Universal quantum control in irreducible state-space sectors: application to bosonic and spin-boson systems

We analyze the dynamical-algebraic approach to universal quantum control introduced in P. Zanardi, S. Lloyd, quant-ph/0305013. The quantum state-space $\cal H$ encoding information decomposes into irreducible sectors and subsystems associated to the group of available evolutions. If this group coincides with the unitary part of the group-algebra \CC{\cal K} of some group $\cal K$ then universal control is achievable over the ${\cal K}$-irreducible components of $\cal H$. This general strategy is applied to different kind of bosonic systems. We first consider massive bosons in a double-well and show how to achieve universal control over all finite-dimensional Fock sectors. We then discuss a multi-mode massless case giving the conditions for generating the whole infinite-dimensional multi-mode Heisenberg-Weyl enveloping-algebra. Finally we show how to use an auxiliary bosonic mode coupled to finite-dimensional systems to generate high-order non-linearities needed for universal control.Comment: 10 pages, LaTeX, no figure

Quantum Chains of Hopf Algebras with Quantum Double Cosymmetry

Given a finite dimensional C^*-Hopf algebra H and its dual H^ we construct the infinite crossed product A=... x H x H^ x H ... and study its superselection sectors in the framework of algebraic quantum field theory. A is the observable algebra of a generalized quantum spin chain with H-order and H^-disorder symmetries, where by a duality transformation the role of order and disorder may also appear interchanged. If H=\CC G is a group algebra then A becomes an ordinary G-spin model. We classify all DHR-sectors of A --- relative to some Haag dual vacuum representation --- and prove that their symmetry is described by the Drinfeld double D(H). To achieve this we construct localized coactions \rho: A \to (A \otimes D(H)) and use a certain compressibility property to prove that they are universal amplimorphisms on A. In this way the double D(H) can be recovered from the observable algebra A as a universal cosymmetrty.Comment: Latex, 48 pages, no figures, extended version of hep-th/9507174, but without the field algebra construction, contains full proofs of the slightly shortened article published in Commun.Math.Phys., the revision only concerns some misprints and an update of the literatur

Quantum Mechanics On Spaces With Finite Fundamental Group

We consider in general terms dynamical systems with finite-dimensional, non-simply connected configuration-spaces. The fundamental group is assumed to be finite. We analyze in full detail those ambiguities in the quantization procedure that arise from the non-simply connectedness of the classical configuration space. We define the quantum theory on the universal cover but restrict the algebra of observables \O to the commutant of the algebra generated by deck-transformations. We apply standard superselection principles and construct the corresponding sectors. We emphasize the relevance of all sectors and not just the abelian ones.Comment: 40 Pages, Plain-TeX, no figure

Representations of Conformal Nets, Universal C*-Algebras and K-Theory

We study the representation theory of a conformal net A on the circle from a K-theoretical point of view using its universal C*-algebra C*(A). We prove that if A satisfies the split property then, for every representation \pi of A with finite statistical dimension, \pi(C*(A)) is weakly closed and hence a finite direct sum of type I_\infty factors. We define the more manageable locally normal universal C*-algebra C*_ln(A) as the quotient of C*(A) by its largest ideal vanishing in all locally normal representations and we investigate its structure. In particular, if A is completely rational with n sectors, then C*_ln(A) is a direct sum of n type I_\infty factors. Its ideal K_A of compact operators has nontrivial K-theory, and we prove that the DHR endomorphisms of C*(A) with finite statistical dimension act on K_A, giving rise to an action of the fusion semiring of DHR sectors on K_0(K_A)$. Moreover, we show that this action corresponds to the regular representation of the associated fusion algebra.Comment: v2: we added some comments in the introduction and new references. v3: new authors' addresses, minor corrections. To appear in Commun. Math. Phys. v4: minor corrections, updated reference

Representations of conformal nets, universal C*-algebras and K-theory

We study the representation theory of a conformal net A on the circle from a K-theoretical point of view using its universal C*-algebra C*(A). We prove that if A satisfies the split property then, for every representation pi of A with finite statistical dimension, pi(C*(A)) is weakly closed and hence a finite direct sum of type I_infty factors. We define the more manageable locally normal universal C*-algebra C*_ln(A) as the quotient of C*(A) by its largest ideal vanishing in all locally normal representations and we investigate its structure. In particular, if A is completely rational with n sectors, then C*_ln(A) is a direct sum of n type I_infty factors. Its ideal K_A of compact operators has nontrivial K-theory, and we prove that the DHR endomorphisms of C*(A) with finite statistical dimension act on K_A, giving rise to an action of the fusion semiring of DHR sectors on K_0(K_A)$. Moreover, we show that this action corresponds to the regular representation of the associated fusion algebra