General entropy-like uncertainty relations in finite dimensions

Abstract

We revisit entropic formulations of the uncertainty principle for an arbitrary pair of positive operator-valued measures (POVM) A and B, acting on finite dimensional Hilbert space. Salicrú generalized (h, φ)-entropies, including Rényi and Tsallis ones among others, are used as uncertainty measures associated with the distribution probabilities corresponding to the outcomes of the observables. We obtain a nontrivial lower bound for the sum of generalized entropies for any pair of entropic functionals, which is valid for both pure and mixed states. The bound depends on the overlap triplet (c A , c B , c A,B ) with c A (resp. c B ) being the overlap between the elements of the POVM A (resp. B) and c A,B the overlap between the pair of POVM. Our approach is inspired by that of de Vicente and Sánchez-Ruiz [Phys. Rev. A 77, 042110 (2008)] and consists in a minimization of the entropy sum subject to the Landau-Pollak inequality that links the maximum probabilities of both observables. We solve the constrained optimization problem in a geometrical way and furthermore, when dealing with Rényi or Tsallis entropic formulations of the uncertainty principle, we overcome the Hölder conjugacy constraint imposed on the entropic indices by the Riesz-Thorin theorem. In the case of nondegenerate observables, we show that for given c A,B > 1 √ 2 , the bound obtained is optimal; and that, for Rényi entropies, our bound improves Deutsch one, but Maassen-Uffink bound prevails when c A,B ≤ 1 2 . Finally, we illustrate by comparing our bound with known previous results in particular cases of Rényi and Tsallis entropies.

Geometric approach to extend Landau-Pollak uncertainty relations for positive operator-valued measures

“…A direct consequence of our results is that a previous work [8] dealing with generalized entropies of probability vectors extends very easily in the most general case of POVM representations of observables.…”

“…[4][5][6], and [7] for recent reviews). Recently, some of us [8] extended entropic formulations of the uncertainty principle to the case of a pair of observables with nondegenerate discrete N -dimensional spectra, using generalized informational entropies. The proposed formulation makes use of the Landau-Pollak inequality (LPI) which has been introduced in time-frequency analysis [9], and later on adapted to the quantum mechanics' language [4].…”

Majorization approach to entropic uncertainty relations for coarse-grained observables

“…One decade later, entropic formulation of the uncertainty principle has as well been developed in the discrete settings [2,3]. Even though, the topic of entropic uncertainty relations (EURs) has a long history (for a detailed review see [4,5]), one can observe a recent increase of interest within the quantum information community leading to several improvements [6][7][8][9][10][11][12][13][14][15][16][17] or even a deep asymptotic analysis of different bounds [18]. This is quite understandable, because the entropic uncertainty relations have various applications, for example in entanglement detection [19][20][21][22][23], security of quantum protocols [24,25], quantum memory [26,27] or as an ingredient of Einstein-Podolsky-Rosen steering criteria [28,29].…”

On Entropic Uncertainty Relations for Measurements of Energy and Its “Complement”

“…An alternative viewpoint is that the above uncertainty relations follow from the monotonicity of the quantum relative entropy [65]. For an arbitrary choice of α and β, the problem of obtaining general entropic bounds was examined in [66]. The inequalities (21) and (22) are preparation uncertainty relations formulated in terms of both the Rényi and Tsallis entropies.…”