Tensor Product Structures, Entanglement, and Particle Scattering

Particle systems admit a variety of tensor product structures (TPSs) depending on the complete system of commuting observables chosen for the analysis. Different notions of entanglement are associated with these different TPSs. Global symmetry transformations and dynamical transformations factor into products of local unitary operators with respect to certain TPSs and not with respect to others. Symmetry-invariant and dynamical-invariant TPSs and corresponding measures of entanglement are defined for particle scattering systems.Comment: 7 pages, no figures; v.2 typo in references corrected, submitted to OSID as part of SMP3

Galilean and Dynamical Invariance of Entanglement in Particle Scattering

Particle systems admit a variety of tensor product structures (TPSs) depending on the algebra of observables chosen for analysis. Global symmetry transformations and dynamical transformations may be resolved into local unitary operators with respect to certain TPSs and not with respect to others. Symmetry-invariant and dynamical-invariant TPSs are defined and various notions of entanglement are considered for scattering states.Comment: 4 pages, no figures; v.3 has typos corrected, a new reference, and a revised conclusio

A Solvable Model for Decoupling of Interacting Clusters

We consider M clusters of interacting particles, whose in-group interactions are arbitrary, and inter-group interactions are approximated by oscillator potentials. We show that there are masses and frequencies that decouple the in-group and inter-group degrees of freedom, which reduces the initial problem to M independent problems that describe each of the relative in-group systems. The dynamics of the M center-of-mass coordinates is described by the analytically solvable problem of M coupled harmonic oscillators. This paper derives and discusses these decoupling conditions. Furthermore, to illustrate our findings, we consider a charged impurity interacting with a ring of ions. We argue that the impurity can be used to probe the center-of-mass dynamics of the ions.Comment: Version accepted for publication in EP

The dynamics of digits: Calculating pi with Galperin's billiards

In Galperin billiards, two balls colliding with a hard wall form an analog calculator for the digits of the number $\pi$. This classical, one-dimensional three-body system (counting the hard wall) calculates the digits of $\pi$ in a base determined by the ratio of the masses of the two particles. This base can be any integer, but it can also be an irrational number, or even the base can be $\pi$ itself. This article reviews previous results for Galperin billiards and then pushes these results farther. We provide a complete explicit solution for the balls' positions and velocities as a function of the collision number and time. We demonstrate that Galperin billiard can be mapped onto a two-particle Calogero-type model. We identify a second dynamical invariant for any mass ratio that provides integrability for the system, and for a sequence of specific mass ratios we identify a third dynamical invariant that establishes superintegrability. Integrability allows us to derive some new exact results for trajectories, and we apply these solutions to analyze the systematic errors that occur in calculating the digits of $\pi$ with Galperin billiards, including curious cases with irrational number bases.Comment: 30 pages, 13 figure

Integrable families of hard-core particles with unequal masses in a one-dimensional harmonic trap

We show that the dynamics of particles in a one-dimensional harmonic trap with hard-core interactions can be solvable for certain arrangements of unequal masses. For any number of particles, there exist two families of unequal mass particles that have integrable dynamics, and there are additional exceptional cases for three, four and five particles. The integrable mass families are classified by Coxeter reflection groups and the corresponding solutions are Bethe ansatz-like superpositions of hyperspherical harmonics in the relative hyperangular coordinates that are then restricted to sectors of fixed particle order. We also provide evidence for superintegrability of these Coxeter mass families and conjecture maximal superintegrability.Comment: 9.5+4.5 pages, 5 figures, 2 tables; v3: a few corrections and addition

