Partial Commutation Research Articles

Entanglement Dynamics Governed by Time-Dependent Quantum Generators

In the article, we investigate entanglement dynamics defined by time-dependent linear generators. We consider multilevel quantum systems coupled to an environment that induces decoherence and dissipation, such that the relaxation rates depend on time. By applying the condition of partial commutativity, one can precisely describe the dynamics of selected subsystems. More specifically, we investigate the dynamics of entangled states. The concurrence is used to quantify the amount of two-qubit entanglement in the time domain. The framework appears to be an efficient tool for investigating quantum evolution of entangled states driven by time-local generators. In particular, non-Markovian effects can be included to observe the restoration of entanglement in time.

Classification of diagnostic features of transient signals in the electric power industry

Problems of practical implementation of traveling wave fault location caused by the registration of signals of different nature are considered. Analysis of the experimentally recorded traveling waves made it possible to divide them into 4 groups caused by partial discharges, lightning overvoltages, scheduled switching and fault commutations. The network dispatchers only needs the fault commutation information. Traveling waves recorded near the place of their origin have different meanings of diagnostic signs. The magnitude of the pre-alarm noise, the number of pulses in the signal and the duration of the signal are used as diagnostic indicators. These three diagnostic signs allow one to recognize each of the 4 causes of the travelling waves.

Open quantum systems integrable by partial commutativity

This article provides a framework to solve linear differential equations based on partial commutativity, which is introduced by means of the Fedorov theorem. The framework is applied to specific types of three-level and four-level quantum systems. The efficiency of the method is evaluated and discussed. The Fedorov theorem appears to answer the need for methods that allow us to study dynamical maps corresponding with time-dependent generators. By applying this method, one can investigate countless examples of dissipative systems such that the relaxation rates depend on time.

Right Angled Artin Groups and partial commutation, old and new

We systematically treat algebraic objects with free partially commuting generators and give short and modern proofs of the various relations between them. These objects include right angled Artin groups, polynomial rings, Lie algebras, and restricted Lie algebras in partially commuting free generators. In particular, we compute the p -central and exponent- p series of all right angled Artin groups, and compute the dimensions of their subquotients. We also describe their associated Lie algebras, and relate them to the cohomology ring of the group as well as to polynomial and power series rings in partially commuting variables. We finally show how the growth series of these various objects are related to each other.

Quantum groups with partial commutation relations

We define new noncommutative spheres with partial commutation relations for the coordinates. We investigate the quantum groups acting maximally on them, which yields new quantum versions of the orthogonal group: They are partially commutative in a way such that they do not interpolate between the classical and the free quantum versions of the orthogonal group. Likewise we define non-interpolating, partially commutative quantum versions of the symmetric group recovering Bichon's quantum automorphism groups of graphs. They fit with the mixture of classical and free independence as recently defined by Speicher and Wysoczanski (rediscovering $\Lambda$-freeness of Mlotkowski), due to some weakened version of a de Finetti theorem.

The Generalized Rank of Trace Languages

Given a partially commutative alphabet and a set of words [Formula: see text], the rank of [Formula: see text] expresses the amount of shuffling required to produce a word belonging to [Formula: see text] from two words whose concatenation belongs to the closure of [Formula: see text] with respect to the partial commutation. In this paper, the notion of rank is generalized from concatenations of two words to an arbitrary fixed number of words. In this way, an infinite sequence of non-negative integers and infinity is assigned to every set of words. It is proved that in the case of alphabets defining free commutative monoids, as well as in the more general case of direct products of free monoids, sequences of ranks of regular sets are exactly non-decreasing sequences that are eventually constant. On the other hand, by uncovering a relationship between rank sequences of regular sets and rational series over the min-plus semiring, it is shown that already for alphabets defining free products of free commutative monoids, rank sequences need not be eventually periodic.

Quantum symmetries on noncommutative complex spheres with partial commutation relations

We introduce the notion of noncommutative complex spheres with partial commutation relations for the coordinates. We compute the corresponding quantum symmetry groups of these spheres, and this yields new quantum unitary groups with partial commutation relations. We also discuss some geometric aspects of the quantum orthogonal groups associated with the mixture of classical and free independence discovered by Speicher and Weber. We show that these quantum groups are quantum symmetry groups on some quantum spaces of spherical vectors with partial commutation relations.

Mixtures of classical and free independence

We revive the concept of \({\Lambda}\)-freeness of Mlotkowski (Quant Prob Relat Top 7:27–41, 2004), which describes a mixture of classical and free independence between algebras of random variables. In particular, we give a description of this in terms of cumulants; this will be instrumental in the subsequent paper (Speicher and Weber, Quantum groups with partial commutation relations, 2016) where the quantum symmetries underlying these mixtures of classical and free independences will be considered.

