11,070 research outputs found

    Lie pairs

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    Extending the theory of systems, we introduce a theory of Lie semialgebra ``pairs'' which parallels the classical theory of Lie algebras, but with a ``null set'' replacing 0. A selection of examples is given. These Lie pairs comprise two categories in addition to the universal algebraic definition, one with ``weak Lie morphisms'' preserving null sums, and the other with ``⪯-morphisms'' preserving a surpassing relation ⪯ that replaces equality. We provide versions of the PBW (Poincare-Birkhoff-Witt) Theorem in these three categories

    Unbounded generators of dynamical semigroups

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    The time evolution of a closed quantum system was described quite early in the history of quantum physics. These dynamics are reversible, and the time evolution is implemented by a continuous unitary group, which is in turn generated by a selfadjoint Hamiltonian operator. So, we have a complete mathematical characterization of all such evolutions. For open quantum systems the time evolution is given by dynamical semigroups. In the case of uniform continuity the generator of the dynamical semigroup is a bounded operator in the famous GKLS-form that has been found by V. Gorini, A. Kossakowski, G. Sudarshan and, independently, G. Lindblad. But the problem of characterizing also the merely strongly continuous dynamical semigroups or, equivalently, their unbounded generators, is open. In the first part of this thesis we introduce a standard formfor the generator of quantum dynamical semigroups that is an unbounded version of the GKLS-form. The basis of the standard form are so-called no-event semigroups, describing an evolution of a quantum system, that maps pure states to multiples of pure states, and completely positive perturbations of their generator that correspond to jumps in this evolution, like absorption by a measurement device. We will give examples of standard semigroups, which appear to be probability preserving to first order (i.e., when looking only at the generator on the finite-rank part of its domain) but not for finite times. Additionally, we construct examples of generators not of standard form by modifying the previous examples. In the second part we relate the notion of standardness toW. Arveson’s classification of endomorphism semigroups. He divided them into three classes, Type I, Type II and Type III. We show that a conservative dynamical semigroup is standard if and only if the minimal dilation of its adjoint is of Type I. The key feature is the set of ketbras in the domain of the no-event generator and whether it is a core for the standard generator. With this knowledge, we suggest to extend this classification to (not necessarily conservative) semigroups that are standard or can be constructed as a series of completely positive perturbations of a no-event semigroup. By construction these are either of Type I or Type II

    Opening the system to the environment: new theories and tools in classical and quantum settings

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    The thesis is organized as follows. Section 2 is a first, unconventional, approach to the topic of EPs. Having grown interest in the topic of combinatorics and graph theory, I wanted to exploit its very abstract and mathematical tools to reinterpret something very physical, that is, the EPs in wave scattering. To do this, I build the interpretation of scattering events from a graph theory perspective and show how EPs can be understood within this interpretation. In Section 3, I move from a completely classical treatment to a purely quantum one. In this section, I consider two quantum resonators coupled to two baths and study their dynamics with local and global master equations. Here, the EPs are the key physical features used as a witness of validity of the master equation. Choosing the wrong master equation in the regime of interest can indeed mask physical and fundamental features of the system. In Section 4, there are no EPs. However I transition towards a classical/quantum framework via the topic of open systems. My main contribution in this work is the classical stochastic treatment and simulation of a spin coupled to a bath. In this work, I show how a natural quantum--to--classical transition occurs at all coupling strengths when certain limits of spin length are taken. As a key result, I also show how the coupling to the environment in this stochastic framework induces a classical counterpart to quantum coherences in equilibrium. After this last topic, in Section 5, I briefly present the key features of the code I built (and later extended) for the latter project. This, in the form of a Julia registry package named SpiDy.jl, has seen further applications in branching projects and allows for further exploration of the theoretical framework. Finally, I conclude with a discussion section (see Sec. 5) where I recap the different conclusions gathered in the previous sections and propose several possible directions.Engineering and Physical Sciences Research Council (EPSRC

    Engineering four-qubit fuel states for protecting quantum thermalization machine from decoherence

