UHF flows and the flip automorphism

A UHF flow is an infinite tensor product type action of the reals on a UHF algebra $A$ and the flip automorphism is an automorphism of $A\otimes A$ sending $x\otimes y$ into $y\otimes x$. If $\alpha$ is an inner perturbation of a UHF flow on $A$, there is a sequence $(u_n)$ of unitaries in $A\otimes A$ such that $\alpha_t\otimes \alpha_t(u_n)-u_n$ converges to zero and the flip is the limit of \Ad u_n. We consider here whether the converse holds or not and solve it with an additional assumption: If $A\otimes A\cong A$ and $\alpha$ absorbs any UHF flow $\beta$ (i.e., $\alpha\otimes\beta$ is cocycle conjugate to $\alpha$), then the converse holds; in this case $\alpha$ is what we call a universal UHF flow.Comment: 18 page

Cauchy Problem and Green's Functions for First Order Differential Operators and Algebraic Quantization

Existence and uniqueness of advanced and retarded fundamental solutions (Green's functions) and of global solutions to the Cauchy problem is proved for a general class of first order linear differential operators on vector bundles over globally hyperbolic Lorentzian manifolds. This is a core ingredient to CAR-/CCR-algebraic constructions of quantum field theories on curved spacetimes, particularly for higher spin field equations.Comment: revised version: typos; reordering of sec 2; results unchange

The Measure of a Measurement

While finite non-commutative operator systems lie at the foundation of quantum measurement, they are also tools for understanding geometric iterations as used in the theory of iterated function systems (IFSs) and in wavelet analysis. Key is a certain splitting of the total Hilbert space and its recursive iterations to further iterated subdivisions. This paper explores some implications for associated probability measures (in the classical sense of measure theory), specifically their fractal components. We identify a fractal scale $s$ in a family of Borel probability measures $\mu$ on the unit interval which arises independently in quantum information theory and in wavelet analysis. The scales $s$ we find satisfy $s\in \mathbb{R}_{+}$ and $s\not =1$, some $s 1$. We identify these scales $s$ by considering the asymptotic properties of $\mu(J) /| J| ^{s}$ where $J$ are dyadic subintervals, and $| J| \to0$.Comment: 18 pages, 3 figures, and reference

Endomorphism Semigroups and Lightlike Translations

Certain criteria are demonstrated for a spatial derivation of a von Neumann algebra to generate a one-parameter semigroup of endomorphisms of that algebra. These are then used to establish a converse to recent results of Borchers and of Wiesbrock on certain one-parameter semigroups of endomorphisms of von Neumann algebras (specifically, Type III_1 factors) that appear as lightlike translations in the theory of algebras of local observables.Comment: 9 pages, Late

Validity and failure of some entropy inequalities for CAR systems

Basic properties of von Neumann entropy such as the triangle inequality and what we call MONO-SSA are studied for CAR systems. We show that both inequalities hold for any even state. We construct a certain class of noneven states giving counter examples of those inequalities. It is not always possible to extend a set of prepared states on disjoint regions to some joint state on the whole region for CAR systems. However, for every even state, we have its `symmetric purification' by which the validity of those inequalities is shown. Some (realized) noneven states have peculiar state correlations among subsystems and induce the failure of those inequalities.Comment: 14 pages, latex, to appear in JMP. Some typos are correcte

Wavelets in mathematical physics: q-oscillators

We construct representations of a q-oscillator algebra by operators on Fock space on positive matrices. They emerge from a multiresolution scaling construction used in wavelet analysis. The representations of the Cuntz Algebra arising from this multiresolution analysis are contained as a special case in the Fock Space construction.Comment: (03/11/03):18 pages; LaTeX2e, "article" document class with "letterpaper" option An outline was added under the abstract (p.1), paragraphs added to Introduction (p.2), mat'l added to Proofs in Theorems 1 and 6 (pgs.5&17), material added to text for the conclusion (p.17), one add'l reference added [12]. (04/22/03):"number 1" replace with "term C" (p.9), single sentences reformed into a one paragraph (p.13), QED symbol moved up one paragraph and last paragraph labeled as "Concluding Remarks.

Correlations in Free Fermionic States

We study correlations in a bipartite, Fermionic, free state in terms of perturbations induced by one party on the other. In particular, we show that all so conditioned free states can be modelled by an auxiliary Fermionic system and a suitable completely positive map.Comment: 17 pages, no figure

Algorithms for entanglement renormalization

We describe an iterative method to optimize the multi-scale entanglement renormalization ansatz (MERA) for the low-energy subspace of local Hamiltonians on a D-dimensional lattice. For translation invariant systems the cost of this optimization is logarithmic in the linear system size. Specialized algorithms for the treatment of infinite systems are also described. Benchmark simulation results are presented for a variety of 1D systems, namely Ising, Potts, XX and Heisenberg models. The potential to compute expected values of local observables, energy gaps and correlators is investigated.Comment: 23 pages, 28 figure

Iterated function systems, representations, and Hilbert space

This paper studies a general class of Iterated Function Systems (IFS). No contractivity assumptions are made, other than the existence of some compact attractor. The possibility of escape to infinity is considered. Our present approach is based on Hilbert space, and the theory of representations of the Cuntz algebras O_n, n=2,3,.... While the more traditional approaches to IFS's start with some equilibrium measure, ours doesn't. Rather, we construct a Hilbert space directly from a given IFS; and our construction uses instead families of measures. Starting with a fixed IFS S_n, with n branches, we prove existence of an associated representation of O_n, and we show that the representation is universal in a certain sense. We further prove a theorem about a direct correspondence between a given system S_n, and an associated sub-representation of the universal representation of O_n.Comment: 22 pages, 3 figures containing 7 EPS graphics; LaTeX2e ("elsart" document class); v2 reflects change in Comments onl

Continuity bounds on the quantum relative entropy

The quantum relative entropy is frequently used as a distance, or distinguishability measure between two quantum states. In this paper we study the relation between this measure and a number of other measures used for that purpose, including the trace norm distance. More precisely, we derive lower and upper bounds on the relative entropy in terms of various distance measures for the difference of the states based on unitarily invariant norms. The upper bounds can be considered as statements of continuity of the relative entropy distance in the sense of Fannes. We employ methods from optimisation theory to obtain bounds that are as sharp as possible.Comment: 13 pages (ReVTeX), 3 figures, replaced with published versio