@article {2006,
	title = {Reflection symmetries for multiqubit density operators},
	journal = {J. Math. Phys. 47 (2006) 032104},
	number = {arXiv.org;quant-ph/0405123v2},
	year = {2006},
	abstract = {For multiqubit density operators in a suitable tensorial basis, we show that a number of nonunitary operations used in the detection and synthesis of entanglement are classifiable as reflection symmetries, i.e., orientation changing rotations. While one-qubit reflections correspond to antiunitary symmetries, as is known for example from the partial transposition criterion, reflections on the joint density of two or more qubits are not accounted for by the Wigner Theorem and are well-posed only for sufficiently mixed states. One example of such nonlocal reflections is the unconditional NOT operation on a multiparty density, i.e., an operation yelding another density and such that the sum of the two is the identity operator. This nonphysical operation is admissible only for sufficiently mixed states.},
	doi = {10.1063/1.2181827},
	url = {http://hdl.handle.net/1963/2121},
	author = {Claudio Altafini and Timothy F. Havel}
}
@article {2004,
	title = {Reduction by group symmetry of second order variational problems on a semidirect product of Lie groups with positive definite Riemannian metric},
	journal = {ESAIM: COCV 10 (2004) 526-548},
	year = {2004},
	publisher = {EDP Sciences},
	abstract = {For a Riemannian structure on a semidirect product of Lie groups, the variational problems can be reduced using the group symmetry. Choosing the Levi-Civita connection of a positive definite metric tensor, instead of any of the canonical connections for the Lie group, simplifies the reduction of the variations but complicates the expression for the Lie algebra valued covariant derivatives. The origin of the discrepancy is in the semidirect product structure, which implies that the Riemannian exponential map and the Lie group exponential map do not coincide. The consequence is that the reduced equations look more complicated than the original ones. The main scope of this paper is to treat the reduction of second order variational problems (corresponding to geometric splines) on such semidirect products of Lie groups. Due to the semidirect structure, a number of extra terms appears in the reduction, terms that are calculated explicitely. The result is used to compute the necessary conditions of an optimal control problem for a simple mechanical control system having invariant Lagrangian equal to the kinetic energy corresponding to the metric tensor. As an example, the case of a rigid body on the Special Euclidean group is considered in detail.},
	doi = {10.1051/cocv:2004018},
	url = {http://hdl.handle.net/1963/3521},
	author = {Claudio Altafini}
}
@article {2004,
	title = {Representing multiqubit unitary evolutions via Stokes tensors},
	journal = {Phys. Rev. A 70 (2004) 032331},
	number = {SISSA;91/2004/M},
	year = {2004},
	abstract = {For the Stokes tensor parametrization of a multiqubit density operator, we provide an explicit formulation of the corresponding unitary dynamics at the infinitesimal level. The main advantage of this formalism (clearly reminiscent of the ideas of {\textquoteleft}{\textquoteleft}coherences\\\'\\\' and {\textquoteleft}{\textquoteleft}coupling Hamiltonians\\\'\\\' of spin systems) is that the pattern of correlation between qubits and the pattern of infinitesimal correlation are highlighted simultaneously and can be used constructively for qubit manipulation. For example, it allows to compute explicitly a Rodrigues\\\' formula for the one-parameter orbits of nonlocal Hamiltonians. The result is easily generalizable to orbits of Cartan subalgebras and allows to express the Cartan decomposition of unitary propagators as a linear action directly in terms of the infinitesimal generators.},
	doi = {10.1103/PhysRevA.70.032331},
	url = {http://hdl.handle.net/1963/2307},
	author = {Claudio Altafini}
}
@article {2002,
	title = {The reachable set of a linear endogenous switching system},
	journal = {Systems Control Lett. 47 (2002) 343-353},
	year = {2002},
	publisher = {Elsevier},
	abstract = {In this work, switching systems are named endogenous when their switching pattern is controllable. Linear endogenous switching systems can be considered as a particular class of bilinear control systems. The key idea is that both types of systems are equivalent to polysystems, i.e. to systems whose flow is piecewise smooth. The reachable set of a linear endogenous switching system can be studied consequently. The main result is that, in general, it has the structure of a semigroup, even when the Lie algebra rank condition is satisfied since the logic inputs cannot reverse the direction of the flow. The adaptation of existing controllability criteria for bilinear systems is straightforward.},
	doi = {10.1016/S0167-6911(02)00228-1},
	url = {http://hdl.handle.net/1963/3142},
	author = {Claudio Altafini}
}
