Reply to "Comment on 'Witnessed entanglement and the geometric measure of quantum discord' "
aa r X i v : . [ qu a n t - ph ] A p r Reply to “Comment on ‘Witnessed entanglement and the geometric measure of quantum discord’ ”
Tiago Debarba, ∗ Thiago O. Maciel, and Reinaldo O. Vianna Departamento de F´ısica - ICEx - Universidade Federal de Minas Gerais,Av. Pres. Antˆonio Carlos 6627 - Belo Horizonte - MG - Brazil - 31270-901. (Dated: October 17, 2018)We show that the mistakes pointed out by Rana and Parashar [Phys. Rev. A , 016301 (2013)] do notinvalidate the main conclusion of our work [Phys. Rev. A , 024302 (2012)]. We show that the errors a ff ectedonly a particular application of our general results, and present the correction. PACS numbers: 03.67.Mn, 03.65.Aa
Rana and Parashar [1] claim that our bounds between geo-metrical discord and entanglement [2] are incorrect. They giveexamples of violations of our bounds and suggest it has to dowith non-monotonicity of geometrical discord in the Hilbert-Schmidt norm. The authors started their comment revising ourdefinition of geometrical discord and pointing a typographicalerror in the definition of negativity. We defined negativity asthe sum of the negative eigenvalues of the partial transpose ofthe state, Eq.16 of our work, while some authors further nor-malize this quantity. Their critique about the normalization ofthe geometrical discord in the Hilbert-Schmidt norm is alsoirrelevant, for the normalized geometrical discord is greaterthan ours.The first counterexample which would violate our results isthe maximally entangled state for two qubits ( φ + ). They con-sider the negativity as 1, while the 2-norm geometrical discordis 1 /
2. But it is not correct. Consider Eq.20 , D (2) ( φ + ) ≥ E w T r ( W φ + ) . (1)We have D (2) ( φ + ) = /
2, and E w = T r ( W φ + φ + ) = T r ( P − φ T + ) = /
2, where P − is the projector associated to thenegative eigenvalue of the partial transpose of φ + . T r ( W φ + )is the number of negative eigenvalues of the partial transpose,which is 1. Thus D (2) = / ≥ E w = / ⊗ D (2) ( ρ ) = .
01 and E w / T r [ W ρ ] = . E w is the negativity, and Eq.20is satisfied. However, in the comment the equation taken wasEq.21, and via that relation we get N / ( d − = . T r [ W ρ ] ≤ d −
1, for a system with dimension d ⊗ d ′ and d ≤ d ′ . In the counterexample we have T r [ W ρ ] =
10, i.e.the partial transpose of the state has 10 negative eigenvaluesand not d − =
1, and this is the reason of the wrong vi-olation in Eq.21. In the comment, the authors conclude thatthe violation comes from the fact that D (2) ( ρ ) is not a mono-tonic distance, but monotonicity does not play any role in ourbounds. Finally, the authors claim that Eq.27 is not valid. Equation27 is a particular case of Eq.22, where we get a linear rela-tion between geometrical discord calculated via trace normand witnessed entanglement. This bound is valid only for en-tanglement measures whose optimal entanglement witnesseslive in the domain − I ≤ W ≤ I , and the entanglement wit-ness for the negativity is not in this domain, which explainsthe problem with the bound in Eq.27. An example of entan-glement measure for which this bound is valid is the randomrobustness of entanglement, Eq.28. Equation 27 can be easilycorrected by means of an inequality more general than Eq.22,namely: D (1) ≥ E w || W ρ || ∞ , (2)where || W ρ || ∞ is the greatest eigenvalue of the optimal entan-glement witness of the state ρ [4]. Note that this bound is validfor every witnessed entanglement.In conclusion, the main results of our work are Eq.20 andEq.22, which are rigorously correct. They were calculatedfrom first principles , via well known inequalities for operatorsand properties of entanglement witnesses. We made two mis-takes when specializing for the negativity, as discussed andclarified above. The conjecture proposed by D. Girolami andG. Adesso [3] about the interplay between geometrical quan-tum discord and entanglement is implicit in Eq.20 and Eq.22. ∗ Electronic address: debarba@fisica.ufmg.br[1] S. Rana and P. Parashar, Phys. Rev. A , 016301 (2013).[2] T. Debarba, T.O. Maciel, R.O. Vianna, Phys. Rev. A , 024302(2012).[3] D. Girolami and G. Adesso, Phys. Rev. A , 052110 (2011).[4] Take the well known inequality for operators A and B , || A || q || B || p ≥ | T r [ AB † ] | , for 1 / q + / p =
1. Set A = ρ − ξ , where ξ is ρ ’s nearest non-discordant state, and set B = W ρ , where W ρ isthe optimal entanglement witness of ρ , then follows straightfor-wardly D ( p ) ≥ E w / || W ρ || q . For p =
1, we have q = ∞∞