In this paper, we introduce Schwartz operators as a non-commutative analog of Schwartz functions and provide a detailed discussion of their properties. We equip them, in particular, with a number of different (but equivalent) families of seminorms which turns the space of Schwartz operators into a Frechet space. The study of the topological dual leads to non-commutative tempered distributions which are discussed in detail as well. We show, in particular, that the latter can be identified with a certain class of quadratic forms, therefore making operations like products with bounded (and also some unbounded) operators and quantum harmonic analysis available to objects which are otherwise too singular for being a Hilbert space operator. Finally, we show how the new methods can be applied by studying operator moment problems and convergence properties of fluctuation operators.

Quantum walks have by now been realized in a large variety of different physical settings. In some of these, particularly with trapped ions, the walk is implemented in phase space, where the corresponding position states are not orthogonal. We develop a general description of such a quantum walk and show how to map it into a standard one with orthogonal states, thereby making available all the tools developed for the latter. This enables a variety of experiments, which can be implemented with smaller step sizes and more steps. Tuning the nonorthogonality allows for an easy preparation of extended states such as momentum eigenstates, which travel at a well-defined speed with low dispersion. We introduce a method to adjust their velocity by momentum shifts, which allows us to experimentally probe the dispersion relation, providing a benchmarking tool for the quantum walk, and to investigate intriguing effects such as the analog of Bloch oscillations.

In this note, we consider quantum spin chains and their translationally invariant pure states. We prove Haag duality for quasilocal observables localized in semi-infinite intervals (-∞ , 0] and [1, ∞) when the von Neumann algebra generated by observables localized in [0, ∞) is non-type I.

We review the quantum version of a well known problem of cryptography called coin tossing (" flipping a coin via telephone "). It can be regarded as a game where two remote players (who distrust each other) tries to generate a uniformly distributed random bit which is common to both parties. The only resource they can use to perform this task is a classical or quantum communication channel. In this paper we provide a general overview over such coin tossing protocols, concerning in particular their security.

We consider an infinite spin chain as a bipartite system consisting of the left and right half-chain and analyze entanglement properties of pure states with respect to this splitting. In this context we show that the amount of entanglement contained in a given state is deeply related to the von Neumann type of the observable algebras associated to the halfchains. Only the type I case belongs to the usual entanglement theory which deals with density operators on tensor product Hilbert spaces, and only in this situation separable normal states exist. In all other cases the corresponding state is infinitely entangled in the sense that one copy of the system in such a state is sufficient to distill an infinite amount of maximally entangled qubit pairs. We apply this results to the critical XY model and show that its unique ground state φS provides a particular example for this type of entanglement.

We consider an infinite spin chain as a bipartite system consisting of the left and right half-chains and analyze entanglement properties of pure states with respect to this splitting. In this context, we show that the amount of entanglement contained in a given state is deeply related to the von Neumann type of the observable algebras associated to the half-chains. Only the type I case belongs to the usual entanglement theory which deals with density operators on tensor product Hilbert spaces, and only in this situation separable normal states exist. In all other cases, the corresponding state is infinitely entangled in the sense that one copy of the system in such a state is sufficient to distill an infinite amount of maximally entangled qubit pairs. We apply this results to the critical XY model and show that its unique ground state φ S provides a particular example for this type of entanglement.

In this paper we propose a method to estimate the density matrix ρ of a d-level quantum system by measurements on the N-fold system in the joint state ρ⊗N. The scheme is based on covariant observables and representation theory of unitary groups and it extends previous results concerning pure states and the estimation of the spectrum of ρ. We show that it is consistent (i.e. the original input state ρ is recovered with certainty if N → ∞), analyze its large deviation behavior, and calculate explicitly the corresponding rate function which describes the exponential decrease of error probabilities in the limit N → ∞. Finally, we discuss the question whether the proposed scheme provides the fastest possible decay of error probabilities.