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    This research was funded by the Personal Research Fund of Tokyo International University, Turkish Academy of Sciences (TÜBA)-Outstanding Young Scientist Award (GEBİP), and the Research Fund of the Istanbul Technical University with project codes: MGA-2022-43528, MDK-2021-42849.Decoherence is a major issue in quantum information processing, degrading the performance of tasks or even precluding them. Quantum error-correcting codes, creating decoherence-free subspaces, and the quantum Zeno effect are among the major means for protecting quantum systems from decoherence. Increasing the number of qubits of a quantum system to be utilized in a quantum information task as a resource expands the quantum state space. This creates the opportunity to engineer the quantum state of the system in a way that improves the performance of the task and even to protect the system against decoherence. Here, we consider a quantum thermalization machine and four-qubit atomic states as its resource. Taking into account the realistic conditions such as cavity loss and atomic decoherence due to ambient temperature, we design a quantum state for the atomic resource as a classical mixture of Dicke and W states. We show that using the mixture probability as the control parameter, the negative effects of the inevitable decoherence on the machine performance almost vanish. Our work paves the way for optimizing resource systems consisting of a higher number of atoms.TÜBATürkiye Bilimler AkademisiIstanbul Teknik ÜniversitesiPublisher's VersionWOS:00115302250000

    Quantum-Classical hybrid systems and their quasifree transformations

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    The focus of this work is the description of a framework for quantum-classical hybrid systems. The main emphasis lies on continuous variable systems described by canonical commutation relations and, more precisely, the quasifree case. Here, we are going to solve two main tasks: The first is to rigorously define spaces of states and observables, which are naturally connected within the general structure. Secondly, we want to describe quasifree channels for which both the Schrödinger picture and the Heisenberg picture are well defined. We start with a general introduction to operator algebras and algebraic quantum theory. Thereby, we highlight some of the mathematical details that are often taken for granted while working with purely quantum systems. Consequently, we discuss several possibilities and their advantages respectively disadvantages in describing classical systems analogously to the quantum formalism. The key takeaway is that there is no candidate for a classical state space or observable algebra that can be put easily alongside a quantum system to form a hybrid and simultaneously fulfills all of our requirements for such a partially quantum and partially classical system. Although these straightforward hybrid systems are not sufficient enough to represent a general approach, we use one of the candidates to prove an intermediate result, which showcases the advantages of a consequent hybrid ansatz: We provide a hybrid generalization of classical diffusion generators where the exchange of information between the classical and the quantum side is controlled by the induced noise on the quantum system. Then, we present solutions for our initial tasks. We start with a CCR-algebra where some variables may commute with all others and hence generate a classical subsystem. After clarifying the necessary representations, our hybrid states are given by continuous characteristic functions, and the according state space is equal to the state space of a non-unital C*-algebra. While this C*-algebra is not a suitable candidate for an observable algebra itself, we describe several possible subsets in its bidual which can serve this purpose. They can be more easily characterized and will also allow for a straightforward definition of a proper Heisenberg picture. The subsets are given by operator-valued functions on the classical phase space with varying degrees of regularity, such as universal measurability or strong*-continuity. We describe quasifree channels and their properties, including a state-channel correspondence, a factorization theorem, and some basic physical operations. All this works solely on the assumption of a quasifree system, but we also show that the more famous subclass of Gaussian systems fits well within this formulation and behaves as expected

    Exploring the Limits of Controlled Markovian Quantum Dynamics with Thermal Resources

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    Our aim is twofold: First, we rigorously analyse the generators of quantum-dynamical semigroups of thermodynamic processes. We characterise a wide class of GKSL-generators for quantum maps within thermal operations and argue that every infinitesimal generator of (a one-parameter semigroup of) Markovian thermal operations belongs to this class. We completely classify and visualise them and their non-Markovian counterparts for the case of a single qubit. Second, we use this description in the framework of bilinear control systems to characterise reachable sets of coherently controllable quantum systems with switchable coupling to a thermal bath. The core problem reduces to studying a hybrid control system ("toy model") on the standard simplex allowing for two types of evolution: (i) instantaneous permutations and (ii) a one-parameter semigroup of dd-stochastic maps. We generalise upper bounds of the reachable set of this toy model invoking new results on thermomajorisation. Using tools of control theory we fully characterise these reachable sets as well as the set of stabilisable states as exemplified by exact results in qutrit systems.Comment: 46 pages mai

    On the additive structure of algebraic valuations of polynomial semirings

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    In this paper, we study factorizations in the additive monoids of positive algebraic valuations N0[α]\mathbb{N}_0[\alpha] of the semiring of polynomials N0[X]\mathbb{N}_0[X] using a methodology introduced by D. D. Anderson, D. F. Anderson, and M. Zafrullah in 1990. A cancellative commutative monoid is atomic if every non-invertible element factors into irreducibles. We begin by determining when N0[α]\mathbb{N}_0[\alpha] is atomic, and we give an explicit description of its set of irreducibles. An atomic monoid is a finite factorization monoid (FFM) if every element has only finitely many factorizations (up to order and associates), and it is a bounded factorization monoid (BFM) if for every element there is a bound for the number of irreducibles (counting repetitions) in each of its factorizations. We show that, for the monoid N0[α]\mathbb{N}_0[\alpha], the property of being a BFM and the property of being an FFM are equivalent to the ascending chain condition on principal ideals (ACCP). Finally, we give various characterizations for N0[α]\mathbb{N}_0[\alpha] to be a unique factorization monoid (UFM), two of them in terms of the minimal polynomial of α\alpha. The properties of being finitely generated, half-factorial, and length-factorial are also investigated along the way.Comment: 20 page

