Jamil Daboul

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 dimensions under certain conditions. Finally, we describe a physical realization of these hybrid gates for spin systems.

The hydrogen algebra as centerless twisted Kac-Moody algebra

Abstract We show that the dynamical symmetry of the hydrogen atom leads in a natural way to the infinite-dimensional algebra, which we identify as the positive subalgebras of twisted Kac-Moody algebras of so(4) and so(3,1), which are isomorphic to each other over R . The relevant automorphism can be associated with parity.

Matrix representation of octonions and generalizations

We define a special matrix multiplication among a special subset of 2N×2N matrices, and study the resulting (nonassociative) algebras and their subalgebras. We derive the conditions under which these algebras become alternative nonassociative, and when they become associative. In particular, these algebras yield special matrix representations of octonions and complex numbers; they naturally lead to the Cayley–Dickson doubling process. Our matrix representation of octonions also yields elegant insights into Dirac’s equation for a free particle. A few other results and remarks arise as byproducts.

Quantum bound states with zero binding energy

After reviewing the general properties of zero-energy quantum states, we give the explicit solutions of the Schrodinger equation with E = 0 for the class of potentials V = −|γ|/rν, where −∞ 2, these solutions are normalizable and correspond to bound states, if the angular momentum quantum number l > 0. (These states are normalizable, even for l = 0, if we increase the space dimension, D, beyond 4; i.e for D > 4.) For ν < −2 the above solutions, although unbound, are normalizable. This is true even though the corresponding potentials are repulsive for all r. We discuss the physics of these unusual effects.

Squeezed states, generalized Hermite polynomials and pseudo-diffusion equation

Abstract We show that the wavefunctions 〈pq; λ|n〈, of the harmonic oscillator in the squeezed state representation, have the generalized Hermite polynomials as their natural orthogonal polynomials. These wavefunctions lead to generalized Poisson Distribution Pn(pq;λ), which satisfy an interesting pseudo-diffusion equation: ∂P n p,q;λ) ∂λ = 1 4 [ ∂ 2 ∂p 2 −( 1 λ 2 ) ∂ 2 ∂q 2 ]P 2 (p,q;λ) , in which the squeeze parameter λ plays the role of time. Th entropies Sn(λ) have minima at the unsqueezed states (λ=1), which means that squeezing or stretching decreases the correlation between momentum p and position q.

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,

EXACT, E=0, QUANTUM SOLUTIONS FOR GENERAL POWER-LAW POTENTIALS

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Exact, E =0, classical solutions for general power-law potentials

). 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.

Radiation reaction for a rotating sphere with rigid surface charge

For zero energy, E=0, we derive exact, quantum solutions for all power-law potentials, V(r)=−γ/rν, with γ>0 and −∞ 2 and l≥1 the solutions are normalizable. Surprisingly, the solutions for ν 2 the partial-wave Hamiltonians, Hl, have overcomplete sets of normalizable eigensolutions. We discuss how to obtain self-adjoint extensions of Hl such that the above E=0 solutions become included in their domains. When 2>ν≥−2 the E=0 solutions are not square-integrable. The ν=2 solutions are also unnormalizable, but are exceptional solutions. We also find that, by increasing the dimension of the Schrodinger equation beyond 4, an effective centrifugal barrier is created which is sufficient to cause binding when E=0 and ν>2, even for l=0. We discuss the physics of the above solutions and compare them to the corresponding classical solutions, which are derived elsewhere.

O(N) symmetries, sum rules for generalized Hermite polynomials and squeezed states

For zero energy,