In quantum mechanics the statistics of the outcomes of a measuring apparatus is described by a positive operator valued measure (POVM). A quantum channel transforms POVMs into POVMs, generally irreversibly, thus losing some of the information retrieved from the measurement. This poses the problem of which POVMs are “undisturbed,” i.e., they are not irreversibly connected to another POVM. We will call such POVMs clean. In a sense, the clean POVMs would be “perfect,” since they would not have any additional “extrinsical” noise. Quite unexpectedly, it turns out that such a “cleanness” property is largely unrelated to the convex structure of POVMs, and there are clean POVMs that are not extremal and vice versa. In this article we solve the cleannes classification problem for number n of outcomes n⩽d (d dimension of the Hilbert space), and we provide a set of either necessary or sufficient conditions for n>d, along with an iff condition for the case of informationally complete POVMs for n=d2.

For states in infinite dimensional Hilbert spaces entanglement quantities like the entanglement of distillation can become infinite. This leads naturally to the question, whether one system in such an infinitely entangled state can serve as a resource for tasks like the teleportation of arbitrarily many qubits. We show that appropriate states cannot be obtained by density operators in an infinite dimensional Hilbert space. However, using techniques for the description of infinitely many degrees of freedom from field theory and statistical mechanics, such states can nevertheless be constructed rigorously. We explore two related possibilities, namely an extended notion of algebras of observables, and the use of singular states on the algebra of bounded operators. As applications we construct the essentially unique infinite analogue of maximally entangled states, and the singular state used heuristically in the fundamental paper of Einstein, Rosen and Podolsky.

In this paper we give a self-contained introduction to the conceptional and mathematical foundations of quantum information theory. In the first part we introduce the basic notions like entanglement, channels, teleportation, etc. and their mathematical description. The second part is focused on a presentation of the quantitative aspects of the theory. Topics discussed in this context include: entanglement measures, channel capacities, relations between both, additivity and continuity properties and asymptotic rates of quantum operations. Finally, we give an overview on some recent developments and open questions.

We consider a quantum version of a well-known statistical decision problem, whose solution is, at first sight, counter-intuitive to many. In the quantum version a continuum of possible choices (rather than a finite set) has to be considered. It can be phrased as a two person game between a player P and a quiz master Q. Then P always has a strategy at least as good as in the classical case, while Q's best strategy results in a game having the same value as the classical game. We investigate the consequences of Q storing his information in classical or quantum ways. It turns out that Q's optimal strategy is to use a completely entangled quantum notepad, on which to encode his prior information.

We review the quantum version of a well known problem of cryptography called coin tossing ("flipping a coin via telephone"). It can be regarded as a game where two remote players (who distrust each other) try to generate a uniformly distributed random bit which is common to both parties. The only resource they can use to perform this task is a classical or quantum communication channel. In this paper we provide a general overview over such coin tossing protocols, concerning in particular their security.

The theory of quantum error correction is a cornerstone of quantum information processing. It shows that quantum data can be protected against decoherence effects, which otherwise would render many of the new quantum applications practically impossible. In this paper we give a self contained introduction to this theory and to the closely related concept of quantum channel capacities. We show, in particular, that it is possible (using appropriate error correcting schemes) to send a nonvanishing amount of quantum data undisturbed (in a certain asymptotic sense) through a noisy quantum channel T , provided the errors produced by T are small enough. This text is part of a volume entitled: “Coherent evolution in noisy environments” to be published in Lecture notes in physics, Springer Verlag, http://link.springer.de/series/lnpp/, Copyright: Springer Verlag, Berlin, Heidelberg, New York Electronic Mail: m.keyl@tu-bs.de Electronic Mail: r.werner@tu-bs.de

Given N quantum systems prepared according to the same density operator \rho, we propose a measurement on the N-fold system which approximately yields the spectrum of \rho. The projections of the proposed observable decompose the Hilbert space according to the irreducible representations of the permutations on N points, and are labeled by Young frames, whose relative row lengths estimate the eigenvalues of \rho in decreasing order. We show convergence of these estimates in the limit N\to\infty, and that the probability for errors decreases exponentially with a rate we compute explicitly.

