Beyond Landau--Pollak and entropic inequalities: geometric bounds imposed on uncertainties sum

21/09/2014
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Beyond Landau--Pollak and entropic inequalities: geometric bounds imposed on uncertainties sum

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Beyond Landau–Pollak and entropic inequalities: geometric bounds imposed on uncertainties sum S. Zozor1,2, G. M. Bosyk2,1, M. Portesi2,1, T. M. Os´an3, and P. W. Lamberti3 1 GIPSA-Lab, Domaine universitaire, 38402 St Martin d’H`eres, France 2 IFLP, CONICET & Dpto de F´ısica, UNLP, 1900 La Plata, Argentina 3 CONICET & FaMAF, Universidad Nacional de C´ordoba, X5000HUA C´ordoba, Argentina steeve.zozor@gipsa-lab.inpg.fr, {gbosyk,portesi}@fisica.unlp.edu.ar, {tosan,lamberti}@famaf.unc.edu.ar Abstract In this contribution we propose generalized inequalities to quantify the uncertainty principle, which establishes a limitation on the simultaneous predictability of incompatible observables. Quantitative formulations based on variances and other moments as well as on entropies have been studied earlier (see [1,2] for recent reviews, and also [3-7] among other references). Here we make use of geometric considerations in order to obtain bounds to suitably defined measures of uncertainty. Let A and B be two observables with finite, discrete spectra, and let ρ represents a mixed state; we denote by p(A; ρ) and p(B; ρ) the probability vectors associated with the observables when the system is in the state ρ (i.e., the probabilities of observing the possible outcomes of A and B, respectively). We focus on relations of the form U(A; ρ) + U(B; ρ) ≥ B(A, B) where U is a measure of uncertainty attached to an observable for a specified state, given as a function of the probability vector associated to it; and B is a non-trivial state-independent bound for the uncertainty sum. Landau–Pollak inequality takes that form when setting U(A; ρ) = arccos( maxi pi(A; ρ)) (related to Wooters metric). This inequality was initially proved in the signal processing domain, for time and frequency representation (equivalent to the pair of quantum observables) of a signal (the correlate of a pure state), and then adapted in quantum mechanics for non-degenerate observables (i.e., those which decompose onto an orthonormal basis) and pure states. We propose here an extension of the Landau–Pollak inequality using functions more general than the arccosine, which applies for arbitrary quantum states (pure or mixed) and for general description of observables (POVM). We obtain a family of inequalities by means of a geometrical approach: when setting U from well suited metrics, our generalization comes out as a consequence of the triangle inequality. Then, we propose an entropic formulation of the uncertainty principle. Here the uncertainty measure is based on generalized entropies of the R´enyi or Havrda-Charv´at-Tsallis type: U(A; ρ) = f( P i[pi(A;ρ)]α ) 1−α , assuming certain conditions for f and any positive index α. Our approach consists of two steps: (i) minimization of each entropy separately, subject to a given maximal probability; (ii) minimization of the entropy sum restricted by the Landau-Pollak inequality previously derived. We will give the detailed proofs of both formulations of the uncertainty principle in the final version, as well as some examples of interest numerically simulated. References: [1] S. Wehner and A. Winter, New Journal of Physics, 12: 025009 (2010). [2] I. Bialynicki-Birula and L. Rudnicki, “Statistical Complexities: Application to Electronic Struc- ture”, Ch. 1, Springer (2010). [3] H. J. Landau and H. O. Pollak, The Bell System Technical Journal, 40: 65-84 (1961). [4] H. Maassen and J. B. M. Uffink, Physical Review Letters, 60: 1103-1106 (1988). [5] S. Zozor, M. Portesi, P. S´anchez-Moreno, and J.S. Dehesa, AIP Conf. Proc. of the 30th Int. Workshop on Bayesian Inference and Maximum Entropy Methods, 1305: 184-191 (2010). [6] S. Zozor, G. M. Bosyk, and M. Portesi, arXiv:1311.5602 [quant-ph]. [7] G. M. Bosyk, T. M. Os´an, P. W. Lamberti, and M. Portesi, Physical Review A, 89: 034101 (2014). Key Words: Generalized Heisenberg relation, Landau–Pollak-type inequalities, entropic uncertainty relation, pure and mixed states, POVM descriptions