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At least 1,117 records · Page 62Linked to original sources
arXiv2013

Quantum Side Information: Uncertainty Relations, Extractors, Channel Simulations

In the first part of this thesis, we discuss the algebraic approach to classical and quantum physics and develop information theoretic concepts within this setup. In the second part, we discuss the uncertainty principle in quantum mechanics. The principle states that even if we have full classical information about the state of a quantum system, it is impossible to deterministically predict the outcomes of all possible measurements. In comparison, the perspective of a quantum observer allows to have quantum information about the state of a quantum system. This then leads to an interplay between uncertainty and quantum correlations. We provide an information theoretic analysis by discussing entropic uncertainty relations with quantum side information. In the third part, we discuss the concept of randomness extractors. Classical and quantum randomness are an essential resource in information theory, cryptography, and computation. However, most sources of randomness exhibit only weak forms of unpredictability, and the goal of randomness extraction is to convert such weak randomness into (almost) perfect randomness. We discuss various constructions for classical and quantum randomness extractors, and we examine especially the performance of these constructions relative to an observer with quantum side information. In the fourth part, we discuss channel simulations. Shannon's noisy channel theorem can be understood as the use of a noisy channel to simulate a noiseless one. Channel simulations as we want to consider them here are about the reverse problem: simulating noisy channels from noiseless ones. Starting from the purely classical case (the classical reverse Shannon theorem), we develop various kinds of quantum channel simulation results. We achieve this by using classical and quantum randomness extractors that also work with respect to quantum side information.

quant-ph↗
arXiv2013

Satisfiability threshold for random regular NAE-SAT

We consider the random regular $k$-NAE-SAT problem with $n$ variables each appearing in exactly $d$ clauses. For all $k$ exceeding an absolute constant $k_0$, we establish explicitly the satisfiability threshold $d_*=d_*(k)$. We prove that for $d d_*$ the problem is unsatisfiable with high probability. If the threshold $d_*$ lands exactly on an integer, we show that the problem is satisfiable with probability bounded away from both zero and one. This is the first result to locate the exact satisfiability threshold in a random constraint satisfaction problem exhibiting the condensation phenomenon identified by Krzakala et al. (2007). Our proof verifies the one-step replica symmetry breaking formalism for this model. We expect our methods to be applicable to a broad range of random constraint satisfaction problems and combinatorial problems on random graphs.

math.PR↗
arXiv2013

Maximum independent sets on random regular graphs

We determine the asymptotics of the independence number of the random $d$-regular graph for all $d \ge d_0$. It is highly concentrated, with constant-order fluctuations around $nα_* - c_*\log n$ for explicit constants $α_*(d)$ and $c_*(d)$. Our proof rigorously confirms the one-step replica symmetry breaking heuristics for this problem, and we believe the techniques will be more broadly applicable to the study of other combinatorial properties of random graphs.

math.PR↗
arXiv2013

Electromagnetic wave propagation in random waveguides

We study long range propagation of electromagnetic waves in random waveguides with rectangular cross-section and perfectly conducting boundaries. The waveguide is filled with an isotropic linear dielectric material, with randomly fluctuating electric permittivity. The fluctuations are weak, but they cause significant cumulative scattering over long distances of propagation of the waves. We decompose the wave field in propagating and evanescent transverse electric and magnetic modes with random amplitudes that encode the cumulative scattering effects. They satisfy a coupled system of stochastic differential equations driven by the random fluctuations of the electric permittivity. We analyze the solution of this system with the diffusion approximation theorem, under the assumption that the fluctuations decorrelate rapidly in the range direction. The result is a detailed characterization of the transport of energy in the waveguide, the loss of coherence of the modes and the depolarization of the waves due to cumulative scattering.

math-ph↗
arXiv2013

Generalized geometry, T-duality, and renormalization group flow

We interpret the physical $B$-field renormalization group flow in the language of Courant algebroids, clarifying the sense in which this flow is the natural "Ricci flow" for generalized geometry. Next we show that the $B$-field renormalization group flow preserves T-duality in a natural sense. As corollaries we obtain new long time existence results for the $B$-field renormalization group flow.

math.DG↗
arXiv2013

Vertex Operators Arising from Linear ODEs

The Heisenberg Oscillator Algebra admits irreducible representations both on the ring $B$ of polynomials in infinitely many indeterminates (the {\em bosonic representation}) and on a graded-by-{\em charge} vector space, the {\em semi-infinite} exterior power of an infinite-dimensional ${\mathbf Q}$-vector space $V$ (the {\em fermionic representation}). Our main observation is that $V$ can be realized as the ${\mathbf Q}$-vector space generated by the solutions to a generic linear ODE of {\em infinite order}. Within this framework, the well known {\em boson-fermion} correspondence for the zero charge fermionic space is a consequence of the formula expressing each solution to a linear ODE as a linear combination of the elements of the universal basis of solutions. In this paper we extend the picture for linear ODEs of finite order. Vertex operators are defined and fully described in this case.

math.AG↗
arXiv2013

Vector cross product in n-dimensional vector space

The definition of vector cross product (VCP) introduced by Eckmann only exists in thethree- and the seven- dimensional vector space. In this paper, according to the orthogonal completeness, magnitude of basis vector cross product and all kinds of combinations of basis vector $\hat{e}_i$, the generalized definition of VCP in the odd n-dimensional vector space is given by introducing a cross term $X_{AB}$. In addition, the definition is validated by reducing the generalization definition to the fundamental three- and seven-dimensional vector space.

