On the Protection Against Noise for Measurement-Based Quantum Computation
Here we establish conditions for some pairs of quantum logic gates which operate on one qubit to be protected against crosstalk.
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arXiv subjects
Valentin Vankov Iliev
Publications and source records attributed to Valentin Vankov Iliev.
The Noiseless Quantum Computer Does Not Exist
In this note we show that any logic gates in a quantum computer is informationally dependent on another quantum logic gate.
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On an Internal Dependence of Simultaneous Measurements
In this paper we show that there exists an internal dependence of the simultaneous measurements made by the two pairs of linear polarizers operated in each leg of the apparatus in Aspect's version of Einstein-Podolsky-Rosen-Bohm Gedankenexperiment. The corresponding Shannon-Kolmogorov's information flow linking a polarizer from one leg to a polarizer from the other leg is proportional to the absolute value of this function of dependence. It turns out that if Bell's inequality is violated, then this information flow is strictly positive, that is, the experiment performed at one leg is informationally dependent on the experiment at the other leg. By throwing out the sign of absolute value, we define the signed information flow linking a polarizer from one leg to a polarizer from the other leg which, in turn, reproduces the probabilities of the four outcomes of the simultaneous measurements, predicted by quantum mechanics. We make an attempt to illustrate the seeming random relation between the total information flow, the total signed information flow, and the violation of Bell's inequality in terms of a kind of uncertainty principle.
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On Markowitz Geometry
By Markowitz geometry we mean the intersection theory of ellipsoids and affine subspaces in a real finite-dimensional linear space. In the paper we give a meticulous and self-contained treatment of this arch-classical subject, which lays a solid mathematical groundwork of Markowitz mean-variance theory of efficient portfolios in economics.
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On the Number of Rumer Diagrams
In this paper we both review the famous article "Eine fur die Valenztheorie geeignete Basis der binaren Vektorinvarianten" of G. Rumer, E. Teller, and H. Weyl on Rumer diagrams and use a powerful old result from algebraic geometry to enumerate these diagrams with given numbers of atoms and valence bonds.
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On a Generalization of Markowitz Preference Relation
Given two families of continuous functions $u$ and $v$ on a topological space $X$, we define a preorder $R=R(u,v)$ on $X$ by the condition that any member of $u$ is an $R$-increasing and any member of $v$ is an $R$-decreasing function. It turns out that if the topological space $X$ is quasi-compact and sequentially compact, then any element of $X$ is $R$-dominated by an $R$-maximal element of $X$. In particular, since the $(n-1)$-dimensional simplex is a compact subset of the real $n$-dimensional vector space, then considering its members as portfolios consisting of $n$ financial assets, we obtain the classical 1952 result of Harry Markowitz that any portfolio is dominated by an efficient portfolio. Moreover, several other examples of possible application of this general setup are presented.
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On the Preference Relations with Negatively Transitive Asymmetric Part. I
Given a linearly ordered set I, every surjective map p: A --> I endows the set A with a structure of set of preferences by "replacing" the elements of I with their inverse images via p considered as "balloons" (sets endowed with an equivalence relation), lifting the linear order on A, and "agglutinating" this structure with the balloons. Every ballooning A of a structure of linearly ordered set I is a set of preferences whose preference relation (not necessarily complete) is negatively transitive and every such structure on a given set A can be obtained by ballooning of certain structure of a linearly ordered set I, intrinsically encoded in A. In other words, the difference between linearity and negative transitivity is constituted of balloons. As a consequence of this characterization, under certain natural topological conditions on the set of preferences A furnished with its interval topology, the existence of a continuous generalized utility function on A is proved.
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On certain permutation representations of the braid group. Part II
In arXiv:0910.1727 we find certain finite homomorphic images of Artin braid group into appropriate symmetric groups, which a posteriori are extensions of the symmetric group on n letters by an abelian group. The main theorem of this paper characterizes completely the extensions of this type that are split.
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A note on certain finite-dimensional representations of the braid group
In this paper the author finds explicitly all finite-dimensional irreducible representations of a series of finite permutation groups that are homomorphic images of Artin braid group.
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On certain permutation representations of the braid group
This paper is devoted to the proof of a structural theorem, concerning certain homomorphic images of Artin braid group on $n$ strands in finite symmetric groups. It is shown that any one of these permutation groups is an extension of the symmetric group on $n$ letters by an appropriate abelian group, and in "half" of the cases this extension splits.
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Semi-symmetric algebras: General Constructions. Part II
In our paper Semi-symmetric Algebras: General Constructions, J. Algebra, 148 (1992), pp. 479-496, we present the construction of the semi-symmetric algebra of a module over a commutative ring with unit, which generalizes the tensor algebra, the symmetric algebra, and the exterior algebra, deduce some of its functorial properties, and prove a classification theorem. In the present paper we continue the study of the semi-symmetric algebra and discuss its graded dual, the corresponding canonical bilinear form, its coalgebra structure, as well as left and right inner products. Here we present a unified treatment of these topics whose exposition in N. Bourbaki, Alg\` ebre, Chapitres 1--3, Hermann, Paris 1970, is made simultaneously for the above three particular (and, without a shadow of doubt - most important) cases.
