Operator Algebraic Framework Research Articles

Quantum Wasserstein distance of order 1 between channels

We set up a general theory leading to a quantum Wasserstein distance of order 1 between channels in an operator algebraic framework. This gives a metric on the set of channels from one composite system to another, which is deeply connected to reductions of the channels. The additivity and stability properties of this metric are studied.

Ginkgo: A Modern Linear Operator Algebra Framework for High Performance Computing

In this article, we presentGinkgo, a modern C++ math library for scientific high performance computing. While classical linear algebra libraries act on matrix and vector objects,Ginkgo’s design principle abstracts all functionality as “linear operators,” motivating the notation of a “linear operator algebra library.”Ginkgo’s current focus is oriented toward providing sparse linear algebra functionality for high performance graphics processing unit (GPU) architectures, but given the library design, this focus can be easily extended to accommodate other algorithms and hardware architectures. We introduce this sophisticated software architecture that separates core algorithms from architecture-specific backends and provide details on extensibility and sustainability measures. We also demonstrateGinkgo’s usability by providing examples on how to use its functionality inside the MFEM and deal.ii finite element ecosystems. Finally, we offer a practical demonstration ofGinkgo’s high performance on state-of-the-art GPU architectures.

Reproduced Computational Results Report for “Ginkgo: A Modern Linear Operator Algebra Framework for High Performance Computing”

The article titled “Ginkgo: A Modern Linear Operator Algebra Framework for High Performance Computing” by Anzt et al. presents a modern, linear operator centric, C++ library for sparse linear algebra. Experimental results in the article demonstrate that Ginkgo is a flexible and user-friendly framework capable of achieving high-performance on state-of-the-art GPU architectures. In this report, the Ginkgo library is installed and a subset of the experimental results are reproduced. Specifically, the experiment that shows the achieved memory bandwidth of the Ginkgo Krylov linear solvers on NVIDIA A100 and AMD MI100 GPUs is redone and the results are compared to what presented in the published article. Upon completion of the comparison, the published results are deemed reproducible.

Higher rank matricial ranges and hybrid quantum error correction

ABSTRACT We introduce and initiate the study of a family of higher rank matricial ranges, taking motivation from hybrid classical and quantum error correction coding theory and its operator algebra framework. In particular, for a noisy quantum channel, a hybrid quantum error correcting code exists if and only if a distinguished special case of the joint higher rank matricial range of the error operators of the channel is non-empty. We establish bounds on Hilbert space dimension in terms of properties of a tuple of operators that guarantee a matricial range is non-empty and hence additionally guarantee the existence of hybrid codes for a given quantum channel. We also discuss when hybrid codes can have advantages over quantum codes and present a number of examples.

The fidelity of density operators in an operator-algebraic framework

Josza’s definition of fidelity [R. Jozsa, J. Mod. Opt. 41(12), 2315–2323 (1994)] for a pair of (mixed) quantum states is studied in the context of two types of operator algebras. The first setting is mainly algebraic in that it involves unital C∗-algebras A that possess a faithful trace functional τ. In this context, the role of quantum states (that is, density operators) in the classical quantum-mechanical framework is assumed by positive elements ρ ∈ A for which τ(ρ) = 1. The second setting is more operator theoretic: by fixing a faithful normal semifinite trace τ on a semifinite von Neumann algebra M, we define and consider the fidelity of pairs of positive operators in M of unit trace. The main results of this paper address monotonicity and preservation of fidelity under the action of certain trace-preserving positive linear maps of A or of the predual M∗. Our results in the von Neumann algebra setting are novel in that we focus on the Schrödinger picture rather than the Heisenberg picture, and they also yield a new proof of a theorem of Molnár [Rep. Math. Phys. 48(3), 299–303 (2001)] on the structure of fidelity-preserving quantum channels on the trace-class operators.

Multicloning and Multibroadcasting in Operator Algebras

We investigate multicloning and multibroadcasting in the general operator algebra framework in arbitrary dimension, generalizing thus results obtained in this framework for simple cloning and broadcasting.

Cloning by positive maps in von Neumann algebras

We investigate cloning in the general operator algebra framework in arbitrary dimension assuming only positivity instead of strong positivity of the cloning operation, generalizing thus results obtained so far under that stronger assumption. The weaker positivity assumption turns out quite natural when considering cloning in the general $$C^*$$ -algebra framework.

