Abstract
We study universal uncertainty relations and present a method called joint probability distribution diagram to improve the majorization bounds constructed independently in [Phys. Rev. Lett. 111, 230401 (2013)] and [J. Phys. A. 46, 272002 (2013)]. The results give rise to state independent uncertainty relations satisfied by any nonnegative Schur-concave functions. On the other hand, a remarkable recent result of entropic uncertainty relation is the direct-sum majorization relation. In this paper, we illustrate our bounds by showing how they provide a complement to that in [Phys. Rev. A. 89, 052115 (2014)].
Highlights
Uncertainty relations[1] are of profound significance in quantum mechanics and quantum information theory
In order to overcome the drawback in the product form of variance base uncertainty relations, Deutsch[8] introduced the entropic uncertainty relations, which were later improved by Maassen and Uffink[9]: H(A) +H(B) ≥−2 log c(A, B), where H is the Shannon entropy, c (A, B) = max m,n am bn is maximum overlap between the basis elements {|am〉} and {|bn〉} of the eigenbases of A and B, respectively
We first introduce a scheme called “joint probability distribution diagram” (JPDD) to consider the optimization problem involved in calculating Ωk
Summary
Uncertainty relations[1] are of profound significance in quantum mechanics and quantum information theory. The well-known form of the Heisenberg’s uncertainty relations, given by Robertson[7], says that the standard deviations of the observables ΔA and ΔB satisfy the following inequality,. The uncertainty relations provide a limitation on how much information one can obtain by measuring a physical system, and can be characterized in terms of the probability distributions of the measurement outcomes. The Maassen-Uffink bound has been surprisingly improved by Coles and Piani[10], Rudnicki, Puchała and Z yczkowski[11], for a review on entopic uncertainty relations see refs 12 and 13. Gheorghiu and Gour[14] proposed a new concept called “universal uncertainty relations” which are not limited to considering only the well-known entropic functions such as Shannon entropy, Renyi entropy and Tsallis entropy, and any nonnegative Schur-concave functions. Z yczkowski[15] independently used majorization technique to establish entropic uncertainty relations similar to “universal uncertainty relations”
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