Nonlinear variants of quantum mechanics can solve tasks that are impossible in standard quantum theory, such as perfectly distinguishing nonorthogonal states. Here we derive the optimal protocol for distinguishing two states of a qubit using the Gross-Pitaevskii equation, a model of nonlinear quantum mechanics that arises as an effective description of Bose-Einstein condensates. Using this protocol, we present an algorithm for unstructured search in the Gross-Pitaevskii model, obtaining an exponential improvement over a previous algorithm of Meyer and Wong. This result establishes a limitation on the effectiveness of the Gross-Pitaevskii approximation. More generally, we demonstrate similar behavior under a family of related nonlinearities, giving evidence that the ability to quickly discriminate nonorthogonal states and thereby solve unstructured search is a generic feature of nonlinear quantum mechanics. arXiv:1507.06334v2 [quant-ph]

We analyze a sample of 73 old long-period comets (LPCs) (orbital periods 200 < P < 1000 yr) with perihelion distances q < 2.5 au, discovered in the period 1850-2014. We cloned the observed comets and also added fictitious LPCs with perihelia in the Jupiter's zone. We consider both a purely dynamical evolution and a physico-dynamical one with different physical lifetimes. We can fit the computed energy distribution of comets with q < 1.3 au to the observed one only within the energy range 0.01 < x < 0.04 au −1 (or periods 125 < P < 1000 yr), where the "energy" is taken as the inverse of the semimajor axis a, namely x ≡ 1/a. The best results are obtained for physical lifetimes of about 200-300 revolutions (for a comet with a standard q = 1 au). We find that neither a purely dynamical evolution, nor a physico-dynamical one can reproduce the long tail of larger binding energies (x > ∼ 0.04 au −1 ) that correspond to most Halley-type comets (HTCs) and Jupiter-family comets. We conclude that most HTCs are not the end states of the evolution of LPCs, but come from a different source, a flattened one that we identify with the Centaurs that are scattered to the inner planetary region from the transNeptunian belt. These results also show that the boundary between LPCs and HTCs should be located at an energy x ∼ 0.04 au −1 (P ∼ 125 yr), rather than the conventional classical boundary at P = 200 yr.

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