Quantum Blackwell–Sherman–Stein Theorem and Related Results

We investigate the problem of comparing quantum statistical models in the general operator algebra framework in arbitrary dimension, thus generalizing results obtained so far in finite dimension, and for the full algebra of operators on a Hilbert space. In particular, the quantum Blackwell–Sherman–Stein theorem is obtained, and informational subordination of quantum information structures is characterized.

Quantum Sufficiency in the Operator Algebra Framework

The paper is devoted to the investigation of the notion of sufficiency in quantum statistics. Three kinds of this notion are considered: plain sufficiency (called simply: sufficiency), Petz’s sufficiency, and Umegaki’s sufficiency. The problem of the existence and structure of the minimal sufficient subalgebra is analyzed in some detail, conditions yielding equivalence of the three modes of sufficiency are considered, and quantum Basu’s theorem is obtained. Moreover, it is shown that an interesting “factorization theorem” of Jencova and Petz needs some corrections to hold true.

Geometry of Interaction V: Logic in the hyperfinite factor

Geometry of Interaction is a transcendental syntax developed in the framework of operator algebras. This fifth installment of the program takes place inside a von Neumann algebra, the hyperfinite factor. It provides a built-in interpretation of cut-elimination as well as an explanation for light, i.e., complexity sensitive, logics.

On Lehner’s ‘free’ noncommutative analogue of de Finetti’s theorem

Inspired by Lehner’s results on exchangeability systems, we define ‘weak conditional freeness’ and ‘conditional freeness’ for stationary processes in an operator algebraic framework of noncommutative probability. We show that these two properties are equivalent, and thus the process embeds into a von Neumann algebraic amalgamated free product over the fixed point algebra of the stationary process.

KMS States and the Chemical Potential for Disordered Systems

We extend the theory of the chemical potential associated to a compact separable gauge group, to the case of disordered quantum systems. This is done in the framework of operator algebras. Among the other results, we show that the chemical potential does not depend on the disorder. The situation of the n–torus is treated in some detail. Indeed, the chemical potential is intrinsically defined in terms of the direct integral decomposition of the Connes–Radon–Nikodym cocycle associated to the (normal extension of the) KMS state ω and its trasforms ω○ρ by the (localized) automorphisms ρ of the observable algebra, carrying the Abelian charges of the model under consideration. This description parallels the analogous one relative to the usual (i.e. non disordered) quantum models.

On the Structure of KMS States of Disordered Systems

We study the set of KMS states of spin systems with random interactions. This is done in the framework of operator algebras by investigating Connes and Borchers Γ–invariants of W*–dynamical systems. In the case of KMS states exhibiting a property of invariance with respect to the spatial translations, some interesting properties emerge naturally. Such a situation covers KMS states obtained by infinite–volume limits of finite–volume Gibbs states, that is the quenched disorder. This analysis can be considered as a step towards fully understanding the very complicated structure of the set of temperature states of quantum spin glasses, and its connection with the breakdown of the symmetry for replicas.

Zeno dynamics in quantum statistical mechanics

We study the quantum Zeno effect in quantum statistical mechanics within the operator algebraic framework. We formulate a condition for the appearance of the effect in W*-dynamical systems, in terms of the short-time behaviour of the dynamics. Examples of quantum spin systems show that this condition can be effectively applied to quantum statistical mechanical models. Further, we derive an explicit form of the Zeno generator, and use it to construct Gibbs equilibrium states for the Zeno dynamics. As a concrete example, we consider the X-Y model, for which we show that a frequent measurement at a microscopic level, e.g. a single lattice site, can produce a macroscopic effect in changing the global equilibrium.

Quasicrystals, aperiodic order, and groupoid von Neumann algebras

We introduce tight binding operators for quasicrystals that are parametrized by Delone sets. These operators can be regarded in a natural operator algebra framework that encodes the long range aperiodic order. This algebraic point of view allows us to study spectral theoretic properties. In particular, the integrated density of states of the tight binding operators is related to a canonical trace on the associated von Neumann algebra. To cite this article: D. Lenz, P. Stollmann, C. R. Acad. Sci. Paris, Ser. I 334 (2002) 1131–1136.

