Comment on "Quantum Kaniadakis entropy under projective measurement".

G M Bosyk,F Holik,P W Lamberti,S Zozor,M Portesi

doi:10.1103/physreve.94.026103

Abstract

We comment on the main result given by Ourabah etal. [Phys. Rev. E 92, 032114 (2015)PLEEE81539-375510.1103/PhysRevE.92.032114], noting that it can be derived as a special case of the more general study that we have provided in [Quantum Inf Process 15, 3393 (2016)10.1007/s11128-016-1329-5]. Our proof of the nondecreasing character under projective measurements of so-called generalized (h,ϕ) entropies (that comprise the Kaniadakis family as a particular case) has been based on majorization and Schur-concavity arguments. As a consequence, we have obtained that this property is obviously satisfied by Kaniadakis entropy but at the same time is fulfilled by all entropies preserving majorization. In addition, we have seen that our result holds for any bistochastic map, being a projective measurement a particular case. We argue here that looking at these facts from the point of view given in [QuantumInf Process 15, 3393 (2016)10.1007/s11128-016-1329-5] not only simplifies the demonstrations but allows for a deeper understanding of the entropic properties involved.

Highlights

Let a quantum system be described by a density operator ρ, that is, a positive-semidefinite operator acting on an Ndimensional Hilbert space, with trace 1
In [4] it was proven that the quantum Kaniadakis entropy cannot decrease under the action of projective measurements
Due to the fact that our proof is based on majorization and the Schur-concavity property, our findings revealed that this result holds for a more general family of entropies, with the Kaniadakis entropy being a particular case

Summary

Introduction

Let a quantum system be described by a density operator ρ, that is, a positive-semidefinite operator acting on an Ndimensional Hilbert space, with trace 1. It is to be expected that some of the properties of the von Neumann entropy remain valid for the Kaniadakis entropy. In [4] it was proven that the quantum Kaniadakis entropy cannot decrease under the action of projective measurements.

Results

Conclusion