439 research outputs found
Commuting Pauli Hamiltonians as maps between free modules
We study unfrustrated spin Hamiltonians that consist of commuting tensor
products of Pauli matrices. Assuming translation-invariance, a family of Hamiltonians
that belong to the same phase of matter is described by a map between modules over
the translation-group algebra, so homological methods are applicable. In any dimension
every point-like charge appears as a vertex of a fractal operator, and can be isolated with
energy barrier at most logarithmic in the separation distance. For a topologically ordered
system in three dimensions, there must exist a point-like nontrivial charge. A connection
between the ground state degeneracy and the number of points on an algebraic set is
discussed. Tools to handle local Clifford unitary transformations are given
Inner products in integrable Richardson-Gaudin models
We present the inner products of eigenstates in integrable Richardson-Gaudin
models from two different perspectives and derive two classes of Gaudin-like
determinant expressions for such inner products. The requirement that one of
the states is on-shell arises naturally by demanding that a state has a dual
representation. By implicitly combining these different representations, inner
products can be recast as domain wall boundary partition functions. The
structure of all involved matrices in terms of Cauchy matrices is made explicit
and used to show how one of the classes returns the Slavnov determinant
formula. This framework provides a further connection between two different
approaches for integrable models, one in which everything is expressed in terms
of rapidities satisfying Bethe equations, and one in which everything is
expressed in terms of the eigenvalues of conserved charges, satisfying
quadratic equations.Comment: 21+16 pages, minor revisions compared to the previous versio
Constant Curvature Algebras and Higher Spin Action Generating Functions
The algebra of differential geometry operations on symmetric tensors over
constant curvature manifolds forms a novel deformation of the sl(2,R)
[semidirect product] R^2 Lie algebra. We present a simple calculus for
calculations in its universal enveloping algebra. As an application, we derive
generating functions for the actions and gauge invariances of massive,
partially massless and massless (for both bose and fermi statistics) higher
spins on constant curvature backgrounds. These are formulated in terms of a
minimal set of covariant, unconstrained, fields rather than towers of auxiliary
fields. Partially massless gauge transformations are shown to arise as
degeneracies of the flat, massless gauge transformation in one dimension
higher. Moreover, our results and calculus offer a considerable simplification
over existing techniques for handling higher spins. In particular, we show how
theories of arbitrary spin in dimension d can be rewritten in terms of a single
scalar field in dimension 2d where the d additional dimensions correspond to
coordinate differentials. We also develop an analogous framework for
spinor-tensor fields in terms of the corresponding superalgebra.Comment: 44 pages, LaTeX, 2 .eps figure
Quantum Coding with Entanglement
Quantum error-correcting codes will be the ultimate enabler of a future
quantum computing or quantum communication device. This theory forms the
cornerstone of practical quantum information theory. We provide several
contributions to the theory of quantum error correction--mainly to the theory
of "entanglement-assisted" quantum error correction where the sender and
receiver share entanglement in the form of entangled bits (ebits) before
quantum communication begins. Our first contribution is an algorithm for
encoding and decoding an entanglement-assisted quantum block code. We then give
several formulas that determine the optimal number of ebits for an
entanglement-assisted code. The major contribution of this thesis is the
development of the theory of entanglement-assisted quantum convolutional
coding. A convolutional code is one that has memory and acts on an incoming
stream of qubits. We explicitly show how to encode and decode a stream of
information qubits with the help of ancilla qubits and ebits. Our
entanglement-assisted convolutional codes include those with a
Calderbank-Shor-Steane structure and those with a more general structure. We
then formulate convolutional protocols that correct errors in noisy
entanglement. Our final contribution is a unification of the theory of quantum
error correction--these unified convolutional codes exploit all of the known
resources for quantum redundancy.Comment: Ph.D. Thesis, University of Southern California, 2008, 193 pages, 2
tables, 12 figures, 9 limericks; Available at
http://digitallibrary.usc.edu/search/controller/view/usctheses-m1491.htm
The SIC Question: History and State of Play
Recent years have seen significant advances in the study of symmetric
informationally complete (SIC) quantum measurements, also known as maximal sets
of complex equiangular lines. Previously, the published record contained
solutions up to dimension 67, and was with high confidence complete up through
dimension 50. Computer calculations have now furnished solutions in all
dimensions up to 151, and in several cases beyond that, as large as dimension
844. These new solutions exhibit an additional type of symmetry beyond the
basic definition of a SIC, and so verify a conjecture of Zauner in many new
cases. The solutions in dimensions 68 through 121 were obtained by Andrew
Scott, and his catalogue of distinct solutions is, with high confidence,
complete up to dimension 90. Additional results in dimensions 122 through 151
were calculated by the authors using Scott's code. We recap the history of the
problem, outline how the numerical searches were done, and pose some
conjectures on how the search technique could be improved. In order to
facilitate communication across disciplinary boundaries, we also present a
comprehensive bibliography of SIC research.Comment: 16 pages, 1 figure, many references; v3: updating bibliography,
dimension eight hundred forty fou
- ā¦