Wavefront Sets in Algebraic Quantum Field Theory

The investigation of wavefront sets of n-point distributions in quantum field theory has recently acquired some attention stimulated by results obtained with the help of concepts from microlocal analysis in quantum field theory in curved spacetime. In the present paper, the notion of wavefront set of a distribution is generalized so as to be applicable to states and linear functionals on nets of operator algebras carrying a covariant action of the translation group in arbitrary dimension. In the case where one is given a quantum field theory in the operator algebraic framework, this generalized notion of wavefront set, called "asymptotic correlation spectrum", is further investigated and several of its properties for physical states are derived. We also investigate the connection between the asymptotic correlation spectrum of a physical state and the wavefront sets of the corresponding Wightman distributions if there is a Wightman field affiliated to the local operator algebras. Finally we present a new result (generalizing known facts) which shows that certain spacetime points must be contained in the singular supports of the 2n-point distributions of a non-trivial Wightman field.

Limit laws for Random matrices and free products

In earlier articles we studied a kind of probability theory in the framework of operator algebras, with the tensor product replaced by the free product. We prove here that free random variables naturally arise as limits of random matrices and that Wigner's semicircle law is a consequence of the central limit theorem for free random variables. In this way we obtain a non-commutative limit distribution of a general gaussian random matrix as an operator in a certain operator algebra, Wigner's law being given by the trace of the spectral measure of the selfadjoint component of this operator

Bell's Inequalities, Relativistic Quantum Field Theory and the Problem of Hidden Variables

Based partly on proving that algebraic relativistic quantum field theory (ARQFT) is a stochastic Einstein local (SEL) theory in the sense of SEL which was introduced by Hellman (1982b) and which is adapted in this paper to ARQFT, the recently proved maximal and typical violation of Bell's inequalities in ARQFT (Summers and Werner 1987a–c) is interpreted in this paper as showing that Bell's inequalities are, in a sense, irrelevant for the problem of Einstein local stochastic hidden variables, especially if this problem is raised in connection with ARQFT. This leads to the question of how to formulate the problem of local hidden variables in ARQFT. By giving a precise definition of hidden-variable theory within the operator algebraic framework of quantum mechanics, it will be argued that the aim of hidden-variable investigations is to determine those classes of quantum theories whose elements represent a statistical content that cannot be reduced in a given way. In some particular way to be stated, a proposition will be stated which distinguishes quantum field theories whose statistical content cannot be reduced without violating some relativistic locality principle.

Conditions excluding the existence of approximate hidden variables

The problem of approximate hidden variables is redefined in the operator algebraic framework of quantum mechanics and a theorem containing negative results is formulated.

Sufficiency, KMS condition and relative entropy in von Neumann algebras

Introduction* Since the investigation of sufficient statistics in abstract measure theoretic terms was initiated by Halmos and Savage (10), the concept of sufficiency has been developed by many mathe- matical statisticians in terms of various relations given by compar- ison of experiments, risk functions within the framework of stati- stical decision problems and so on. A characterization of sufficiency was given in (12) through the measure of Kullback-Leib ler informa- tion. The concept of sufficiency was first generalized by Umegaki (22, 23) to the noncommutative case of semi-finite von Neumann algebras with some extension of the Kullback-Leibler information (usually called the relative entropy). Later the related discussions especially concerning the relative entropy for quantum systems have been made by several authors, e.g., Araki (2, 3), Gudder and Marchand (7), and Lindblad (13). As defined precisely and explained in §§ 1 and 4 of this paper, the concept of sufficiency is more or less considered through the informativity of a certain subalgebra with respect to a given alge- bra for a dynamical system of interest. Namely, in the case that such a subalgebra is sufficient, the relative entropy on the subalgebra is equal to that on the given algebra. This fact may or may not be a reason why the concept of sufficiency has not been entered into analysis of physical systems, in which the change of entropy is thought of more relevant. The Kubo-Martin-Schwinger (KMS) condition was introduced by these three authors (11, 14) as a boundary condition of the thermal Green function. Haag, Hugenholtz and Winnink (8) showed that in the operator algebraic framework this condition is a fundamental one describing thermal equilibrium of quantum systems. The KMS condition through the Tomita-Takesaki theory now becomes a core of studying von Neumann algebras. Under the above historical basis, our main motivation of this