Session typing and asynchronous subtyping for the higher-order π-calculus

This paper proposes a session typing system for the higher-order π-calculus (the HOπ-calculus) with asynchronous communication subtyping, which allows partial commutativity of actions in higher-order processes. The system enables two complementary kinds of optimisation, mobile code and asynchronous permutation of session actions, within processes that utilise structured, typed communications. Our first contribution is a session typing system for the HOπ-calculus using techniques from the linear λ-calculus. Integration of arbitrary higher-order code mobility and sessions leads to technical difficulties in type soundness, because linear usage of session channels and completion of sessions are required. Our second contribution is to introduce an asynchronous subtyping system which uniformly deals with type-manifested asynchrony and linear functions. The most technical challenge for subtyping is to prove the transitivity of the subtyping relation. We also demonstrate the expressiveness of our typing system with an e-commerce example, where optimised processes can interact respecting the expected sessions.

Regular languages and partial commutations

The closure of a regular language under a [partial] commutation I has been extensively studied. We present new advances on two problems of this area: (1) When is the closure of a regular language under [partial] commutation still regular? (2) Are there any robust classes of languages closed under [partial] commutation? We show that the class Pol(G) of polynomials of group languages is closed under commutation, and under partial commutation when the complement of I in A2 is a transitive relation. We also give a sufficient graph theoretic condition on I to ensure that the closure of a language of Pol(G) under I-commutation is regular. We exhibit a very robust class of languages W which is closed under commutation. This class contains Pol(G), is decidable and can be defined as the largest positive variety of languages not containing (ab)⁎. It is also closed under intersection, union, shuffle, concatenation, quotients, length-decreasing morphisms and inverses of morphisms. If I is transitive, we show that the closure of a language of W under I-commutation is regular. The proofs are nontrivial and combine several advanced techniques, including combinatorial Ramsey type arguments, algebraic properties of the syntactic monoid, finiteness conditions on semigroups and properties of insertion systems.

The Tutte-Grothendieck Group of an Alphabetic Rewriting System

The two operations, deletion and contraction of an edge, on multigraphs directly lead to the Tutte polynomial which satisfies a universal problem. As observed by Brylawski (1972) in terms of order relations, these operations may be interpreted as a particular instance of a general theory which involves universal invariants like the Tutte polynomial and a universal group, called the Tutte-Grothendieck group. In this contribution, Brylawski’s theory is extended in two ways: first of all, the order relation is replaced by a string rewriting system, and secondly, commutativity by partial commutations (that permits a kind of interpolation between noncommutativity and full commutativity). This allows us to clarify the relations between the semigroup subject to rewriting and the Tutte-Grothendieck group: the latter is actually the Grothendieck group completion of the former, up to the free adjunction of a unit (this was not even mentioned by Brylawski), and normal forms may be seen as universal invariants. Moreover we prove that such universal constructions are also possible in case of a nonconvergent rewriting system, outside the scope of Brylawski’s work.

ON APPLICATIONS OF PI-ALGEBRAS IN THE ANALYSIS OF QUANTUM CHANNELS

In this paper, we discuss some constructive procedures which can be used in characterizations of linear transformations which preserve the set of states of a fixed quantum system. Our methods are based on analyzing an explicit form of a linear positive map in its Kraus representation. In particular, we discuss the so-called partial commutativity of operators and its applications to investigation of decoherence-free subspaces. These subspaces can also be considered as a special class of quantum error correcting codes. Using the concept of standard polynomials and Amitsur–Levitzki theorem and other ideas from the so-called polynomial identity algebras (PI-algebras) we discuss some effective algorithms for analyzing properties of quantum channels.

Rotation-invariant t-norms: The rotation invariance property revisited

Focusing on conjunctive, left-continuous, increasing [ 0 , 1 ] 2 → [ 0 , 1 ] functions T we redefine the rotation invariance property in terms of contour lines. Under the assumption of the existence of a neutral element e ∈ ] 0 , 1 ] , this rotation invariance property requires some partial commutativity and associativity. The functions that are rotation invariant w.r.t. all of their contour lines are characterized.

Partially commutative inverse monoids

Free partially commutative inverse monoids are investigated. As in the case of free partially commutative monoids or groups (trace monoids or graph groups), free partially commutative inverse monoids are defined as quotients of free inverse monoids modulo a partially defined commutation relation on the generators. A quasi linear time algorithm for the word problem is presented. More precisely, we give an \(\mathcal {O}(n\log(n))\) algorithm for a RAM. \(\mathsf {NP}\) -completeness of the submonoid membership problem (also known as the generalized word problem) and the membership problem for rational sets is shown. Moreover, free partially commutative inverse monoids modulo a finite idempotent presentation are studied. It turns out that the word problem is decidable if and only if the complement of the partial commutation relation is transitive.

Semi-commutations and Partial commutations

The aim of this paper is to show that a semi-commutation function can be expressed as the compound of a sequential transformation, a partial commutation function, and the reverse transformation. Moreover, we give a necessary and sufficient condition for the image of a regular language to be computed by the compound of two sequential functions and a partial commutation function.

