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A QUANTIZATION PROCEDURE BASED ON COMPLETELY POSITIVE MAPS AND MARKOV OPERATORS

Carlos F. Lardizabal

Published Aug 16, 2016• 2 citations
OpenAlex

Abstract

We describe $ω$-limit sets of completely positive (CP) maps over finite-dimensional spaces. In such sets and in its corresponding convex hulls, CP maps present isometric behavior and the states contained in it commute with each other. Motivated by these facts, we describe a quantization procedure based on CP maps which are induced by Markov (transfer) operators. Classical dynamics are described by an action over essentially bounded functions. A non-expansive linear map, which depends on a choice of a probability measure, is the centerpiece connecting phenomena over function and matrix spaces.

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We describe ωωω-limit sets of completely positive (CP) maps over finite-dimensional spaces. In such sets and in its corresponding convex hulls, CP maps present isometric behavior and the states contained in it commute with each other. Motivated by these facts, we describe a quantization procedure based on CP maps which are induced by Markov (transfer) operators. Classical dynamics are described by an action over essentially bounded functions. A non-expansive linear map, which depends on a choice of a probability measure, is the centerpiece connecting phenomena over function and matrix spaces.
A well-known theorem of W. Arveson states that a completely positive (CP) map dominated (difference is CP) by a given CP map is described through a positive contraction in the commutant of the homomorphism of Stinespring representation. This contraction plays the role of Radon-Nikodym derivative of measure theory. We look at implications of this to one parameter semigroups of unital completely positive maps, known as quantum Markov semigroups. We focus our attention to quantum Markov semigroups of B(H), the algebra of all bounded operators on a Hilbert space. These semigroups can be dilated to semigroups of unital endomorphisms (E₀-semigroups) of B(K) for some other Hilbert space K. Here we see that dominated CP semigroups dilate to dominated semigroups of the dilation. Moreover at the level of Eo-semigroups the dominated CP semigroups are described through positive contractive local cocycles, which are really families of contractions of 'Radon-Nikodym derivatives' mentioned above. We analyze the structure for quantum semigroups with bounded generators.
Let {o^,..., a) n ] and {<TI,..., a n ] be finite sets of normal states over the bounded linear operators acting over some infinite-dimensional separable Hilbert-space.In the paper mutually equivalent sets of necessary and sufficient conditions are derived that there exists a completely positive, unity preserving linear map Ttransforming 0-1,..., a n simultaneously into o)i..., o) n : a) k = a k ° T,k = l,... 9 n.Of particular interest is this 6 '/z-tuple-problem" in case of pairs of density operators (normal states), i.e. for n=2.In this situation possible connections with the notion of "generalized transition probability" are analyzed, and at least in case of normal states over the bounded operators a characterization of functionals of this type is proposed and applied. § lo Main Results, ExamplesLet jtf be a unital C*-algebra ? with unit 7, topological dual jtf*, and group of unitary elements <%(j&).To each Fe ^(j/) let us associate a bounded linear operator T v over the dual by T ¥ (a))(X) = co(VXV*) = co v (X), for all XEJ^ and any COG jtf*.On the set of bounded linear operators ^(j/*) over j/* we may introduce the weak operator topology (w-topology) which is characterized by the system of semi-norms {q^xi coe.s/*,XEJ/} given by q (0 ,x(T) = \T(a>)(X)\, with Te^(X*).Then, by % u (<xf} the weak closure of the convex hull of the operators T v will be denoted: ^M(j^) = conv{T F : VE<%(JZ?)} W .In this paper as an essential result the following theorem is proved (proofs of most of the assertions of this part are given In § §3-5): LI Ttoeoremo Let 3? be an infinite-dimensional separable complex
We investigate the set (a) of positive, trace preserving maps acting on density matrices of size N and a sequence of its nested subsets: the sets of maps which are (b) decomposable, (c) completely positive, and (d) extended by identity impose positive partial transpose and (e) are superpositive. Working with the Hilbert–Schmidt (Euclidean) measure, we derive tight explicit two-sided bounds for the volumes of all five sets. A sample consequence is the fact that, as N increases, a generic positive map becomes not decomposable and, a fortiori, not completely positive. Due to the Jamiołkowski isomorphism, the results obtained for quantum maps are closely connected to similar relations between the volume of the set of quantum states and the volumes of its subsets (such as states with positive partial transpose or separable states) or supersets. Our approach depends on the systematic use of duality to derive quantitative estimates and on various tools of classical convexity, high-dimensional probability, and geometry of Banach spaces, some of which are not standard.
We investigate some particular completely positive maps which admit a stable commutative Von Neumann subalgebra. The restriction of such maps to the stable algebra is then a Markov operator. In the first part of this article, we propose a recipe in order to find a quantum extension of a given Markov operator in the above sense. We show that the existence of such an extension is linked with the existence of a special form of dilation for the Markov operator studied by Attal in \cite{Att1}, reducing the problem to the extension of dynamical system. We then apply our method to the same problem in continuous time, proving the existence of a quantum extension for Lévy processes. In the second part of this article, we focus on the case where the commutative algebra is isomorphic to $\Acal=l^\infty(1,...,N)$ with NNN either finite or infinite. We propose a classification of the CP maps leaving $\Acal$ stable, producing physical examples of each classes.
We develop a theory of integration with respect to quantum instruments through two complementary approaches: a vector measure formulation based on Bartle's integration theory and a tensor product construction. As a principal application, we establish a CP map-valued Lyapunov theorem by characterizing the convexity of the range of non-atomic completely positive instruments via the associated integration map. This extends the work of Plosker and Ramsey~\cite{Plosker_Ramsey} from POVMs to completely positive instruments and gives a general characterization of the phenomenon exhibited in their setting. We also establish a correspondence between completely positive instruments and completely positive maps, prove a Krein--Milman type theorem for the C∗C^*C∗-convex set of unital completely positive instruments, and show that the tensor product construction connects the theory of CP instruments with the integration theory for POVMs due to Farenick \emph{et al}(\cite{douglus_plosker_ramsey_povmintegration_1}).