Volume 55, Issue 6, June 2014

A powerful tool for studying geometrical problems in Hilbert spaces is developed. We demonstrate the convergence and robustness of our method in every dimension by considering dynamical systems theory. This method provides numerical solutions to hard problems involving many coupled nonlinear equations in low and high dimensions (e.g., quantum tomography problem, existence and classification of Pauli partners, mutually unbiased bases, complex Hadamard matrices, equiangular tight frames, etc.). Additionally, this tool can be used to find analytical solutions and also to implicitly prove the existence of solutions. Here, we develop the theory for the quantum pure state tomography problem in finite dimensions but this approach is straightforwardly extended to the rest of the problems. We prove that solutions are always attractive fixed points of a nonlinear operator explicitly given. As an application, we show that the statistics collected from three random orthonormal bases is enough to reconstruct pure states from experimental (noisy) data in every dimension d ⩽ 32.
 ARTICLES

 Partial Differential Equations

Asymptotic solution of light transport problems in optically thick luminescent media
View Description Hide DescriptionWe study light transport in optically thick luminescent random media. Using radiative transport theory for luminescent media and applying asymptotic and computational methods, a corrected diffusion approximation is derived with the associated boundary conditions and boundary layer solution. The accuracy of this approach is verified for a planeparallel slab problem. In particular, the reduced system models accurately the effect of reabsorption. The impacts of varying the Stokes shift and using experimentally measured luminescence data are explored in detail. The results of this study have application to the design of luminescent solar concentrators, fluorescence medical imaging, and optical cooling using antiStokes fluorescence.

Multiplicity and concentration of solutions for a quasilinear Choquard equation
View Description Hide DescriptionIn this paper, we study a quasilinear Choquard equation involving the plaplacian operator and a potential function V. Under suitable assumptions on V and the nonlinearity, we prove the existence, multiplicity, and concentration of solutions for the equation by variational methods.
 Representation Theory and Algebraic Methods

The Euclidean algebra in rank 2 classical Lie algebras
View Description Hide DescriptionWe classify the embeddings of the Euclidean algebra into the rank 2 semisimple, classical Lie algebras. The classifications are up to inner automorphism. We also examine the finitedimensional, irreducible representations of the rank 2 semisimple, classical Lie algebras restricted to with respect to various embeddings. All Lie algebras and representations are over the complex numbers.

Braid representations from unitary braided vector spaces
View Description Hide DescriptionWe investigate braid group representations associated with unitary braided vector spaces, focusing on a conjecture that such representations should have virtually abelian images in general and finite image provided the braiding has finite order. We verify this conjecture for the two infinite families of Gaussian and grouptype braided vector spaces, as well as the generalization to quasibraided vector spaces of grouptype.

Four types of (super)conformal mechanics: Dmodule reps and invariant actions
View Description Hide Description(Super)conformal mechanics in one dimension is induced by parabolic or hyperbolic/trigonometric transformations, either homogeneous (for a scaling dimension λ) or inhomogeneous (at λ = 0, with ρ an inhomogeneity parameter). Four types of (super)conformal actions are thus obtained. With the exclusion of the homogeneous parabolic case, dimensional constants are present. Both the inhomogeneity and the insertion of λ generalize the construction of Papadopoulos [Class. Quant. Grav.30, 075018 (2013); eprint arXiv:1210.1719]. Inhomogeneous Dmodule reps are presented for the d = 1 superconformal algebras osp(12), sl(21), B(1, 1), and A(1, 1). For centerless superVirasoro algebras, Dmodule reps are presented (in the homogeneous case for ; in the inhomogeneous case for ). The four types of d = 1 superconformal actions are derived for systems. When , the homogeneously induced actions are D(2, 1; α)invariant (α is critically linked to λ); the inhomogeneously induced actions are A(1, 1)invariant.

Possible central extensions of nonrelativistic conformal algebras in 1+1
View Description Hide DescriptionWe investigate possibility of central extension for nonrelativistic conformal algebras in 1+1 dimension. Three different forms of charges can be suggested. A trivial charge for temporal part of the algebra exists for all elements of lGalilei algebra class. In attempt to find a central extension as of conformal Galilean algebra for other elements of the lGalilei class, possibility for such extension was excluded. For integer and half integer elements of the class, we can have an infinite extension of the generalized mass charge for the Virasorolike extended algebra. For finite algebras, a regular charge inspired by Schrödinger central extension is possible.
 ManyBody and Condensed Matter Physics

Manyparticle quantum graphs and BoseEinstein condensation
View Description Hide DescriptionIn this paper, we propose quantum graphs as onedimensional models with a complex topology to study BoseEinstein condensation and phase transitions in a rigorous way. We first investigate noninteracting manyparticle systems on quantum graphs and provide a complete classification of systems that exhibit BoseEinstein condensation. We then consider models of interacting particles that can be regarded as a generalisation of the wellknown TonksGirardeau gas. Here, our principal result is that no phase transitions occur in bosonic systems with repulsive hardcore interactions, indicating an absence of BoseEinstein condensation.
 Quantum Mechanics

