In this paper the angle-angular momentum entropic lower bound is proved by using Tsallis-like entropies and Riesz theorem for the quantum scattering of the spinless particles. Numerical estimations of the scattering entropies, as well as an experimental test of the state-independent entropic lower bound, are obtained by using the amplitude reconstruction from the available phase shift analyses for the pionnucleus scatterings. A standard interpretation of these results in terms of the optimal state dominance is presented. Then, it is shown that experimental pion-nucleus entropies are well described by optimal entropies and that the experimental data are consistent with the principle of minimum distance in the space of scattering states. [S0031-9007 (98)08033-8] PACS numbers: 03.65.Ca, 25.80.DjOver the past two decades there has been increasing interest [1] in the investigation of the entropic uncertainty relations introduced in Ref. [2]. In this paper the entropic lower bound for the quantum scattering [3] is investigated in a more general form by introducing Tsallis-like entropies (see Refs. [4,5]). Hence, by using Riesz theorem [6], the state-independent angle-angular momentum entropic lower bound is proved for the scattering of spinless particles. The results on the first experimental test of this state independent entropic bound in pion-nucleus scattering are obtained by calculations of the scattering entropies from the experimental available phase shifts [7][8][9][10][11]. Moreover, comparisons of these results with the optimal state [12] prediction are presented.Angular entropy S u .-The informational angular entropy

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The angle-angular momentum entropic upper bounds are proved in a more general form by using Tsallis-like entropies for the quantum scattering of the spinless particles. So, the problem to find an upper bound for all Tsallis-like scattering entropy when the elastic integrated cross section s el and the forward differential cross section ds dV ͑1͒ are fixed is completely solved in terms of the optimal entropies obtained from the principle of minimum distance in the space of states. The results on experimental tests of these optimal upper bounds are presented for both extensive and nonextensive statistics cases. PACS numbers: 03.65.Ca, 05.70.Ce, 13.75.Gx, 25.80.Dj Entropy [1] is a concept which appears in many fields of science being in the center of interest not only in classical and quantum physics but also in mathematical and communication theories. Any study leading to an understanding of entropy must proceed via probability theory. The heritage of quantum entropies from quantum mechanics is in fact their strong relationships to the Hilbert spaces of quantum states. So, the entropy of a quantum state describing a physical system is a quantity expressing the uncertainty or randomness of the system. This uncertainty attached to a physical system can be regarded as the amount of information carried by the system, so that the entropy of a state can be interpreted as the information carried by the state. In recent years there has been an increasing interest (see Ref.[1]) in the investigation of quantum entropy not only by proving new entropic uncertainty relations (see, e.g., Refs. [1-4]) for the standard additive system but also by a generalization of such results to nonextensive statistics [5][6][7][8]. In Ref.[3] (the angle and angular momentum) information entropies as well as the entropic angle-angular momentum uncertainty relations were introduced. Using Tsallislike entropies and the Riesz theorem [9] in Ref.[4], the state independent angle-angular momentum entropic lower bounds were proved for the quantum scattering of the spinless particles. Then, it was shown that the experimental pion-nucleus scattering entropies are well described by the optimal entropies corresponding to the optimal states which were introduced in Ref.[10]-via reproducing kernel Hilbert space methods (RKHS). Also, in Refs. [3,4], it was suggested that extremal properties of entropy, such as maximum entropy, can be important for the characterization of the optimal states derived from the principle of minimum distance in the space of states (Ref.[10]).In this Letter, we report the results on the investigation of some fundamental physical questions concerning entropic bounds, once the basic experimental information is fixed and used as constraints in the variational procedure. The optimal entropic upper bounds are proved by using the Lagrange multipliers method and Tsallis-like entropies [5] for quantum scattering of spinless particles. Hence, the problem is to find an upper bound for each Tsallis-like scattering entropy S a ͑q͒ (a ϵ u, L...