Entropic uncertainty minimum for angle and angular momentum

Alison M Yao,Electra Eleftheriadou,Miles J Padgett,Thomas Brougham,Stephen M Barnett

doi:10.1088/2040-8978/16/10/105404

Abstract

Uncertainty relations are key components in the understanding of the nature of quantum mechanics. In particular, entropic relations are preferred in the study of angular position and angular momentum states. We propose a new form of angle–angular momentum state that provides, for all practical purposes, a lower bound on the entropic uncertainty relation, , for any given angular uncertainty, thus improving upon previous bounds. We establish this by comparing this sum with the absolute minimum value determined by a global numerical search. These states are convenient to work with both analytically and experimentally, which suggests that they may be of use for quantum information purposes.

Highlights

Current demands for secure, high-bandwidth communications have resulted in a lot of interest in new protocols in quantum information based on spatial modes carrying orbital angular momentum (OAM) [1,2,3]
The potential for each photon to carry more than one bit of information [4] means that in applications such as quantum key distribution (QKD) is the data transfer rate increased in proportion to the number of bits of information carried by each photon but the security of the protocol is increased
Security proofs for QKD are expressed in terms of information or entropy bounds [13]. This suggests that the entropic uncertainty relation for angle and angular momentum may find application in secure communications based on OAM [14]

Summary

Introduction

Current demands for secure, high-bandwidth communications have resulted in a lot of interest in new protocols in quantum information based on spatial modes carrying orbital angular momentum (OAM) [1,2,3]. Security proofs for QKD are expressed in terms of information or entropy bounds [13] This suggests that the entropic uncertainty relation for angle and angular momentum may find application in secure communications based on OAM [14]. An important point to note is that for linear position and momentum, the lower bound on the entropy sum (2), is satisfied for the Gaussian wavefunctions that satisfy the equality in the more familiar Heisenberg uncertainty principle. These states are referred to as the intelligent states [12, 20] and, have been demonstrated experimentally [10].

Analysis

Conclusion

Findings

Entropic minimum