulate a theorem that connects SUSY QM to a set of solutions of the PV equation through specific PHA.

Two different exactly solvable systems are constructed using the supersymmetric quantum mechanics formalism and a pseudoscalar one-dimensional version of the Dirac- Moshinsky oscillator as a departing system. One system is built using a first-order SUSY transformation. The second is obtained through the confluent supersymmetry algorithm. The two of them are explicitly designed to have the same spectrum as the departing system and pseudoscalar potentials.

The nonlinear supercoherent states, associated with a nonlinear generalization of the Kornbluth-Zypman (KZ) supersymmetric annihilation operator (SAO) of the supersymmetric harmonic oscillator, will be studied. We discuss as well the Heisenberg uncertainty relation σ²_xσ²_p for a special case which will allow us to compare our results with those obtained for the KZ linear supercoherent states.

Supersymmetry transformations will be used to obtain new exactly solvable potentials from the complex oscillator. The corresponding Hamiltonians are ruled by polynomial Heisenberg algebras. By applying a mechanism to reduce to second the order of these algebras, the connection with the Painlevé IV equation is achieved, supplying us with an algorithm to generate solutions to such equation.

Gaussian Klauder coherent states are discussed in the context of the infinite well quantum model, otherwise known as the particle in a box. A supersymmetric partner system is also presented, as well as a construction of coherent states in this new system. We show that these states can be chosen, in both systems to have many properties usually expected for coherent states. In particular, they yield highly localised wave packets for a short period of time, which evolve in a quasi-classical manner and which saturate approximately Heisenberg uncertainty relation. These studies are elaborated in one- and two-dimensional contexts. Finally, some relations are established between the Gaussian states being mostly used here and the generalised coherent states, which are more standardly found in the literature.

Quantum systems described by second and third order polynomial Heisenberg algebras are obtained applying supersymmetric quantum mechanics to the harmonic oscillator with an infinite potential barrier. These systems are linked with the Painlevé IV and V equations, respectively, thus several solutions of these non-linear second-order differential equations will be found, along with a chain of Bäcklund transformations connecting such solutions.

Light propagation processes in a homogeneous dielectric slab are considered from the point of view of the optical analog of the Schrödinger equation. The reflection and transmission coefficients and longitudinal shifts of the wavefront are determined by considering the analytical properties of the external field amplitudes as functions of the wave number. An analytic approach to obtain the resonances in the "long lifetime" limit is also presented.

The properties of the Kronecker product of 2 x 2 matrices are reviewed in terms of Hubbard operators. This framework constitutes a shorthand notation to deal with the tensor algebra of operators acting on multiqubit states. As an application we derive some analytical expressions related to the geometric measure of entanglement of a certain class of multiqubit invariant permutation states. Our results can be straightaway extended for systems of larger dimensions.

The probability of finding a particle in the interaction zone of a scattering process is analyzed in one dimension. The quantum state of the particle is described by a Gaussian wave packet that is narrow in the momentum distribution. We find the intervals of time that are required in order to get a maximum probability in the interaction zone as a lower bound of the duration of the scattering process. As an immediate application, times involved in the scattering associated with symmetric potentials (as the rectangular barriers and wells) are calculated.

A concise study of ternary and cubic algebras with Z₃ grading is presented. We discuss some underlying ideas leading to the conclusion that the discrete symmetry group of permutations of three objects, S₃, and its abelian subgroup Z₃ may play an important role in quantum physics. We show then how most of important algebras with Z₂ grading can be generalized with ternary composition laws combined with a Z₃ grading.

We investigate in particular a ternary, Z₃-graded generalization of the Heisenberg algebra.

It turns out that introducing a non-trivial cubic root of unity, , one can define two types of creation operators instead of one, accompanying the usual annihilation operator. The two creation operators are non-hermitian, but they are mutually conjugate. Together, the three operators form a ternary algebra, and some of their cubic combinations generate the usual Heisenberg algebra.

An analogue of Hamiltonian operator is constructed by analogy with the usual harmonic oscillator, and some properties of its eigenfunctions are briefly discussed.