Conditional entropic uncertainty relations for Tsallis entropies

Dariusz Kurzyk,Zbigniew Puchała,Łukasz Pawela

doi:10.1007/s11128-018-1955-1

Abstract

The entropic uncertainty relations are a very active field of scientific inquiry. Their applications include quantum cryptography and studies of quantum phenomena such as correlations and non-locality. In this work we find entanglement-dependent entropic uncertainty relations in terms of the Tsallis entropies for states with a fixed amount of entanglement. Our main result is stated as Theorem 1. Taking the special case of von Neumann entropy and utilizing the concavity of conditional von Neumann entropies, we extend our result to mixed states. Finally we provide a lower bound on the amount of extractable key in a quantum cryptographic scenario.

Highlights

Formulated by Heisenberg [1], the uncertainty relation gives insight into differences between classical and quantum mechanics
Berta et al [28] showed that a bound on the uncertainties of the measurement outcomes depends on the amount of entanglement between measured particle and the quantum memory
In order to study the case of general Tsallis entropies Tq, we introduce the following proposition

Summary

Introduction

Formulated by Heisenberg [1], the uncertainty relation gives insight into differences between classical and quantum mechanics. Some results were generalized; entropic formulations of the uncertainty relation in terms of Rényi entropies are included in [16]. Berta et al [28] showed that a bound on the uncertainties of the measurement outcomes depends on the amount of entanglement between measured particle and the quantum memory As a consequence, they formulated a conditional uncertainty relation given as. This is a consequence of the fact that S(A|B) is negative for an entangled state ρAB Another field of application of entropic uncertainty relations with the presence of quantum memory is quantum cryptography [9]. Our results apply to states with a fixed amount of entanglement, described by parameter λ This allows us to find non-trivial bounds for the entropic uncertainty relation.

Qubit conditional uncertainty relations

Analytical minima

Bounding the conditional entropies

Security of quantum key distribution protocols

Conclusion