Quantum Inequalities in Quantum Mechanics

We study a phenomenon occurring in various areas of quantum physics, in which an observable density (such as an energy density) which is classically pointwise non-negative may assume arbitrarily negative expectation values after quantization, even though the spatially integrated density remains non-negative. Two prominent examples which have previously been studied are the energy density (in quantum field theory) and the probability flux of rightwards-moving particles (in quantum mechanics). However, in the quantum field context, it has been shown that the magnitude and space-time extension of negative energy densities are not arbitrary, but restricted by relations which have come to be known as quantum inequalities. In the present work, we explore the extent to which such quantum inequalities hold for typical quantum mechanical systems. We derive quantum inequalities of two types. The first are kinematical quantum inequalities where spatially averaged densities are shown to be bounded below. Specifically, we obtain such kinematical quantum inequalities for the current density in one spatial dimension (imposing constraints on the backflow phenomenon) and for the densities arising in Weyl–Wigner quantization. The latter quantum inequalities are direct consequences of sharp Gårding inequalities. The second type are dynamical quantum inequalities where one obtains bounds from below on temporally averaged densities. We derive such quantum inequalities in the case of the energy density in general quantum mechanical systems having suitable decay properties on the negative spectral axis of the total energy.Furthermore, we obtain explicit numerical values for the quantum inequalities on the one-dimensional current density, using various spatial averaging weight functions. We also improve the numerical value of the related backflow constant previously investigated by Bracken and Melloy. In many cases our numerical results are controlled by rigorous error estimates.submitted 27/01/04, accepted 05/05/04     

Quantum Ergodicity for Quantum Graphs without Back-Scattering

We give an estimate of the quantum variance for d-regular graphs quantised with boundary scattering matrices that prohibit back-scattering. For families of graphs that are expanders, with few short cycles, our estimate leads to quantum ergodicity for these families of graphs. Our proof is based on a uniform control of an associated random walk on the bonds of the graph. We show that recent constructions of Ramanujan graphs, and asymptotically almost surely, random d-regular graphs, satisfy the necessary conditions to conclude that quantum ergodicity holds.     

Stability of Quantum Electrodynamics and Quantum Chromodynamics

We consider the vacuum energy in QED viewed as in a system of charged fermions and bosons and in QCD viewed as in a system of quarks (fermions) and gluons (bosons) in a self-dual field with a constant strength. We show that the cause of instability is the instability of bosons in the self-dual vacuum field. For the global stability of a system consisting of fermions and bosons, the number of fermions should be sufficiently large. The nonzero self-dual field leading to the confinement of fermions realizes the minimum of the vacuum energy in the case where the boson has the smallest mass in the system. Confinement therefore does not arise in QED, where the fermion (electron) has the smallest mass, and does arise in QCD, where the boson (gluon) has the smallest mass.     

Quantum Spacetime,Quantum Geometry and Planck scales

The Principles of Quantum Mechanics and of Classical General Relativity indicate that Spacetime in the small (Planck scale) ought to be described by a noncommutative C* Algebra, implementing spacetime uncertainty relations. A model C* algebra of Quantum Spacetime and its Quantum Geometry is described. Interacting Quantum Field Theory on such a background is discussed, with open problems and recent progress. Applications to cosmology suggest that the Planck scale ought to depend upon dynamics, and possible consequences in the large of the quantum structure in the small are outlined.     

Quantum B-splines

Quantum splines are piecewise polynomials whose quantum derivatives (i.e. certain discrete derivatives or equivalently certain divided differences) agree up to some order at the joins. Just like classical splines, quantum splines admit a canonical basis with compact support: the quantum B-splines. These quantum B-splines are the q-analogues of classical B-splines. Here quantum B-spline bases and quantum B-spline curves are investigated, using a new variant of the blossom: the q (quantum)-blossom. The q-blossom of a degree d polynomial is the unique symmetric, multiaffine function in d variables that reduces to the polynomial along the q-diagonal. By applying the q-blossom, algorithms and identities for quantum B-spline bases and quantum B-spline curves are developed, including quantum variants of the de Boor algorithms for recursive evaluation and quantum differentiation, knot insertion procedures for converting from quantum B-spline to piecewise quantum Bézier form, and a quantum variant of Marsden’s identity.     

Quantum B-algebras

The concept of quantale was created in 1984 to develop a framework for non-commutative spaces and quantum mechanics with a view toward non-commutative logic. The logic of quantales and its algebraic semantics manifests itself in a class of partially ordered algebras with a pair of implicational operations recently introduced as quantum B-algebras. Implicational algebras like pseudo-effect algebras, generalized BL- or MV-algebras, partially ordered groups, pseudo-BCK algebras, residuated posets, cone algebras, etc., are quantum B-algebras, and every quantum B-algebra can be recovered from its spectrum which is a quantale. By a two-fold application of the functor “spectrum”, it is shown that quantum B-algebras have a completion which is again a quantale. Every quantale Q is a quantum B-algebra, and its spectrum is a bigger quantale which repairs the deficiency of the inverse residuals of Q. The connected components of a quantum B-algebra are shown to be a group, a fact that applies to normal quantum B-algebras arising in algebraic number theory, as well as to pseudo-BCI algebras and quantum BL-algebras. The logic of quantum B-algebras is shown to be complete.     