Synchronization languages

Synchronization expressions introduced in [7] within the framework of the ParC project are a high-level construct which allow a programmer to express minimal synchronization constraints of a program in a distributed context. The study and the implementation of these expressions are based on their associated synchronization languages. Synchronization languages have been introduced in [9, 10]. They give a way to implement synchronization expressions and specify their semantic. These languages represent distributed systems whose behaviour respects synchronization constraints expressed by the programmer with synchronization expressions. So these languages describe all the correct executions of a program. In [9, 10] Guo, Salomaa and Yu propose a characterization of synchronization languages based on a rewriting system named R which generalizes partial commutations. This system gives a way to rewrite a word representing a parallel execution into a word with a lower or equal degree of parallelism. Guo et al. show that every synchronization language is closed under the system R and they conjecture that it is sufficient for a regular language to be closed under R to be a synchronization language. We show that the conjecture is true in the particular case of languages expressing the synchronization between two distinct actions. We also show that the conjecture is false in the general case.

Normal Forms for Basis-Conjugating Automorphisms of a Free Group

International Journal of Algebra and ComputationVol. 08, No. 06, pp. 631-669 (1998) No AccessNormal Forms for Basis-Conjugating Automorphisms of a Free GroupMauricio Gutiérrez and Sava KrstićMauricio GutiérrezDepartment of Mathematics, Tufts University, Medford, MA 02155, USA and Sava KrstićDepartment of Mathematics, Tufts University, Medford, MA 02155, USAhttps://doi.org/10.1142/S0218196798000314Cited by:6 PreviousNext AboutSectionsPDF/EPUB ToolsAdd to favoritesDownload CitationsTrack CitationsRecommend to Library ShareShare onFacebookTwitterLinked InRedditEmail AMSC: 20F32, 20F28, 20F38 Remember to check out the Most Cited Articles! Check out Algebra & Computation books in the Mathematics 2021 catalogue. FiguresReferencesRelatedDetailsCited By 6Finite-type invariants of w-knotted objects, I: w-knots and the Alexander polynomialDror Bar-Natan and Zsuzsanna Dancso26 April 2016 | Algebraic & Geometric Topology, Vol. 16, No. 2AUTOMORPHISMS OF PARTIALLY COMMUTATIVE GROUPS II: COMBINATORIAL SUBGROUPSANDREW J. DUNCAN and VLADIMIR N. REMESLENNIKOV3 December 2012 | International Journal of Algebra and Computation, Vol. 22, No. 07NORMAL FORMS FOR AUTOMORPHISMS OF UNIVERSAL COXETER GROUPS AND PALINDROMIC AUTOMORPHISMS OF FREE GROUPSADAM PIGGOTT and KIM RUANE30 April 2012 | International Journal of Algebra and Computation, Vol. 20, No. 08PROPER ACTIONS OF AUTOMORPHISM GROUPS OF FREE PRODUCTS OF FINITE GROUPSYUQING CHEN, HENRY H. GLOVER, and CRAIG A. JENSEN20 November 2011 | International Journal of Algebra and Computation, Vol. 15, No. 02A rational noncommutative invariant of boundary linksStavros Garoufalidis and Andrew Kricker8 February 2004 | Geometry & Topology, Vol. 8, No. 1The Pure Symmetric Automorphisms of a Free Group Form a Duality GroupNoel Brady, Jon McCammond, John Meier and Andy Miller1 Dec 2001 | Journal of Algebra, Vol. 246, No. 2 Recommended Vol. 08, No. 06 Metrics History Received 8 August 1997 PDF download

A note on the commutative closure of star-free languages

We show that the commutative closure of a star-free language is either star-free or not regular anymore. Actually, this property is shown to hold exactly for the closure with respect to a partial commutation corresponding to a transitive dependence relation. Moreover, the question whether the closure of a star-free language remains star-free is decidable precisely for transitive partial commutation relations.

∏ 11-universality of some propositional logics of concurrent programs

It is proved that validity problems for two variants of propositional dynamic logic (PDL) connected with concurrent programming are highly undecidable (∏ 1 1-universal). These variants are an extension of PDL by asynchronous programming constructs shuffle and iterated shuffle and a variant of PDL with a partial commutativity relation on primitive programs. In both cases propositional variables are not used.

On the Starheight of Some Rational Subsets Closed under Partial Commutations

Let A be a finite alphabet, A * the free monoid it generates, and θ = θ −1 ⊆ A × A a relation of partial commutations. Denote by ≍ the congruence on the free monoid generated by the relators ab ≍ ba for all ( a , b )∈θ. It has been shown that if a word w ∈ A * contains all letters of A then [ w *] = { u ∈ A *| u ≍ w n for some n ≥ 0} is a rational subset if and only if the graph of the complement θ of θ is connected. We prove:(1) if θ is not connected then [ w *] has starheight one (2) if θ and θ are connected then for all integers n there exists a word w∈A* of length O(n) such that [w*] is a rational subset of starheight n.