Exactly solvable potentials with finitely many discrete eigenvalues of arbitrary choice
View Description Hide DescriptionWe address the problem of possible deformations of exactly solvable potentials having finitely many discrete eigenvalues of arbitrary choice. As Kay and Moses showed in 1956, reflectionless potentials in one dimensional quantum mechanics are exactly solvable. With an additional time dependence these potentials are identified as the soliton solutions of the Korteweg de Vries (KdV) hierarchy. An Nsoliton potential has the time t and 2N positive parameters, k 1 < ⋯ < k N and {c j }, j = 1, …, N, corresponding to N discrete eigenvalues . The eigenfunctions are elementary functions expressed by the ratio of determinants. The DarbouxCrumKreinAdler transformations or the AbrahamMoses transformations based on eigenfunction deletions produce lower soliton number potentials with modified parameters . We explore various identities satisfied by the eigenfunctions of the soliton potentials, which reflect the uniqueness theorem of Gel'fandLevitanMarchenko equations for separable (degenerate) kernels.

Homogenization limit for a multiband effective mass model in heterostructures
View Description Hide DescriptionWe study the homogenization limit of a multiband model that describes the quantum mechanical motion of an electron in a quasiperiodic crystal. In this approach, the distance among the atoms that constitute the material (lattice parameter) is considered a small quantity. Our model include the description of materials with variable chemical composition, intergrowth compounds, and heterostructures. We derive the effective multiband evolution system in the framework of the kp approach. We study the well posedness of the mathematical problem. We compare the effective mass model with the standard kp models for uniform and nonuniforms crystals. We show that in the limit of vanishing lattice parameter, the particle density obtained by the effective mass model, converges to the exact probability density of the particle.

Quantum tomography meets dynamical systems and bifurcations theory
View Description Hide DescriptionA powerful tool for studying geometrical problems in Hilbert spaces is developed. We demonstrate the convergence and robustness of our method in every dimension by considering dynamical systems theory. This method provides numerical solutions to hard problems involving many coupled nonlinear equations in low and high dimensions (e.g., quantum tomography problem, existence and classification of Pauli partners, mutually unbiased bases, complex Hadamard matrices, equiangular tight frames, etc.). Additionally, this tool can be used to find analytical solutions and also to implicitly prove the existence of solutions. Here, we develop the theory for the quantum pure state tomography problem in finite dimensions but this approach is straightforwardly extended to the rest of the problems. We prove that solutions are always attractive fixed points of a nonlinear operator explicitly given. As an application, we show that the statistics collected from three random orthonormal bases is enough to reconstruct pure states from experimental (noisy) data in every dimension d ⩽ 32.

Shorttime asymptotics of a rigorous path integral for N = 1 supersymmetric quantum mechanics on a Riemannian manifold
View Description Hide DescriptionFollowing Feynman's prescription for constructing a path integral representation of the propagator of a quantum theory, a shorttime approximation to the propagator for imaginarytime, N = 1 supersymmetric quantum mechanics on a compact, evendimensional Riemannian manifold is constructed. The path integral is interpreted as the limit of products, determined by a partition of a finite time interval, of this approximate propagator. The limit under refinements of the partition is shown to converge uniformly to the heat kernel for the Laplacede Rham operator on forms. A version of the steepest descent approximation to the path integral is obtained, and shown to give the expected shorttime behavior of the supertrace of the heat kernel.

Generalized space and linear momentum operators in quantum mechanics
View Description Hide DescriptionWe propose a modification of a recently introduced generalized translation operator, by including a qexponential factor, which implies in the definition of a Hermitian deformed linear momentum operator , and its canonically conjugate deformed position operator . A canonical transformation leads the Hamiltonian of a positiondependent mass particle to another Hamiltonian of a particle with constant mass in a conservative force field of a deformed phase space. The equation of motion for the classical phase space may be expressed in terms of the generalized dual qderivative. A positiondependent mass confined in an infinite square potential well is shown as an instance. Uncertainty and correspondence principles are analyzed.

Minimal time trajectories for twolevel quantum systems with two bounded controls
View Description Hide DescriptionIn this paper we consider the minimum time population transfer problem for a two level quantum system driven by two external fields with bounded amplitude. The controls are modeled as real functions and we do not use the Rotating Wave Approximation. After projection on the Bloch sphere, we treat the timeoptimal control problem with techniques of optimal synthesis on 2D manifolds. Based on the Pontryagin Maximum Principle, we characterize a restricted set of candidate optimal trajectories. Properties on this set, crucial for complete optimal synthesis, are illustrated by numerical simulations. Furthermore, when the two controls have the same bound and this bound is small with respect to the difference of the two energy levels, we get a complete optimal synthesis up to a small neighborhood of the antipodal point of the initial condition.
 Quantum Information and Computation

Inequalities for quantum marginal problems with continuous variables
View Description Hide DescriptionWe consider a mixed continuousvariable bosonic quantum system and present inequalities which must be satisfied between principal values of the covariances of a complete set of observables of the whole system and the principal values of the covariances of a complete set of observables of a subsystem. We use several classical results for the proof: the CourantFischerWeyl minmax theorem for Hermitian operators and its consequence, the Cauchy interlacing theorem, and prove their analogues in the symplectic setting. For the case of passive transformations of Gaussian mixed states we also prove that the obtained inequalities are, in a sense, the best possible. The obtained mathematical results are applied to the system of n uncorrelated thermal modes of the electromagnetic field. Finally, we present the results of numerical simulations of the problem, suggesting avenues of further research.

