Majorization approach to entropic uncertainty relations for coarse-grained observables

Abstract

We improve the entropic uncertainty relations for position and momentum coarse-grained measurements. We derive the continuous, coarse-grained counterparts of the discrete uncertainty relations based on the concept of majorization. The obtained entropic inequalities involve two Rényi entropies of the same order, and thus go beyond the standard scenario with conjugated parameters. In a special case describing the sum of two Shannon entropies the majorization-based bounds significantly outperform the currently known results in the regime of larger coarse graining, and might thus be useful for entanglement detection in continuous variables.

Uncertainty, joint uncertainty, and the quantum uncertainty principle

“…The joint uncertainty measures considered here are of the form p q  r r Å ( ( ) ( )). A nontrivial u v , ( ) for this restricted class can be found using the methods in [25,26].…”

“…Pioneered by Hirschman [4], many works [11][12][13][14][15][16][17][18] 2 have used entropies to quantify uncertainty, culminating in a recent surge of quantum information-theoretic treatments of the uncertainty principle [19][20][21][22][23][24][25][26][27][28][29][30]. An important contribution of these recent works is the formulation of uncertainty relations applicable on a quantum system correlated with a quantum memory; such relations are used to strengthen the security proofs of cryptographic tasks [31,32].…”

“…The first two axioms are inspired by earlier approaches [23][24][25][26] to measure-independent notions of uncertainty, wherein the connection between uncertainty and a mathematical concept called majorization [42] was utilized. Majorization is a hierarchy among probability distributions, induced by the action of a class of transformations called doubly stochastic maps.…”

“…Another contribution of this paper is a deeper understanding of so-called universal uncertainty relations found in [23][24][25][26]: pairs of vectors u v , ( ) such that u v , ( )provides a nontrivial bound (like the c of equation (1)) for a whole class of measures, J  Î . We find that no universal relations exist if J includes all possible measures; however, restricting to specific operational frameworks (using the (1) and (2) types of combined experiments discussed in the previous paragraph) is what makes the nontrivial universal relations found in [23][24][25][26] possible.…”

Entropic uncertainty relations for successive measurements of canonically conjugate observables

“…Questions of their optimality are addressed in [14]. Other approaches are based on the sum of variances [15,16] and on majorization relations [17][18][19][20][21]. The variance-based formulation was recently applied to noise and disturbance [22].…”