83 research outputs found
Zero-energy states for a class of quasi-exactly solvable rational potentials
Quasi-exactly solvable rational potentials with known zero-energy solutions
of the Schro\" odinger equation are constructed by starting from exactly
solvable potentials for which the Schr\" odinger equation admits an so(2,1)
potential algebra. For some of them, the zero-energy wave function is shown to
be normalizable and to describe a bound state.Comment: LaTeX, 13 pages, 2 figures on request, to appear in Phys. Lett.
Infinite loop superalgebras of the Dirac theory on the Euclidean Taub-NUT space
The Dirac theory in the Euclidean Taub-NUT space gives rise to a large
collection of conserved operators associated to genuine or hidden symmetries.
They are involved in interesting algebraic structures as dynamical algebras or
even infinite-dimensional algebras or superalgebras. One presents here the
infinite-dimensional superalgebra specific to the Dirac theory in manifolds
carrying the Gross-Perry-Sorkin monopole. It is shown that there exists an
infinite-dimensional superalgebra that can be seen as a twisted loop
superalgebra.Comment: 16 pages, LaTeX, references adde
Unitary transformations for testing Bell inequalities
It is shown that optical experimental tests of Bell inequality violations can
be described by SU(1,1) transformations of the vacuum state, followed by photon
coincidence detections. The set of all possible tests are described by various
SU(1,1) subgroups of Sp(8,). In addition to establishing a common
formalism for physically distinct Bell inequality tests, the similarities and
differences of post--selected tests of Bell inequality violations are also made
clear. A consequence of this analysis is that Bell inequality tests are
performed on a very general version of SU(1,1) coherent states, and the
theoretical violation of the Bell inequality by coincidence detection is
calculated and discussed. This group theoretical approach to Bell states is
relevant to Bell state measurements, which are performed, for example, in
quantum teleportation.Comment: 3 figure
Exact, E=0, Solutions for General Power-Law Potentials. I. Classical Orbits
For zero energy, , we derive exact, classical solutions for {\em all}
power-law potentials, , with and . When the angular momentum is non-zero, these solutions lead to
the orbits , for all . When , the orbits are bound and go through the origin.
This leads to discrete discontinuities in the functional dependence of
and , as functions of , as the orbits pass through the origin. We
describe a procedure to connect different analytic solutions for successive
orbits at the origin. We calculate the periods and precessions of these bound
orbits, and graph a number of specific examples. Also, we explain why they all
must violate the virial theorem. The unbound orbits are also discussed in
detail. This includes the unusual orbits which have finite travel times to
infinity and also the special case.Comment: LaTeX, 27 pages with 12 figures available from the authors or can be
generated from Mathematica instructions at end of the fil
Universal amplitude ratios of two-dimensional percolation from field theory
We complete the determination of the universal amplitude ratios of
two-dimensional percolation within the two-kink approximation of the form
factor approach. For the cluster size ratio, which has for a long time been
elusive both theoretically and numerically, we obtain the value 160.2, in good
agreement with the lattice estimate 162.5 +/- 2 of Jensen and Ziff.Comment: 8 page
Quantum superintegrability and exact solvability in N dimensions
A family of maximally superintegrable systems containing the Coulomb atom as
a special case is constructed in N-dimensional Euclidean space. Two different
sets of N commuting second order operators are found, overlapping in the
Hamiltonian alone. The system is separable in several coordinate systems and is
shown to be exactly solvable. It is solved in terms of classical orthogonal
polynomials. The Hamiltonian and N further operators are shown to lie in the
enveloping algebra of a hidden affine Lie algebra
Quantum gates on hybrid qudits
We introduce quantum hybrid gates that act on qudits of different dimensions.
In particular, we develop two representative two-qudit hybrid gates (SUM and
SWAP) and many-qudit hybrid Toffoli and Fredkin gates. We apply the hybrid SUM
gate to generating entanglement, and find that operator entanglement of the SUM
gate is equal to the entanglement generated by it for certain initial states.
We also show that the hybrid SUM gate acts as an automorphism on the Pauli
group for two qudits of different dimension under certain conditions. Finally,
we describe a physical realization of these hybrid gates for spin systems.Comment: 8 pages and 1 figur
Entangling power and operator entanglement in qudit systems
We establish the entangling power of a unitary operator on a general
finite-dimensional bipartite quantum system with and without ancillas, and give
relations between the entangling power based on the von Neumann entropy and the
entangling power based on the linear entropy. Significantly, we demonstrate
that the entangling power of a general controlled unitary operator acting on
two equal-dimensional qudits is proportional to the corresponding operator
entanglement if linear entropy is adopted as the quantity representing the
degree of entanglement. We discuss the entangling power and operator
entanglement of three representative quantum gates on qudits: the SUM, double
SUM, and SWAP gates.Comment: 8 pages, 1 figure. Version 3: Figure was improved and the MS was a
bit shortene
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