Conditions for uniqueness of product representations for separable quantum channels and separable quantum states
View Description Hide DescriptionWe give a sufficient condition that an operator sum representation of a separable quantum channel in terms of product operators is the unique product representation for that channel, and then provide examples of such channels for any number of parties. This result has implications for efforts to determine whether or not a given separable channel can be exactly implemented by local operations and classical communication. By the ChoiJamiolkowski isomorphism, it also translates to a condition for the uniqueness of product state ensembles representing a given quantum state. These ideas follow from considerations concerning whether or not a subspace spanned by a given set of product operators contains at least one additional product operator.

Properties of subentropy
View Description Hide DescriptionSubentropy is an entropylike quantity that arises in quantum information theory; for example, it provides a tight lower bound on the accessible information for pure state ensembles, dual to the von Neumann entropy upper bound in Holevo's theorem. Here we establish a series of properties of subentropy, paralleling the welldeveloped analogous theory for von Neumann entropy. Further, we show that subentropy is a lower bound for minentropy. We introduce a notion of conditional subentropy and show that it can be used to provide an upper bound for the guessing probability of any classicalquantum state of two qubits; we conjecture that the bound applies also in higher dimensions. Finally, we give an operational interpretation of subentropy within classical information theory.

Geometry and topology of CC and CQ states
View Description Hide DescriptionWe show that mixed bipartite CC and CQ states are geometrically and topologically distinguished in the space of states. They are characterized by nonvanishing EulerPoincaré characteristics on the topological side and by the existence of symplectic structures on the geometric side.
 Relativistic Quantum Mechanics, Quantum Field Theory, Quantum Gravity, and String Theory

Modesum construction of the twopoint functions for the Stueckelberg vector fields in the Poincaré patch of de Sitter space
View Description Hide DescriptionWe perform canonical quantization of the Stueckelberg Lagrangian for massive vector fields in the conformally flat patch of de Sitter space in the BunchDavies vacuum and find their Wightman twopoint functions by the modesum method. We discuss the zeromass limit of these twopoint functions and their limits where the Stueckelberg parameter ξ tends to zero or infinity. It is shown that our results reproduce the standard flatspace propagator in the appropriate limit. We also point out that the classic work of Allen and Jacobson [“Vector twopoint functions in maximally symmetric spaces,” Commun. Math. Phys.103, 669 (1986)] for the twopoint function of the Proca field and a recent work by Tsamis and Woodard [“Maximally symmetric vector propagator,” J. Math. Phys.48, 052306 (2007)] for that of the transverse vector field are two limits of our twopoint function, one for ξ → ∞ and the other for ξ → 0. Thus, these two works are consistent with each other, contrary to the claim by the latter authors.

A deformation of quantum affine algebra in squashed WessZuminoNovikovWitten models
View Description Hide DescriptionWe proceed to study infinitedimensional symmetries in twodimensional squashed WessZuminoNovikovWitten models at the classical level. The target space is given by squashed S^{3} and the isometry is SU (2)L × U(1)R. It is known that SU (2)L is enhanced to a couple of Yangians. We reveal here that an infinitedimensional extension of U(1)R is a deformation of quantum affine algebra, where a new deformation parameter is provided with the coefficient of the WessZumino term. Then we consider the relation between the deformed quantum affine algebra and the pair of Yangians from the viewpoint of the leftright duality of monodromy matrices. The integrable structure is also discussed by computing the r/smatrices that satisfy the extended classical YangBaxter equation. Finally, two degenerate limits are discussed.

Construction of dynamics and timeordered exponential for unbounded nonsymmetric Hamiltonians
View Description Hide DescriptionWe prove under certain assumptions that there exists a solution of the Schrödinger or the Heisenberg equation of motion generated by a linear operator H acting in some complex Hilbert space , which may be unbounded, not symmetric, or not normal. We also prove that, under the same assumptions, there exists a time evolution operator in the interaction picture and that the evolution operator enjoys a useful series expansion formula. This expansion is considered to be one of the mathematically rigorous realizations of socalled “timeordered exponential,” which is familiar in the physics literature. We apply the general theory to prove the existence of dynamics for the mathematical model of Quantum Electrodynamics quantized in the Lorenz gauge, the interaction Hamiltonian of which is not even symmetric or normal.