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On Lunn-Senior's Mathematical Model of Isomerism in Organic Chemistry. Part II
The second part of this paper is devoted to the following important question in organic chemistry: given two isomers of a molecule, how to identify them with their structural formulae using only type properties of that molecule? A classical answer of this question is given for benzene by the identification of its di-substituted (para, ortho, and meta), and tri-substituted (asymmetric, vicinal, and symmetric) derivatives via the Korner substitution reactions among them. Here we develop a machinery within the framework of the Lunn-Senior's mathematical model of isomerism in organic chemistry, which, in principle, answers this question. In particular, it is shown that the members of a chiral pair cannot be distinguished via substitution reactions. The examples of ethene, benzene, and cyclopropane are discussed.
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On certain organic compounds with one mono-substitution and at least three di-substitution homogeneous derivatives
In this paper parent substances with molecules which can be divided into a skeleton and six univalent substituents, and that have the properties mentioned in the title, are considered. Two instances are the molecules of benzene and cyclopropane. The Lunn-Senior's symmetry groups of substitution isomerism of these compounds are described and upper bounds of the numbers of their di-substitution and tri-substitution homogeneous derivatives are found. Lists of the possible simple substitution reactions among di-substitution homogeneous derivatives, on one hand, and di-substitution heterogeneous, and tri-substitution homogeneous derivatives, on the other, are given. These substitution reactions allow for some derivatives to be identified with their structural formulae.
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On Lunn-Senior's Mathematical Model of Isomerism in Organic Chemistry. Part I
The aim of this paper is to present a generalization of Lunn-Senior's mathematical model of isomerism in organic chemistry. The main idea of Lunn and Senior is that if the type of isomerism is fixed, a molecule with a fixed skeleton and d univalent substituents has a symmetry group $W\leq S_d$ which is generally not the molecule's 3-dimensional symmetry group. The unit character of W induces a representation of the symmetric group $S_d$ which governs the combinatorics of the isomers of the given molecule. Lunn-Senior's thesis is that certain non-negative integers established by this representation are upper boundaries of the corresponding numbers, yielded by the experiment (and often coincide with them). Moreover, the authors define (in a particular case) a partial order among the objects of the model, such that some simple substitution reactions correspond to inequalities. These two groups of data determine the group W, and produce so called "type properties" of the molecule (properties which do not depend on the nature of the univalent substituents). Our hypothesis is that if we replace the unit character of $W$ by any one-dimensional character of $W$ (thus we count only a part of the isomers - those having a maximum property), we also get a type property of the molecule. An instance of that is the inventory of the stereoisomers called chiral pairs. The formalism can be generalized naturally and produces some preliminary chemical results. Especially the partial order is defined and studied in the general case and indicates the possible genetic relations among the corresponding molecules. An important result of E. Ruch which connects the dominance order among partitions and the existence of chiral pairs is obtained as a consequence of a more general statement.
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A Generalization of Redfield's Master Theorem
Generalizations of Redfield's master theorem and superposition theorem are proved by using decomposition of the tensor product of several induced monomial representations of the symmetric group $S_d$ into transitive constituents. As direct consequences, one obtains several graphical corollaries. Given graphs $Γ_1,\hdots ,Γ_k$, with $d$ vertices, together with their automorphism groups $W_1\leq S_d,\hdots, W_k\leq S_d$, one can find the number of superpositions of $Γ_1,\hdots ,Γ_k$, whose automorphism groups satisfy one of the following conditions: (1) the groups consist of even permutations; (2) the groups are trivial, in case at least one of $W_m$'s is cyclic; (3) the groups are of odd order, in case at least one of $W_m$'s is dihedral and its order is not divisible by 4; (4) the groups are of order dividing a natural number $r$, in case at least one of $W_m$'s has a normal solvable subgroup of order $r$, such that the corresponding factor-group is cyclic of order relatively prime to $r$; (5) the groups are $q$-groups ($q$ is a prime), in case at least one of $W_m$'s has a normal $q$-subgroup such that the corresponding factor-group is cyclic of order relatively prime to $q$.
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A Generalization of Polya's Enumeration Theorem or the Secret Life of Certain Index Sets
Polya's fundamental enumeration theorem is generalized in terms of Schur-Macdonald's theory (S-MT) of invariant matrices. Given a permutation group $W\leq S_d$ and a one-dimensional character $χ$ of $W$, the polynomial functor $F_χ$ corresponding via S-MT to the induced monomial representation $U_χ= ind_W^{S_d}(χ)$ of $S_d$, is studied. It turns out that the characteristic $ch(F_χ)$ is the weighted inventory of some set $J(χ)$ of $W$-orbits in the integer-valued hypercube $[0,\infty)^d$. The elements of $J(χ) can be distinguished among all $W$-orbits by a maximum property. The identity $ch(F_χ) = ch(U_χ)$ of both characteristics is a consequence of S-MT. Polya's theorem can be obtained from the above identity by specialization $χ=1_W$, where $1_W$ is the unit character of $W$.
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