Smoothed Analysis of Tensor Decompositions

Low rank tensor decompositions are a powerful tool for learning generative models, and uniqueness results give them a significant advantage over matrix decomposition methods. However, tensors pose significant algorithmic challenges and tensors analogs of much of the matrix algebra toolkit are unlikely to exist because of hardness results. Efficient decomposition in the overcomplete case (where rank exceeds dimension) is particularly challenging. We introduce a smoothed analysis model for studying these questions and develop an efficient algorithm for tensor decomposition in the highly overcomplete case (rank polynomial in the dimension). In this setting, we show that our algorithm is robust to inverse polynomial error -- a crucial property for applications in learning since we are only allowed a polynomial number of samples. While algorithms are known for exact tensor decomposition in some overcomplete settings, our main contribution is in analyzing their stability in the framework of smoothed analysis. Our main technical contribution is to show that tensor products of perturbed vectors are linearly independent in a robust sense (i.e. the associated matrix has singular values that are at least an inverse polynomial). This key result paves the way for applying tensor methods to learning problems in the smoothed setting. In particular, we use it to obtain results for learning multi-view models and mixtures of axis-aligned Gaussians where there are many more "components" than dimensions. The assumption here is that the model is not adversarially chosen, formalized by a perturbation of model parameters. We believe this an appealing way to analyze realistic instances of learning problems, since this framework allows us to overcome many of the usual limitations of using tensor methods.Comment: 32 pages (including appendix

Understanding entangled spins in QED

The stability of two entangled spins dressed by electrons is studied by calculating the scattering phase shifts. The interaction between electrons is interpreted by fully relativistic QED and the screening effect is described phenomenologically in the Debye exponential form $e^{-\alpha r}$. Our results show that if the (Einstein-Podolsky-Rosen-) EPR-type states are kept stable under the interaction of QED, the spatial wave function must be parity-dependent. The spin-singlet state $s=0$ and the polarized state $\frac 1{\sqrt{2}}(\mid +-> -\mid -+>)$ along the z-axis\QTR{bf}{\}give rise to two different kinds of phase shifts\QTR{bf}{.} Interestingly, the interaction between electrons in the spin-singlet pair is found to be attractive. Such an attraction could be very useful when we extract the entangled spins from superconductors. A mechanism to filter the entangled spins is also discussed.Comment: 6 pages, 3 figures. changes adde

Theory of High-Tc Superconductivity: Accurate Predictions of Tc

The superconducting transition temperatures of high-Tc compounds based on copper, iron, ruthenium and certain organic molecules are discovered to be dependent on bond lengths, ionic valences, and Coulomb coupling between electronic bands in adjacent, spatially separated layers [1]. Optimal transition temperature, denoted as T_c0, is given by the universal expression $k_BT_c0 = e^2 \Lambda / \ell\zeta$; $\ell$ is the spacing between interacting charges within the layers, \zeta is the distance between interacting layers and \Lambda is a universal constant, equal to about twice the reduced electron Compton wavelength (suggesting that Compton scattering plays a role in pairing). Non-optimum compounds in which sample degradation is evident typically exhibit Tc < T_c0. For the 31+ optimum compounds tested, the theoretical and experimental T_c0 agree statistically to within +/- 1.4 K. The elemental high Tc building block comprises two adjacent and spatially separated charge layers; the factor e^2/\zeta arises from Coulomb forces between them. The theoretical charge structure representing a room-temperature superconductor is also presented.Comment: 7 pages 5 references, 6 figures 1 tabl

Developing a Research Mentorship Program: The American Society of Pediatric Nephrology's Experience

Background: Most pediatric nephrologists work in academia. Mentor-mentee relationships provide support and guidance for successful research career. Mentorship program implementation is valuable in medical fields for providing research opportunities to young faculty. Methods: The American Society of Pediatric Nephrology (ASPN) established a research mentorship program to (a) assist with matching of appropriate mentor-mentee dyads and (b) establish metrics for desirable mentor-mentee outcomes with two independent components: (1) the grants review workshop, a short-term program providing mentor feedback on grant proposals, and (2) the longitudinal program, establishing long-term mentor-mentee relationships. Regular surveys of both mentors and mentees were reviewed to evaluate and refine the program. Results: Twelve mentees and 17 mentors participated in the grant review workshop and 19 mentees were matched to mentors in the longitudinal program. A review of NIH RePORTER data indicated that since 2014, 13 NIH grants have been awarded. Mentees in the longitudinal program reported that the program helped most with identifying an outside mentor, improving grant research content, and with general career development. Mentors perceived themselves to be most helpful in assisting with overall career plans. Email communications were preferred over phone or face-to-face communications. Mentees endorsed strong interest in staying in touch with their mentors and 100% of mentors expressed their willingness to serve in the future. Conclusion: This mentorship program was initiated and supported by a relatively small medical society and has shown early success in cultivating mentoring relationships for a future generation of clinician-scientists