Pauli graphs when the Hilbert space dimension contains a square: why the Dedekind psi function ?

We study the commutation relations within the Pauli groups built on all decompositions of a given Hilbert space dimension $q$, containing a square, into its factors. Illustrative low dimensional examples are the quartit ($q=4$) and two-qubit ($q=2^2$) systems, the octit ($q=8$), qubit/quartit ($q=2\times 4$) and three-qubit ($q=2^3$) systems, and so on. In the single qudit case, e.g. $q=4,8,12,...$, one defines a bijection between the $\sigma (q)$ maximal commuting sets [with $\sigma[q)$ the sum of divisors of $q$] of Pauli observables and the maximal submodules of the modular ring $\mathbb{Z}_q^2$, that arrange into the projective line $P_1(\mathbb{Z}_q)$ and a independent set of size $\sigma (q)-\psi(q)$ [with $\psi(q)$ the Dedekind psi function]. In the multiple qudit case, e.g. $q=2^2, 2^3, 3^2,...$, the Pauli graphs rely on symplectic polar spaces such as the generalized quadrangles GQ(2,2) (if $q=2^2$) and GQ(3,3) (if $q=3^2$). More precisely, in dimension $p^n$ ($p$ a prime) of the Hilbert space, the observables of the Pauli group (modulo the center) are seen as the elements of the $2n$-dimensional vector space over the field $\mathbb{F}_p$. In this space, one makes use of the commutator to define a symplectic polar space $W_{2n-1}(p)$ of cardinality $\sigma(p^{2n-1})$, that encodes the maximal commuting sets of the Pauli group by its totally isotropic subspaces. Building blocks of $W_{2n-1}(p)$ are punctured polar spaces (i.e. a observable and all maximum cliques passing to it are removed) of size given by the Dedekind psi function $\psi(p^{2n-1})$. For multiple qudit mixtures (e.g. qubit/quartit, qubit/octit and so on), one finds multiple copies of polar spaces, ponctured polar spaces, hypercube geometries and other intricate structures. Such structures play a role in the science of quantum information.Comment: 18 pages, version submiited to J. Phys. A: Math. Theo

On the Veldkamp Space of GQ(4, 2)

The Veldkamp space, in the sense of Buekenhout and Cohen, of the generalized quadrangle GQ(4, 2) is shown not to be a (partial) linear space by simply giving several examples of Veldkamp lines (V-lines) having two or even three Veldkamp points (V-points) in common. Alongside the ordinary V-lines of size five, one also finds V-lines of cardinality three and two. There, however, exists a subspace of the Veldkamp space isomorphic to PG(3, 4) having 45 perps and 40 plane ovoids as its 85 V-points, with its 357 V-lines being of four distinct types. A V-line of the first type consists of five perps on a common line (altogether 27 of them), the second type features three perps and two ovoids sharing a tricentric triad (240 members), whilst the third and fourth type each comprises a perp and four ovoids in the rosette centered at the (common) center of the perp (90). It is also pointed out that 160 non-plane ovoids (tripods) fall into two distinct orbits -- of sizes 40 and 120 -- with respect to the stabilizer group of a copy of GQ(2, 2); a tripod of the first/second orbit sharing with the GQ(2, 2) a tricentric/unicentric triad, respectively. Finally, three remarkable subconfigurations of V-lines represented by fans of ovoids through a fixed ovoid are examined in some detail.Comment: 6 pages, 7 figures; v2 - slightly polished, subsection on fans of ovoids and three figures adde

Qudits of composite dimension, mutually unbiased bases and projective ring geometry

The $d^2$ Pauli operators attached to a composite qudit in dimension $d$ may be mapped to the vectors of the symplectic module $\mathcal{Z}_d^{2}$ ($\mathcal{Z}_d$ the modular ring). As a result, perpendicular vectors correspond to commuting operators, a free cyclic submodule to a maximal commuting set, and disjoint such sets to mutually unbiased bases. For dimensions $d=6,~10,~15,~12$, and 18, the fine structure and the incidence between maximal commuting sets is found to reproduce the projective line over the rings $\mathcal{Z}_{6}$, $\mathcal{Z}_{10}$, $\mathcal{Z}_{15}$, $\mathcal{Z}_6 \times \mathbf{F}_4$ and $\mathcal{Z}_6 \times \mathcal{Z}_3$, respectively.Comment: 10 pages (Fast Track communication). Journal of Physics A Mathematical and Theoretical (2008) accepte

Spatial Analyses of the Flow of Slaughter Livestock in 1955 and 1960

In this study attention is focused on the spatial aspects of slaughter livestock movements from production to slaughtering. Given the regional levels of production, slaughtering and the costs of moving one unit of various types of slaughter livestock from any one region to another region, this study is concerned with ascertaining the regional price differentials, and the volume and direction of regional imports and exports that are consistent with minimizing the total cost of moving the livestock from production to slaughter. In addition, questions about the consequences of changes in the existing structure of the livestock economy may be evaluated with regard to their impact on regional prices and slaughter livestock flows

Joint Spatial Analysis of Regional Slaughter and the Flows and Pricing of Livestock and Meat

The purpose of this study is to (1) develop a model to handle the simultaneous solution for the processing and flow problem, (2) develop estimates of slaughtering capacity for cattle and hogs in each region, and (3) apply the model using estimates of regional levels of production, regional levels of consumption, regional slaughtering capacities, and transportation costs of live slaughter animals and meats. Attention is focused at determining what regional levels of slaughter and directions and levels of interregional livestock and meat flows satisfy the regional production consumption, and capacity constraints and make the total cost of transportation of live slaughter animals and meat a minimum. The analysis is broadened to also obtain the impacts of alternative regional slaughter capacity restrictions and .regional differences in the labor cost of slaughtering livestock

