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Physical Review E

Quantum Kaniadakis entropy under projective measurement

Kamel Ourabah, Amel Hiba Hamici-Bendimerad, Mouloud Tribeche

Published Sep 10, 2015• 17 citations
Publisher

Abstract

It is well known that the von Neumann entropy of a quantum state does not decrease with a projective measurement. This property holds for Tsallis and Rényi entropies as well. We show that the recently introduced quantum version of the Kaniadakis entropy preserves this property.
DOI
10.1103/physreve.92.032114

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Comment on “Quantum Kaniadakis entropy under projective measurement”

G. M. Bosyk, Steeve Zozor, Federico Holik +2 more

We comment on the main result given by Ourabah et al. [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 [Quantum Inf 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.

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Reply to “Comment on ‘Quantum Kaniadakis entropy under projective measurement’ ”

Kamel Ourabah, Mouloud Tribeche

We rely on our proof of the nondecreasing character of quantum Kaniadakis entropy under projective measurement [Phys. Rev. E 92, 032114 (2015)PLEEE81539-375510.1103/PhysRevE.92.032114], and we put it into perspective with the results of Bosyk et al. [Quantum Inf Process 15, 3393 (2016)10.1007/s11128-016-1329-5]. Our method, adopted for the proof that Kaniadakis entropy does not decrease under a projective measurement, is based on Jensen's inequalities, while the method proposed by the authors of the Comment represents another alternative and clearly correct method to prove the same thing. Furthermore, we clarify that our interest in Kaniadakis entropy is due to the fact that this entropy has a transparent physical significance, emerging within the special relativity.

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We comment on the main result given by Ourabah et al. [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 [Quantum Inf 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.
We rely on our proof of the nondecreasing character of quantum Kaniadakis entropy under projective measurement [Phys. Rev. E 92, 032114 (2015)PLEEE81539-375510.1103/PhysRevE.92.032114], and we put it into perspective with the results of Bosyk et al. [Quantum Inf Process 15, 3393 (2016)10.1007/s11128-016-1329-5]. Our method, adopted for the proof that Kaniadakis entropy does not decrease under a projective measurement, is based on Jensen's inequalities, while the method proposed by the authors of the Comment represents another alternative and clearly correct method to prove the same thing. Furthermore, we clarify that our interest in Kaniadakis entropy is due to the fact that this entropy has a transparent physical significance, emerging within the special relativity.
A first use of Kaniadakis entropy in the context of quantum information is presented. First we show that (as all smooth and concave trace-form entropies) it exhibits some properties allowing it to be a possible candidate for a generalized quantum information theory. We then use it to determine the degree of entanglement. The influence of the parameter κ , that underpins Kaniadakis entropy, on the mutual information measure is then highlighted. It is shown that Kaniadakis entropy reduces the mutual information, which is always smaller than its usual von Neumann counterpart. Our results may contribute to the ongoing investigation involving generalized entropies in the context of quantum information.
Von Neumann proposed two quantum entropies, one in 1927 and the other in 1929. He intended to define thermodynamic entropy for quantum states in both cases. The 1927 entropy is the well-known von Neumann entropy, which is zero for all pure states. The 1929 entropy, which is nonzero for almost all pure states, has largely been ignored. After reviewing the history and properties of both entropies, we argue that the 1927 entropy is not thermodynamic entropy and should be used simply as a measure of entanglement, whereas the 1929 entropy is the true thermodynamic entropy for quantum states.