scholarly journals Complete homotopy invariants for translation invariant symmetric quantum walks on a chain

Quantum ◽  
2018 ◽  
Vol 2 ◽  
pp. 95 ◽  
Author(s):  
C. Cedzich ◽  
T. Geib ◽  
C. Stahl ◽  
L. Velázquez ◽  
A. H. Werner ◽  
...  
Keyword(s):  
Practical Method ◽  
Quantum Walks ◽  
Translation Invariance ◽  
Translation Invariant ◽  
Symmetric Quantum ◽  
Space Projection ◽  
Exponential Behaviour ◽  
A Chain ◽  
The One ◽  
Locality Conditions

We provide a classification of translation invariant one-dimensional quantum walks with respect to continuous deformations preserving unitarity, locality, translation invariance, a gap condition, and some symmetry of the tenfold way. The classification largely matches the one recently obtained (arXiv:1611.04439) for a similar setting leaving out translation invariance. However, the translation invariant case has some finer distinctions, because some walks may be connected only by breaking translation invariance along the way, retaining only invariance by an even number of sites. Similarly, if walks are considered equivalent when they differ only by adding a trivial walk, i.e., one that allows no jumps between cells, then the classification collapses also to the general one. The indices of the general classification can be computed in practice only for walks closely related to some translation invariant ones. We prove a completed collection of simple formulas in terms of winding numbers of band structures covering all symmetry types. Furthermore, we determine the strength of the locality conditions, and show that the continuity of the band structure, which is a minimal requirement for topological classifications in terms of winding numbers to make sense, implies the compactness of the commutator of the walk with a half-space projection, a condition which was also the basis of the general theory. In order to apply the theory to the joining of large but finite bulk pieces, one needs to determine the asymptotic behaviour of a stationary Schrödinger equation. We show exponential behaviour, and give a practical method for computing the decay constants.

scholarly journals Quantum Walks: Schur Functions Meet Symmetry Protected Topological Phases

2021 ◽  
Author(s):  
C. Cedzich ◽  
T. Geib ◽  
F. A. Grünbaum ◽  
L. Velázquez ◽  
A. H. Werner ◽  
...  
Keyword(s):  
Schur Functions ◽  
Complete Classification ◽  
Quantum Walks ◽  
Topological Indices ◽  
Translation Invariance ◽  
Topological Phases ◽  
Translation Invariant ◽  
Finite Dimensional ◽  
Complete Set ◽  
Symmetry Protected

AbstractThis paper uncovers and exploits a link between a central object in harmonic analysis, the so-called Schur functions, and the very hot topic of symmetry protected topological phases of quantum matter. This connection is found in the setting of quantum walks, i.e. quantum analogs of classical random walks. We prove that topological indices classifying symmetry protected topological phases of quantum walks are encoded by matrix Schur functions built out of the walk. This main result of the paper reduces the calculation of these topological indices to a linear algebra problem: calculating symmetry indices of finite-dimensional unitaries obtained by evaluating such matrix Schur functions at the symmetry protected points $$\pm 1$$ ± 1 . The Schur representation fully covers the complete set of symmetry indices for 1D quantum walks with a group of symmetries realizing any of the symmetry types of the tenfold way. The main advantage of the Schur approach is its validity in the absence of translation invariance, which allows us to go beyond standard Fourier methods, leading to the complete classification of non-translation invariant phases for typical examples.

scholarly journals Emulating the local Kuramoto model with an injection-locked photonic crystal laser array

2021 ◽  
Vol 11 (1) ◽  
Author(s):  
Naotomo Takemura ◽  
Kenta Takata ◽  
Masato Takiguchi ◽  
Masaya Notomi
Keyword(s):  
Photonic Crystal ◽  
Nearest Neighbor ◽  
Multiple Quantum Well ◽  
Kuramoto Model ◽  
Multiple Quantum ◽  
Laser Array ◽  
Novel Strategy ◽  
A Chain ◽  
The One ◽  
The Kuramoto Model

AbstractThe Kuramoto model is a mathematical model for describing the collective synchronization phenomena of coupled oscillators. We theoretically demonstrate that an array of coupled photonic crystal lasers emulates the Kuramoto model with non-delayed nearest-neighbor coupling (the local Kuramoto model). Our novel strategy employs indirect coupling between lasers via additional cold cavities. By installing cold cavities between laser cavities, we avoid the strong coupling of lasers and realize ideal mutual injection-locking with effective non-delayed dissipative coupling. First, after discussing the limit cycle interpretation of laser oscillation, we demonstrate the synchronization of two indirectly coupled lasers by numerically simulating coupled-mode equations. Second, by performing a phase reduction analysis, we show that laser dynamics in the proposed device can be mapped to the local Kuramoto model. Finally, we briefly demonstrate that a chain of indirectly coupled photonic crystal lasers actually emulates the one-dimensional local Kuramoto chain. We also argue that our proposed structure, which consists of periodically aligned cold cavities and laser cavities, will best be realized by using state-of-the-art buried multiple quantum well photonic crystals.