math-ph↗
arXiv2013

Quantum Locally Testable Codes

We initiate the study of quantum Locally Testable Codes (qLTCs). We provide a definition together with a simplification, denoted sLTCs, for the special case of stabilizer codes, together with some basic results using those definitions. The most crucial parameter of such codes is their soundness, $R(δ)$, namely, the probability that a randomly chosen constraint is violated as a function of the distance of a word from the code ($δ$, the relative distance from the code, is called the proximity). We then proceed to study limitations on qLTCs. In our first main result we prove a surprising, inherently quantum, property of sLTCs: for small values of proximity, the better the small-set expansion of the interaction graph of the constraints, the less sound the qLTC becomes. This phenomenon, which can be attributed to monogamy of entanglement, stands in sharp contrast to the classical setting. The complementary, more intuitive, result also holds: an upper bound on the soundness when the code is defined on poor small-set expanders (a bound which turns out to be far more difficult to show in the quantum case). Together we arrive at a quantum upper-bound on the soundness of stabilizer qLTCs set on any graph, which does not hold in the classical case. Many open questions are raised regarding what possible parameters are achievable for qLTCs. In the appendix we also define a quantum analogue of PCPs of proximity (PCPPs) and point out that the result of Ben-Sasson et. al. by which PCPPs imply LTCs with related parameters, carries over to the sLTCs. This creates a first link between qLTCs and quantum PCPs.

quant-ph↗
arXiv2013

New addition formula for the little $q$-Bessel functions

Starting from the addition formula for little $q$-Jacobi polynomials, we derive a new addition formula for the little $q$-Bessel functions. The result is obtained by the use of a limit transition. We also establish a product formula for little $q$-Bessel functions with a positive and symmetric kernel.

math-ph↗
arXiv2013

Rigorous numerics for NLS: bound states, spectra, and controllability

In this paper it is demonstrated how rigorous numerics may be applied to the one-dimensional nonlinear Schrödinger equation (NLS); specifically, to determining bound--state solutions and establishing certain spectral properties of the linearization. Since the results are rigorous, they can be used to complete a recent analytical proof [6] of the local exact controllability of NLS.

math.DS↗
arXiv2013

A study of obliquely propagating longitudinal shear waves in a periodic laminate

The basic purpose of this work is to demonstrate, by quite simple and explicit calculation, the possibility that a simple laminate composed of ordinary materials can display the kind of response associated with a "metamaterial". Specifically, a wave of Floquet--Bloch type is analyzed, taking careful account of the requirements of causality and passivity. Then, the generation of such a wave by transmission, from ordinary homogeneous material, into a half-space composed of the laminate is considered. As is well-known for any dispersive material, phase velocity is distinct from group velocity, and information is carried by the latter. Thus, in line with the discussion of photonic crystals by Notomi [1], refraction is considered in terms of the group velocity. Negative refraction appears never to occur, if the interface between ordinary material and composite is parallel to the interfaces in the composite but negative refraction is possible, over an appreciable frequency range, if the interface between ordinary material and composite is orthogonal to the interface in the composite.

cond-mat.mtrl-sci↗
arXiv2013

Geometric Spinors, Generalized Dirac Equation and Mirror Particles

It is shown that since the geometric spinors are elements of Clifford algebras, they must have the same transformation properties as any other Clifford number. In general, a Clifford number $Φ$ transforms into a new Clifford number $Φ'$ according to $Φ\to Φ' = {\rm{R}}\,Φ\,{\rm{S}}$, i.e., by the multiplication from the left and from the right by two Clifford numbers ${\rm R}$ and ${\rm S}$. We study the case of $Cl(1,3)$, which is the Clifford algebra of the Minkowski spacetime. Depending on choice of ${\rm R}$ and ${\rm S}$, there are various possibilities, including the transformations of vectors into 3-vectors, and the transformations of the spinors of one minimal left ideal of $Cl(1,3)$ into another minimal left ideal. This, among others, has implications for understanding the observed non-conservation of parity.

hep-th↗
arXiv2013

Entropy, subentropy and the elementary symmetric functions

We use complex contour integral techniques to study the entropy H and subentropy Q as functions of the elementary symmetric polynomials, revealing a series of striking properties. In particular for these variables, derivatives of -Q are equal to derivatives of H of one higher order and the first derivatives of H and Q are seen to be completely monotone functions. It then follows that exp (-H) and exp(-Q) are Laplace transforms of infinitely divisible probability distributions.

quant-ph↗
arXiv2013

Exact solutions and zero modes in scalar field theory

We provide a set of exact solutions in field theory of scalar fields with $Z_2$ symmetry that involve Jacobi elliptic functions. These solutions have the interesting property to provide massive waves even if one starts from a massless equation. We analyze them classically providing also exact solutions to the corresponding equations for the Green functions needed to completely solve them in a strong coupling limit. This is accomplished using a functional expansion into powers of the current. It is shown that the spontaneous breaking of the $Z_2$ symmetry is due to the existence of a zero mode that persists also in the case of the Higgs mechanism. In this latter case, the zero mode seems to play a role similar to the Goldstone boson in the breaking of a continuous symmetry and so, it should be important at lower momenta as a long range excitation.

math-ph↗
arXiv2013

A classification of 2-dim Lattice Theory

A unified classification and analysis is presented of two dimensional Dirac operators of QCD-like theories in the continuum as well as in a naive lattice discretization. Thereby we consider the quenched theory in the strong coupling limit. We do not only consider the case of a lattice which has an even number of lattice sites in both directions and is thus equivalent to the case of staggered fermions. We also study lattices with one or both directions with an odd parity to understand the general mechanism of changing the universality class via a discretization. Furthermore we identify the corresponding random matrix ensembles sharing the global symmetries of these QCD-like theories. Despite the Mermin-Wagner-Coleman theorem we find good agreement of lattice data with our random matrix predictions.

hep-lat↗