Purification is a process in which decoherence is partially reversed by using several input systems which have been subject to the same noise. The purity of the outputs generally increases with the number of input systems, and decreases with the number of required output systems. We construct the optimal quantum operations for this task, and discuss their asymptotic behaviour as the number of inputs goes to infinity. The rate at which output systems may be generated depends crucially on the type of purity requirement. If one tests the purity of the output systems one at a time, the rate is infinite: this fidelity may be made to approach 1, while at the same time the number of outputs goes to infinity arbitrarily fast. On the other hand, if one also requires the correlations between outputs to decrease, the rate is zero: if fidelity with the pure product state is to go to 1, the number of outputs per input goes to zero. However, if only a fidelity close to 1 is required, the optimal purifier achieves a positive rate, which we compute.

Trends in Quantum Mechanics, pp. 1-300 (2000) No AccessTRENDS IN QUANTUM MECHANICSProceedings of the International SymposiumH.-D. Doebner, S.T. Ali, M. Keyl, and R.F. WernerH.-D. DoebnerTechnical University of Clausthal, Germany, S.T. AliConcordia University, Canada, M. KeylTechnical University of Braunschweig, German, and R.F. WernerTechnical University of Braunschweig, Germanhttps://doi.org/10.1142/9789814527071Cited by:0 AboutSectionsPDF/EPUB ToolsAdd to favoritesDownload CitationsTrack CitationsRecommend to Library ShareShare onFacebookTwitterLinked InRedditEmail Abstract: The Table of Contents for the full book PDF is as follows: Preface List of Participants Quantization and Deformations Quantization on the Cylinder Some Basic Properties of Nonstandard Deformations The Coulomb-Oscillator Relation on the N-Sphere Wigner Quantization and All That Relationships between q-Deformations, Typical Length Scales and Lower Measurability Bounds Description of Kerr States via Deformed Bosons Quantum Mechanics on Phase Spaces ZN × ZN Measurement: Theory and Experiment Continuous Fuzzy Measurement of Energy: Realization and Application Decoherence and the Final Pointer Basis On Hybrid Dynamics of the Copenhagen Dichotomic World Reality, Viewed from Quantum Mechanics Storage and Read-Out of Quantum-State Information via Interference Theoretical Challenges in Atom Optics: Atomic and Molecular Diffraction by Transmission Gratings ls There a Gravitational Collapse of the Wave-Packet? Iteration of Non-Destructive Measurements Operators and Maps Affiliated to EPR Channels Reconstruction of Quantum States and Its Conceptual Implications Time Asymmetry and Decay Spectral Characterization of Decay in Quantum Mechanics The Phenomenological Preparation - Registration Arrow of Time and Its Semigroup Representation in the RHS Quantum Theory Three Complimentary Descriptions of a Resonance Resonances, Rigged Hilbert Spaces and Irreversible Quantum Mechanics Nonlinear Quantum Mechanics Geometric Formulation of Nonlinear Quantum Mechanics for Density Matrices Fundamental Principles of Quantum Mechanics and Non(linearity) Nonlinear von Neumann-Type Equations Some Aspects of Nonlinearity and Gauge Transformation in Quantum Mechanics Nonlocality in Nonlinear Quantum Mechanics Quantum Field Theory On a Theorem of Ashtekar and Lewandowski in the Mathematical Framework of Canonical Quantization in Quantum Gravity The Fuzzy (Super)sphere and Field Theory Quantum Fields along Worldlines Field Theory Revisited Basic Quantum Theory and Measurement from the Viewpoint of Local Quantum Physics FiguresReferencesRelatedDetails Trends in Quantum MechanicsMetrics History PDF download

Observers and reference frames are central notions in general relativity. In this paper it is discussed how statements on quantum field theories on a curved space-time can be related to an observer. The bacis idea is to characterize an observer by those observables he can observe.