scholarly journals XXIII.—On the Disruptive Discharge of Electricity: An Experimental Thesis for the Degree of Doctor of Science, Department A

1878 ◽  
Vol 28 (2) ◽  
pp. 633-671 ◽  
Author(s):  
Alexander Macfarlane
Keyword(s):  
Copper Wire ◽  
Late Summer ◽  
The Other ◽  
Straight Line ◽  
Summer Session ◽  
The Difference ◽  
A Chain ◽  
The One ◽  
Metallic Parts ◽  
The University

The experiments to which I shall refer were carried out in the physical laboratory of the University during the late summer session. I was ably assisted in conducting the experiments by three students of the laboratory,—Messrs H. A. Salvesen, G. M. Connor, and D. E. Stewart. The method which was used of measuring the difference of potential required to produce a disruptive discharge of electricity under given conditions, is that described in a paper communicated to the Royal Society of Edinburgh in 1876 in the names of Mr J. A. Paton, M. A., and myself, and was suggested to me by Professor Tait as a means of attacking the experimental problems mentioned below.The above sketch which I took of the apparatus in situ may facilitate tha description of the method. The receiver of an air-pump, having a rod capable of being moved air-tight up and down through the neck, was attached to one of the conductors of a Holtz machine in such a manner that the conductor of the machine and the rod formed one conducting system. Projecting from the bottom of the receiver was a short metallic rod, forming one conductor with the metallic parts of the air-pump, and by means of a chain with the uninsulated conductor of the Holtz machine. Brass balls and discs of various sizes were made to order, capable of being screwed on to the ends of the rods. On the table, and at a distance of about six feet from the receiver, was a stand supporting two insulated brass balls, the one fixed, the other having one degree of freedom, viz., of moving in a straight line in the plane of the table. The fixed insulated ball A was made one conductor with the insulated conductor of the Holtz and the rod of the receiver, by means of a copper wire insulated with gutta percha, having one end stuck firmly into a hole in the collar of the receiver, and having the other fitted in between the glass stem and the hollow in the ball, by which it fitted on to the stem tightly. A thin wire similarly fitted in between the ball B and its insulating stem connected the ball with the insulated half ring of a divided ring reflecting electrometer.

scholarly journals Dietary fat intake in healthy adolescents: inverse relationships between the estimated intake of saturated fatty acids and serum cholesterol

2001 ◽  
Vol 85 (3) ◽  
pp. 333-341 ◽  
Author(s):  
Gösta Samuelson ◽  
Lars-Erik Bratteby ◽  
Rawya Mohsen ◽  
Bengt Vessby
Keyword(s):  
Fatty Acids ◽  
Serum Cholesterol ◽  
Milk Fat ◽  
Saturated Fatty Acids ◽  
Saturated Fat ◽  
Dietary Fatty Acids ◽  
Cholesterol Esters ◽  
Fat Intake ◽  
A Chain ◽  
The One

The objective of the present study was to describe the intake of dietary fatty acids among healthy 15-year-old boys and girls and to relate the intake of specific fatty acids and the fatty acid composition of the serum cholesterol esters to serum lipid, apolipoprotein (Apo) and insulin concentrations respectively. Fifty-two girls and forty-two boys were randomly selected from the official population register. Unexpectedly, significant inverse associations were found between the dietary content of saturated fatty acids with a chain length of four to fifteen C atoms, mainly derived from milk fat, as well as the corresponding fatty acids in the serum cholesterol esters, on the one hand and the serum concentrations of cholesterol and ApoB on the other. The estimated dietary intake of 4:0–10:0, 12:0 and 14:0 respectively, were all significantly inversely related to the serum cholesterol (r-0.32,r-0.31,r-0.30, all P<0.05) and ApoB (r-0.42,r-0.42, andr-0.40, all P<0.05) concentrations in girls and 12:0 to the ApoB concentration (r-0.55, P<0.01) in boys. The proportions of 12:0 and 15:0 in the serum cholesterol esters were negatively correlated with the serum cholesterol concentrations in both girls (r-0.34,r-0.32, P<0.05) and boys (r-0.53, P<0.01;r-0.32, P<0.05) and with the ApoB concentrations among boys (r-0.61, P<0.01;r-0.43, P<0.05). It is conceivable that milk fat contains or is associated with some component in the diet, or some other characteristics of the food intake, which counterbalances the expected positive relationships between saturated fat intake and lipid levels.

