The Chevalley-Gras formula over global fields

In this article we give an adelic proof of the Chevalley-Gras formula for global fields, which itself is a generalization of the ambiguous class number formula. The idea is to reduce the formula to the Hasse norm theorem, the local and global reciprocity laws. We also give an adelic proof of the Chevalley-Gras formula for $0$-th divisor class group in the function field case, which extends a result of Rosen.Comment: minor revisio

The pointer basis and the feedback stabilization of quantum systems

The dynamics for an open quantum system can be `unravelled' in infinitely many ways, depending on how the environment is monitored, yielding different sorts of conditioned states, evolving stochastically. In the case of ideal monitoring these states are pure, and the set of states for a given monitoring forms a basis (which is overcomplete in general) for the system. It has been argued elsewhere [D. Atkins et al., Europhys. Lett. 69, 163 (2005)] that the `pointer basis' as introduced by Zurek and Paz [Phys. Rev. Lett 70, 1187(1993)], should be identified with the unravelling-induced basis which decoheres most slowly. Here we show the applicability of this concept of pointer basis to the problem of state stabilization for quantum systems. In particular we prove that for linear Gaussian quantum systems, if the feedback control is assumed to be strong compared to the decoherence of the pointer basis, then the system can be stabilized in one of the pointer basis states with a fidelity close to one (the infidelity varies inversely with the control strength). Moreover, if the aim of the feedback is to maximize the fidelity of the unconditioned system state with a pure state that is one of its conditioned states, then the optimal unravelling for stabilizing the system in this way is that which induces the pointer basis for the conditioned states. We illustrate these results with a model system: quantum Brownian motion. We show that even if the feedback control strength is comparable to the decoherence, the optimal unravelling still induces a basis very close to the pointer basis. However if the feedback control is weak compared to the decoherence, this is not the case