2,976 research outputs found
PT symmetry, Cartan decompositions, Lie triple systems and Krein space related Clifford algebras
Gauged PT quantum mechanics (PTQM) and corresponding Krein space setups are
studied. For models with constant non-Abelian gauge potentials and extended
parity inversions compact and noncompact Lie group components are analyzed via
Cartan decompositions. A Lie triple structure is found and an interpretation as
PT-symmetrically generalized Jaynes-Cummings model is possible with close
relation to recently studied cavity QED setups with transmon states in
multilevel artificial atoms. For models with Abelian gauge potentials a hidden
Clifford algebra structure is found and used to obtain the fundamental symmetry
of Krein space related J-selfadjoint extensions for PTQM setups with
ultra-localized potentials.Comment: 11 page
Time Reversal and n-qubit Canonical Decompositions
For n an even number of qubits and v a unitary evolution, a matrix
decomposition v=k1 a k2 of the unitary group is explicitly computable and
allows for study of the dynamics of the concurrence entanglement monotone. The
side factors k1 and k2 of this Concurrence Canonical Decomposition (CCD) are
concurrence symmetries, so the dynamics reduce to consideration of the a
factor. In this work, we provide an explicit numerical algorithm computing v=k1
a k2 for n odd. Further, in the odd case we lift the monotone to a two-argument
function, allowing for a theory of concurrence dynamics in odd qubits. The
generalization may also be studied using the CCD, leading again to maximal
concurrence capacity for most unitaries. The key technique is to consider the
spin-flip as a time reversal symmetry operator in Wigner's axiomatization; the
original CCD derivation may be restated entirely in terms of this time
reversal. En route, we observe a Kramers' nondegeneracy: the existence of a
nondegenerate eigenstate of any time reversal symmetric n-qubit Hamiltonian
demands (i) n even and (ii) maximal concurrence of said eigenstate. We provide
examples of how to apply this work to study the kinematics and dynamics of
entanglement in spin chain Hamiltonians.Comment: 20 pages, 3 figures; v2 (17pp.): major revision, new abstract,
introduction, expanded bibliograph
Symmetric spaces and Lie triple systems in numerical analysis of differential equations
A remarkable number of different numerical algorithms can be understood and
analyzed using the concepts of symmetric spaces and Lie triple systems, which
are well known in differential geometry from the study of spaces of constant
curvature and their tangents. This theory can be used to unify a range of
different topics, such as polar-type matrix decompositions, splitting methods
for computation of the matrix exponential, composition of selfadjoint numerical
integrators and dynamical systems with symmetries and reversing symmetries. The
thread of this paper is the following: involutive automorphisms on groups
induce a factorization at a group level, and a splitting at the algebra level.
In this paper we will give an introduction to the mathematical theory behind
these constructions, and review recent results. Furthermore, we present a new
Yoshida-like technique, for self-adjoint numerical schemes, that allows to
increase the order of preservation of symmetries by two units. Since all the
time-steps are positive, the technique is particularly suited to stiff
problems, where a negative time-step can cause instabilities
Point vortices on the sphere: a case with opposite vorticities
We study systems formed of 2N point vortices on a sphere with N vortices of
strength +1 and N vortices of strength -1. In this case, the Hamiltonian is
conserved by the symmetry which exchanges the positive vortices with the
negative vortices. We prove the existence of some fixed and relative
equilibria, and then study their stability with the ``Energy Momentum Method''.
Most of the results obtained are nonlinear stability results. To end, some
bifurcations are described.Comment: 35 pages, 9 figure
On quantum error-correction by classical feedback in discrete time
We consider the problem of correcting the errors incurred from sending
quantum information through a noisy quantum environment by using classical
information obtained from a measurement on the environment. For discrete time
Markovian evolutions, in the case of fixed measurement on the environment, we
give criteria for quantum information to be perfectly corrigible and
characterize the related feedback. Then we analyze the case when perfect
correction is not possible and, in the qubit case, we find optimal feedback
maximizing the channel fidelity.Comment: 11 pages, 1 figure, revtex
Symplectic analogs of polar decomposition and their applications to bosonic Gaussian channels
We obtain several analogs of real polar decomposition for even dimensional
matrices. In particular, we decompose a non-degenerate matrix as a product of a
Hamiltonian and an anti-symplectic matrix and under additional requirements we
decompose a matrix as a skew-Hamiltonian and a symplectic matrix. We apply our
results to study bosonic Gaussian channels up to inhomogeneous symplectic
transforms
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