Uncertainty relations for general phase spaces

We describe a setup for obtaining uncertainty relations for arbitrary pairs of observables related by Fourier transform. The physical examples discussed here are standard position and momentum, number and angle, finite qudit systems, and strings of qubits for quantum information applications. The uncertainty relations allow an arbitrary choice of metric for the distance of outcomes, and the choice of an exponent distinguishing e.g., absolute or root mean square deviations. The emphasis of the article is on developing a unified treatment, in which one observable takes values in an arbitrary locally compact abelian group and the other in the dual group. In all cases the phase space symmetry implies the equality of measurement uncertainty bounds and preparation uncertainty bounds, and there is a straightforward method for determining the optimal bounds.Comment: For the proceedings of QCMC 201

Quantum cryptography as a retrodiction problem

We propose a quantum key distribution protocol based on a quantum retrodiction protocol, known as the Mean King problem. The protocol uses a two way quantum channel. We show security against coherent attacks in a transmission error free scenario, even if Eve is allowed to attack both transmissions. This establishes a connection between retrodiction and key distribution.Comment: 5 pages, 1 figur

Empirical risk minimization as parameter choice rule for general linear regularization methods.

We consider the statistical inverse problem to recover f from noisy measurements Y = Tf + sigma xi where xi is Gaussian white noise and T a compact operator between Hilbert spaces. Considering general reconstruction methods of the form (f) over cap (alpha) = q(alpha) (T*T)T*Y with an ordered filter q(alpha), we investigate the choice of the regularization parameter alpha by minimizing an unbiased estiate of the predictive risk E[parallel to T f - T (f) over cap (alpha)parallel to(2)]. The corresponding parameter alpha(pred) and its usage are well-known in the literature, but oracle inequalities and optimality results in this general setting are unknown. We prove a (generalized) oracle inequality, which relates the direct risk E[parallel to f - (f) over cap (alpha pred)parallel to(2)] with the oracle prediction risk inf(alpha>0) E[parallel to T f - T (f) over cap (alpha)parallel to(2)]. From this oracle inequality we are then able to conclude that the investigated parameter choice rule is of optimal order in the minimax sense. Finally we also present numerical simulations, which support the order optimality of the method and the quality of the parameter choice in finite sample situations