Quantum groups

The paper is the expanded text of a report to the International Mathematical Congress in Berkeley (1986). In it a new algebraic formalism connected with the quantum method of the inverse problem is developed.Examples are constructed of noncommutative Hopf algebras and their connection with solutions of the Yang-Baxter quantum identity are discussed.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 155, pp. 18–49, 1986.     

Quantum spaces

In this paper, a more general concept of quantum space is given by modifying the original concept defined by Borceux and Bossche. We show that a quantum space is a topological analogue of a quantale defined by Mulvey, and also a non-commutative generalization of the Zariski spectrum of a commutative ring. But quantum spaces are not good enough to have much of the properties of topological spaces, such as product spaces and quotient spaces.     

Quantum octonions

This paper constructs a quantum deformation of the complex Cayley dgebra. The method uses the representation theory of U _q(sl(2)), the quantized enveloping algebra of the simple complex Lie algebra s/(2). The paper begins by constructing a quantum deforma-tion of the complex quaternion algebra, since this simpler case illustrates all of the necessary steps. As intermediate results, deformations are constructed of sl(2) and the 7-dimensional simple Malcev algebra.     

Quantum teleportation

In the framework of an algebraic approach, we consider a quantum teleportation procedure. It turns out that using the quantum measurement nonlocality hypothesis is unnecessary for describing this procedure. We study the question of what material objects are information carriers for quantum teleportation. __________ Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 157, No. 1, pp. 79–98, October, 2008.     

Quantum groups

An elementary introduction to the notions of the quantum Lie Groups and quantum Lie algebras is given. The approach is based on the fundamental commutation relations which appeared first in the quantum inverse scattering method.     

矩阵在量子计算与量子信息中的应用

介绍了量子计算与量子信息中的一些重要的矩阵及其应用,包括密度矩阵、酉矩阵等.     

Quantum Analog Computing

Quantum analog computing is based upon similarity between mathematical formalism of quantum mechanics and phenomena to be computed. It exploits a dynamical convergence of several competing phenomena to an attractor which can represent an extremum of a function, an image, a solution to a system of ODE, or a stochastic process. In this paper, a quantum version of recurrent neural nets (QRN) as an analog computing device is discussed. This concept is introduced by incorporating classical feedback loops into conventional quantum networks. It is shown that the dynamical evolution of such networks, which interleave quantum evolution with measurement and reset operations, exhibit novel dynamical properties. Moreover, decoherence in quantum recurrent networks is less problematic than in conventional quantum network architectures due to the modest phase coherence times needed for network operation. Application of QRN to simulation of chaos, turbulence, NP-problems, as well as data compression demonstrate computational speedup and exponential increase of information capacity.     

Quantum Floquet functions

The quantum analog of the Floquet solution of the auxiliary linear problem for the lattice model of the nonlinear Schrödinger equation is constructed. Applications to the quantum Gel'fand-Levitan-Marchenko equations are discussed.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 131, pp. 128–141, 1983.The authors are grateful to L. D. Faddeev, P. P. Kulish, E. K. Sklyanin, and M. A. Semenov-Tyan-shanskii for useful and interesting discussions.     

Quantum Bayesian computation

Quantum Bayesian computation is an emerging field that levers the computational gains available from quantum computers. They promise to provide an exponential speed-up in Bayesian computation. Our article adds to the literature in three ways. First, we describe how quantum von Neumann measurement provides quantum versions of popular machine learning algorithms such as Markov chain Monte Carlo and deep learning that are fundamental to Bayesian learning. Second, we describe quantum data encoding methods needed to implement quantum machine learning including the counterparts to traditional feature extraction and kernel embeddings methods. Third, we show how quantum algorithms naturally calculate Bayesian quantities of interest such as posterior distributions and marginal likelihoods. Our goal then is to show how quantum algorithms solve statistical machine learning problems. On the theoretical side, we provide quantum versions of high dimensional regression, Gaussian processes and stochastic gradient descent. On the empirical side, we apply a quantum FFT algorithm to Chicago house price data. Finally, we conclude with directions for future research.     

Conditional Expectations on Compact Quantum Groups and Quantum Double Cosets

We prove that a conditional expectation on a compact quantum group that satisfies certain conditions can be decomposed into a composition of two conditional expectations one of which is associated with quantum double cosets and the other preserves the counit. __________ Translated from Ukrains'kyi Matematychnyi Zhurnal, Vol. 57, No. 5, pp. 644–653, May, 2005.     

QUANTUM GAUSSIAN PROCESSES

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