    Sumsets and Veronese varieties

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    In this paper, to any subset AZn\mathcal{A} \subset \mathbb{Z}^n we explicitly associate a unique monomial projection Yn,dAY_{n, d_{\mathcal{A}}} of a Veronese variety, whose Hilbert function coincides with the cardinality of the tt-fold sumsets tAt \mathcal{A}. This link allows us to tackle the classical problem of determining the polynomial pAQ[t]p_{\mathcal{A}} \in \mathbb{Q}[t] such that tA=pA(t)|t \mathcal{A}|=p_{\mathcal{A}}(t) for all tt0t \geq t_0 and the minimum integer n0(A)t0n_0(\mathcal{A}) \leq t_0 for which this condition is satisfied, i.e. the so-called phase transition of tA|t \mathcal{A}|. We use the Castelnuovo-Mumford regularity and the geometry of Yn,dAY_{n, d_{\mathcal{A}}} to describe the polynomial pA(t)p_{\mathcal{A}}(t) and to derive new bounds for n0(A)n_0(\mathcal{A}) under some technical assumptions on the convex hull of A\mathcal{A}; and vice versa we apply the theory of sumsets to obtain geometric information of the varieties Yn,dAY_{n, d_{\mathcal{A}}}

    Matrix theory for independence algebras

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    Funding: This work was funded by national funds through the FCT - Fundação para a Ciência e a Tecnologia, I.P., under the scope of the projects UIDB/00297/2020, UIDP/00297/2020 (Center for Mathematics and Applications) and PTDC/MAT/PUR/31174/2017.A universal algebra  with underlying set A is said to be a matroid algebra if ⟨A, ⟨•⟩⟩ where ⟨•⟩ denotes the operator subalgebra generated by, is a matroid. A matroid algebra is said to be an independence algebra if every mapping α : X → A defined on a minimal generating set X of  can be extended to an endomorphism of . These algebras are particularly well-behaved generalizations of vector spaces, and hence they naturally appear in several branches of mathematics, such as model theory, group theory, and semigroup theory. It is well known that matroid algebras have a well-defined notion of dimension. Let  be any independence algebra of finite dimension n, with at least two elements. Denote by End() the monoid of endomorphisms of . In the 1970s, Glazek proposed the problem of extending the matrix theory for vector spaces to a class of universal algebras which included independence algebras. In this paper, we answer that problem by developing a theory of matrices for (almost all) finite-dimensional independence algebras. In the process of solving this, we explain the relation between the classification of independence algebras obtained by Urbanik in the 1960s, and the classification of finite independence algebras up to endomorphism-equivalence obtained by Cameron and Szabo in 2000. (This answers another question by experts on independence algebras.) We also extend the classification of Cameron and Szabo to all independence algebras. The paper closes with a number of questions for experts on matrix theory, groups, semigroups, universal algebra, set theory or model theory.PostprintPeer reviewe

    The Tracy-Singh product of solutions of the Yang-Baxter equation

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    Let VV and VV' be vector spaces over the same field with dim(V)=n\operatorname{dim}(V )=n and dim(V)=m\operatorname{dim}(V')=m. Let c:VVVVc:V \otimes V \rightarrow V \otimes V and d:VVVVd:V'\otimes V' \rightarrow V' \otimes V' be solutions of the Yang-Baxter equation. We show that the Tracy-Singh (or block Kronecker) product of the matrices cc and dd with a particular partition into blocks of cc and dd is the representing matrix of a solution of the Yang-Baxter equation, cd~:WWWW\tilde{ c_d}:\mathcal{ W} \otimes \mathcal{ W} \rightarrow \mathcal{ W} \otimes \mathcal{ W} , with dim(W)=nm\operatorname{dim}(\mathcal{ W} )=nm. Iteratively, it is possible to construct from cc and dd an infinite family of solutions of the Yang-Baxter equation.Comment: arXiv admin note: text overlap with arXiv:2212.1380
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