Spatial Analyses of the Meat Marketing Sector in 1955 and 1960

The livestock products sector is a complex composed of the activities of production, farm marketing, slaughtering, distribution and consumption. The level of each of these activities varies spatially and thus regional imbalances are generated which make necessary product flows between the geographical areas. Within this setting this study is concerned with an interregional analysis of the livestock meat sector of the U. S. economy. Thus, spatial slaughter-consumption relations will be basic observations for this analysis. In this study regional demands are reflected by price dependent demand relations or specific estimates of consumption. Regional supplies are dressed carcass weights of livestock slaughter within the regions. In particular for the beef, pork, veal, and lamb and mutton sectors for the years 1955 and 1960 answers will be sought to the following questions: 1. What are the levels of regional demand for each of these meat products? 2. What are the levels of regional supply for each of these products? 3. What is the aggregate interregional trade for each meat product for each year? 4. For each commodity and for each year, what regions import, export or do neither? 5. What are the levels of regional exports and imports for each region, commodity and year? 6. What is the optimum volume and direction of trade between all possible pairs of regions for each commodity and each year? 7. What are the optimum price differentials between regions for each commodity and year? 8. What is the total transport cost for the aggregate trade of each commodity and year? 9. What is the impact of alternative ways of estimating regional meat consumption on the interregional flows and price differentials? In the following pages the results that are generated by these questions will be given and the implications and uses of the results will be discussed

Mermin's Pentagram as an Ovoid of PG(3,2)

Mermin's pentagram, a specific set of ten three-qubit observables arranged in quadruples of pairwise commuting ones into five edges of a pentagram and used to provide a very simple proof of the Kochen-Specker theorem, is shown to be isomorphic to an ovoid (elliptic quadric) of the three-dimensional projective space of order two, PG(3,2). This demonstration employs properties of the real three-qubit Pauli group embodied in the geometry of the symplectic polar space W(5,2) and rests on the facts that: 1) the four observables/operators on any of the five edges of the pentagram can be viewed as points of an affine plane of order two, 2) all the ten observables lie on a hyperbolic quadric of the five-dimensional projective space of order two, PG(5,2), and 3) that the points of this quadric are in a well-known bijective correspondence with the lines of PG(3,2).Comment: 5 pages, 4 figure

Observation of secondary instability of 2/1 magnetic island in compass high density limit plasmas

Density limit disruptions (DLDs) have been observed in tokamak plasmas when high density regimes are explored. The DLDs are harmless in small size tokamaks like COMPASS,larger tokamaks like JET try to avoid them and they are extremely undesirable in ITER sizetokamaks due to the severe structural damages they can cause. It is very important to understand the dynamics of the DLDs so that better strategies to ameliorate or avoid them can bedeveloped. In this work, following detection in JET [1] of a secondary instability (SI) to thewell-known m/n = 2/1 MHD mode (where m and n are the poloidal and toroidal mode numbers, respectively) in the precursor of DLD, we analyse the evolution of the 2/1 magnetic islandin COMPASS DLD to look for the presence of this SI just close to the onset of energy quenchphase of the disruption. The presence of this SI to the magnetic island was associated with theoccurrence of minor disruptions preceding the major disruption and with the major disruptionitself in [1]. The coherence observed between the perturbations caused by the SI in the magneticpoloidal flux and in the electron temperature was very high (above 0.9), allowing to determinethat the SI perturbations came from the same position as the magnetic island. In the work presented here, only the perturbations in the magnetic poloidal flux are analysed since at the time ofthe experiments in COMPASS, no diagnostics was operational for measuring the time evolutionof the electron temperature with high time rate.Nonlinear MHD numerical simulations have also shown that island deformation during itsrapid growth can lead to the secondary magnetic island formation [2]. A recent review [3] ofthe theory of current sheet formation that leads to magnetic reconnection discusses the role ofplasmoids during magnetic island evolution. Since the validity ranges of the mentioned theoretical works are not directly comparable to the experimental conditions, one cannot claimwith certainty that the SI observed in JET [1] and in COMPASS disruptions (reported here)are the same as observed in those numerical works [2, 3]. However, there are some qualitative43rd EPS Conference on Plasma Physics P5.003similarities between them.The main COMPASS [4] diagnostics used for the analysis in the present work, are the threetoroidally separated arrays (A at 32.5◦, B at 212.5◦and C at 257.5◦from the vessel axis) ofMirnov coils (MCs), each with 24 MCs located poloidally. The MC arrays A and C, toroidallyseparated by 135◦, measure the change in poloidal magnetic flux, dBp/dt. The MC array B,toroidally separated by 180◦to the array A, measures the poloidal magnetic field, Bp. Thesemagnetic sensors have good responsivity to high frequency (up to 1 MHz)

Finite Projective Spaces, Geometric Spreads of Lines and Multi-Qubits

Given a (2N - 1)-dimensional projective space over GF(2), PG(2N - 1, 2), and its geometric spread of lines, there exists a remarkable mapping of this space onto PG(N - 1, 4) where the lines of the spread correspond to the points and subspaces spanned by pairs of lines to the lines of PG(N - 1, 4). Under such mapping, a non-degenerate quadric surface of the former space has for its image a non-singular Hermitian variety in the latter space, this quadric being {\it hyperbolic} or {\it elliptic} in dependence on N being {\it even} or {\it odd}, respectively. We employ this property to show that generalized Pauli groups of N-qubits also form two distinct families according to the parity of N and to put the role of symmetric operators into a new perspective. The N=4 case is taken to illustrate the issue.Comment: 3 pages, no figures/tables; V2 - short introductory paragraph added; V3 - to appear in Int. J. Mod. Phys.