scholarly journals Entanglement marginal problems

Quantum ◽  
2021 ◽  
Vol 5 ◽  
pp. 589
Author(s):  
Miguel Navascués ◽  
Flavio Baccari ◽  
Antonio Acín
Keyword(s):  
Quantum State ◽  
System Size ◽  
Separable Extension ◽  
Translation Invariant ◽  
Time Space ◽  
Marginal Problem ◽  
Spatial Dimensions ◽  
A Chain ◽  
Reduced Density ◽  
Star Configuration

We consider the entanglement marginal problem, which consists of deciding whether a number of reduced density matrices are compatible with an overall separable quantum state. To tackle this problem, we propose hierarchies of semidefinite programming relaxations of the set of quantum state marginals admitting a fully separable extension. We connect the completeness of each hierarchy to the resolution of an analog classical marginal problem and thus identify relevant experimental situations where the hierarchies are complete. For finitely many parties on a star configuration or a chain, we find that we can achieve an arbitrarily good approximation to the set of nearest-neighbour marginals of separable states with a time (space) complexity polynomial (linear) on the system size. Our results even extend to infinite systems, such as translation-invariant systems in 1D, as well as higher spatial dimensions with extra symmetries.

scholarly journals Asymmetries in symmetric quantum walks on two-dimensional networks

2005 ◽  
Vol 72 (4) ◽  
Author(s):  
Oliver Mülken ◽  
Antonio Volta ◽  
Alexander Blumen
Keyword(s):  
Quantum Walks ◽  
Two Dimensional ◽  
Symmetric Quantum

Extension of Monotonic Functions and Representation of Preferences

2021 ◽  
Author(s):  
Özgür Evren ◽  
Farhad Hüsseinov
Keyword(s):  
Dominance Relation ◽  
Translation Invariance ◽  
Compact Set ◽  
Continuous Real Function ◽  
Translation Invariant ◽  
Extension Theorems ◽  
Topological Vector Spaces ◽  
Dominance Relations ◽  
Universal Space ◽  
Vector Structure

Consider a dominance relation (a preorder) ≿ on a topological space X, such as the greater than or equal to relation on a function space or a stochastic dominance relation on a space of probability measures. Given a compact set K ⊆ X, we study when a continuous real function on K that is strictly monotonic with respect to ≿ can be extended to X without violating the continuity and monotonicity conditions. We show that such extensions exist for translation invariant dominance relations on a large class of topological vector spaces. Translation invariance or a vector structure are no longer needed when X is locally compact and second countable. In decision theoretic exercises, our extension theorems help construct monotonic utility functions on the universal space X starting from compact subsets. To illustrate, we prove several representation theorems for revealed or exogenously given preferences that are monotonic with respect to a dominance relation.

Preliminary Investigation of Mesh Stability for Linear Systems

1999 ◽  
Author(s):  
Pete Seiler ◽  
Aniruddha Pant ◽  
J. K. Hedrick
Keyword(s):  
Linear Systems ◽  
Mimo Systems ◽  
Preliminary Investigation ◽  
Two Dimensions ◽  
String Stability ◽  
Significant Research ◽  
Simple Class ◽  
The Subject ◽  
A Chain ◽  
The One

Abstract Damping of disturbances as they propagate through a chain of interconnected systems, termed string stability, has been the subject of significant research. In this paper, we investigate mesh stability, which is the two-dimensional extension of string stability. We review the key results used for string stability analysis and then generalize the conditions for MIMO systems. These results are then applied to a simple class of linear systems which form a mesh in two-dimensions. It is shown (as in the one-dimensional case) that communicating the velocity and acceleration of the lead vehicle to all subsystems is sufficient for mesh stability. This result is then verified by simulation.

The non-uniform stationary measure for discrete-time quantum walks in one dimension

2015 ◽  
Vol 15 (11&12) ◽  
pp. 1060-1075
Author(s):  
Norio Konno ◽  
Masato Takei
Keyword(s):  
Eigenvalue Problem ◽  
Discrete Time ◽  
Quantum Walks ◽  
Stationary Measure ◽  
Unitary Matrix ◽  
Uniform Measure ◽  
Simple Argument ◽  
Stationary Measures ◽  
Time Quantum ◽  
The One

We consider stationary measures of the one-dimensional discrete-time quantum walks (QWs) with two chiralities, which is defined by a 2 $\times$ 2 unitary matrix $U$. In our previous paper \cite{Konno2014}, we proved that any uniform measure becomes the stationary measure of the QW by solving the corresponding eigenvalue problem. This paper reports that non-uniform measures are also stationary measures of the QW except when $U$ is diagonal. For diagonal matrices, we show that any stationary measure is uniform. Moreover, we prove that any uniform measure becomes a stationary measure for more general QWs not by solving the eigenvalue problem but by a simple argument.

Sign in / Sign up

Export Citation Format

Share Document