arXiv subjects
Publications and source records attributed to Heide Narnhofer.
The Hugenholz-Boltzmann-evolution is generalized to strongly interacting systems on the lattice. Under appropriate assumptions states stable under this evolution are shown to satify the KMS-condition. How far these assumptions are reasonable is discussed.
Generalizing the relation between spin-systems and Fermi-systems on the lattice we construct for a spin-system with dimension d an algebra for which quasifree time-evolutions exist. With appropriate assumptions the gauge invariant subalgebra common for both algebras is invariant under this time-evolution and on this subalgebra is norm-asymptotically abelian.
We argue that for Fermi systems on lattices or the continuum with interaction invariant under a kind of Galilei transformation the time evolution is either weakly asymptotically abelian or at least $η$-abelian in the tracial state but not norm asymptotically abelian.
We study the behaviour of continuous automorphism groups of quantum spin systems on the lattice. Whereas the shift is norm asymptotically abelian continuous automorphism groups can lead only to delocalization but not to norm asymptotic abelianess. As a consequence the shift does not allow a continuous extension.
We study the continuous extension of discrete shift translations on one-dimensional quantum lattice systems.
For quantum lattice systems a Boltzmann-type evolution arrises according to results of Hugenholtz in the limit of N-scaled time evolution together with an interaction scaled as N^-1/2. According ti Illner-Neunzert this passage to an irreversible dynamics can only happen when the initial state satisfies stronger assumptions than the evolved state. We show that the initial state satisfies space clustering as it is necessary for the construction of a fluctuation algebra whereas the evolved state allows fluctuations only in an appropriate local scaling but not on a mesoscopic level.
We construct the fluctuation algebra for fermions in a quasifree state and its timedependence for quasifree evolutions. We find a Bose-Einstein-condensate and study its stability under interaction.
It is shown that under the assumption of the nuclearity condition for local regions the resulting Doplicher-Longo-algebra between two double cones which due to nuclearity is type I allows to estimate its entropy by nuclearity bounds.
For Fermi systems interacting with a Galilei invariant pair potential with a cut-off for particles with highly different velocities the time evolution corresponds to an automorphism. We prove that all states satisfying the KMS-condition are locally normal with respect to the representation in Fock-space. Removing the cut-off limit states remain locally normal provided the interaction is repulsive or of positive type. The modular automorphism of the von Neumann-algebra of the limit state coincides with the time evolution.
We argue that for Fermi systems with Galilei invariant interaction the time evolution is weakly asymptotically abelian in time invariant states but not norm asymptotically abelian.Consequences for the existence of invariant states are discussed.
We study the emergent dynamics resulting from the infinite volume limit of the mean-field dissipative dynamics of quantum spin chains with clustering, but not time-invariant states. We focus upon three algebras of spin operators: the commutative algebra of mean-field operators, the quasi-local algebra of microscopic, local operators and the collective algebra of fluctuation operators. In the infinite volume limit, mean-field operators behave as time-dependent, commuting scalar macroscopic averages while quasi-local operators, despite the dissipative underlying dynamics, evolve unitarily in a typical non-Markovian fashion. Instead, the algebra of collective fluctuations, which is of bosonic type with time-dependent canonical commutation relations, undergoes a time-evolution that retains the dissipative character of the underlying microscopic dynamics and exhibits non-linear features. These latter disappear by extending the time-evolution to a larger algebra where it is represented by a continuous one-parameter semigroup of completely positive maps. The corresponding generator is not of Lindblad form and displays mixed quantum-classical features, thus indicating that peculiar hybrid systems may naturally emerge at the level of quantum fluctuations in many-body quantum systems endowed with non time-invariant states.
We study the dissipative dynamics of $N$ quantum spins with Lindblad generator consisting of operators scaling as fluctuations, namely with the inverse square-root of $N$. In the large $N$ limit, the microscopic dissipative time-evolution converges to a non-Markovian unitary dynamics on strictly local operators, while at the mesoscopic level of fluctuations it gives rise to a dissipative non-Markovian dynamics. The mesoscopic time-evolution is Gaussian and exhibits either a stable or an unstable asymptotic character; furthermore, the mesoscopic dynamics builds correlations among fluctuations that survive in time even when the original microscopic dynamics is unable to correlate local observables.
We trace back the phenomenon of "delayed-choice entanglement swapping" as it was realized in a recent experiment to the commutativity of the projection operators that are involved in the corresponding measurement process. We also propose an experimental set-up which depends on the order of successive measurements corresponding to noncommutative projection operators. In this case entanglement swapping is used to teleport a quantum state from Alice to Bob, where Bob has now the possibility to examine the noncommutativity within the quantum history.
Each semigroup describing the time evolution of an open quantum system on a finite dimensional Hilbert space is related to a special structure of this space. It is shown how the space can be decomposed into subspaces: One is related to decay, orthogonal subspaces support the stationary states. Specialities where the complete positivity of evolutions is actually needed for analysis, mainly for evolution of coherence, are highlighted. Decompositions are done the same way for evolutions in discrete as in continuous time, but evolutions may show differences, only for discrete semigroups there may appear cases of sudden decay and of perpetual oscillation. Concluding the analysis we identify the relation of the state space structure to the processes of Decay, Decoherence, Dissipation and Dephasing.
We discuss the concept of how entanglement changes with respect to different factorizations of the total algebra which describes the quantum states. Depending on the considered factorization a quantum state appears either entangled or separable. For pure states we always can switch unitarily between separability and entanglement, however, for mixed states a minimal amount of mixedness is needed. We discuss our general statements in detail for the familiar case of qubits, the GHZ states, Werner states and Gisin states, emphasizing their geometric features. As theorists we use and play with this free choice of factorization, which is naturally fixed for an experimentalist. For theorists it offers an extension of the interpretations and is adequate to generalizations, as we point out in the examples of quantum teleportation and entanglement swapping.
We consider an open bipartite quantum system with dissipative Lindblad type dynamics. In order to study the entanglement of the stationary states, we develop a perturbative approach and apply it to the physically significant case when a purely dissipative perturbation is added to the unperturbed generator which by itself would produce reversible unitary dynamics.
Let $ω$ be a factor state on the quasi-local algebra $\cal{A}$ of observables generated by a relativistic quantum field, which in addition satisfies certain regularity conditions (satisfied by ground states and the recently constructed thermal states of the $P(ϕ)_2$ theory). We prove that there exist space and time translation invariant states, some of which are arbitrarily close to $ω$ in the weak* topology, for which the time evolution is weakly asymptotically abelian.
We construct an algebra with twisted commutation relations and equip it with the shift. For appropriate irregularity of the non-local commutation relations we prove that the tracial state is the only translation-invariant state.