Quantum Channels and Representation Theory

In the study of d-dimensional quantum channels $(d \geq 2)$, an assumption which is not very restrictive, and which has a natural physical interpretation, is that the corresponding Kraus operators form a representation of a Lie algebra. Physically, this is a symmetry algebra for the interaction Hamiltonian. This paper begins a systematic study of channels defined by representations; the famous Werner-Holevo channel is one element of this infinite class. We show that the channel derived from the defining representation of SU(n) is a depolarizing channel for all $n$, but for most other representations this is not the case. Since the Bloch sphere is not appropriate here, we develop technology which is a generalization of Bloch's technique. Our method works by representing the density matrix as a polynomial in symmetrized products of Lie algebra generators, with coefficients that are symmetric tensors. Using these tensor methods we prove eleven theorems, derive many explicit formulas and show other interesting properties of quantum channels in various dimensions, with various Lie symmetry algebras. We also derive numerical estimates on the size of a generalized ``Bloch sphere'' for certain channels. There remain many open questions which are indicated at various points through the paper.Comment: 28 pages, 1 figur

Lie Algebras and Suppression of Decoherence in Open Quantum Systems

Since there are many examples in which no decoherence-free subsystems exist (among them all cases where the error generators act irreducibly on the system Hilbert space), it is of interest to search for novel mechanisms which suppress decoherence in these more general cases. Drawing on recent work (quant-ph/0502153) we present three results which indicate decoherence suppression without the need for noiseless subsystems. There is a certain trade-off; our results do not necessarily apply to an arbitrary initial density matrix, or for completely generic noise parameters. On the other hand, our computational methods are novel and the result--suppression of decoherence in the error-algebra approach without noiseless subsystems--is an interesting new direction.Comment: 7 page

On the number of representations providing noiseless subsystems

This paper studies the combinatoric structure of the set of all representations, up to equivalence, of a finite-dimensional semisimple Lie algebra. This has intrinsic interest as a previously unsolved problem in representation theory, and also has applications to the understanding of quantum decoherence. We prove that for Hilbert spaces of sufficiently high dimension, decoherence-free subspaces exist for almost all representations of the error algebra. For decoherence-free subsystems, we plot the function $f_d(n)$ which is the fraction of all $d$-dimensional quantum systems which preserve $n$ bits of information through DF subsystems, and note that this function fits an inverse beta distribution. The mathematical tools which arise include techniques from classical number theory.Comment: 17 pp, 4 figs, accepted for Physical Review

Vacuum Geometry of the N=2 Wess-Zumino Model

We give a mathematically rigorous construction of the moduli space and vacuum geometry of a class of quantum field theories which are N=2 supersymmetric Wess-Zumino models on a cylinder. These theories have been proven to exist in the sense of constructive quantum field theory, and they also satisfy the assumptions used by Vafa and Cecotti in their study of the geometry of ground states. Since its inception, the Vafa-Cecotti theory of topological-antitopological fusion, or tt* geometry, has proven to be a powerful tool for calculations of exact quantum string amplitudes. However, tt* geometry postulates the existence of certain vector bundles and holomorphic sections built from the ground states. Our purpose in the present article is to give a mathematical proof that this postulate is valid within the context of the two-dimensional N=2 supersymmetric Wess-Zumino models. We also give a simpler proof in the case of holomorphic quantum mechanics.Comment: 38 page

Accretion of Planetary Material onto Host Stars

Accretion of planetary material onto host stars may occur throughout a star's life. Especially prone to accretion, extrasolar planets in short-period orbits, while relatively rare, constitute a significant fraction of the known population, and these planets are subject to dynamical and atmospheric influences that can drive significant mass loss. Theoretical models frame expectations regarding the rates and extent of this planetary accretion. For instance, tidal interactions between planets and stars may drive complete orbital decay during the main sequence. Many planets that survive their stars' main sequence lifetime will still be engulfed when the host stars become red giant stars. There is some observational evidence supporting these predictions, such as a dearth of close-in planets around fast stellar rotators, which is consistent with tidal spin-up and planet accretion. There remains no clear chemical evidence for pollution of the atmospheres of main sequence or red giant stars by planetary materials, but a wealth of evidence points to active accretion by white dwarfs. In this article, we review the current understanding of accretion of planetary material, from the pre- to the post-main sequence and beyond. The review begins with the astrophysical framework for that process and then considers accretion during various phases of a host star's life, during which the details of accretion vary, and the observational evidence for accretion during these phases.Comment: 18 pages, 5 figures (with some redacted), invited revie