30,777 research outputs found
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
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