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Revista mexicana de física

versión impresa ISSN 0035-001X

Rev. mex. fis. vol.67 no.5 México sep./oct. 2021  Epub 28-Mar-2022

https://doi.org/10.31349/revmexfis.67.052301 

Corrige el artículo: 10.31349/revmexfis.64.662

Errata

Errata to ‘Local available quantum correlations for Bell diagonal states and Markovian decoherence’

David M. Bellorin R.a 

Hermann L. Albrecht Q.a   

aDepartamento de Física, Universidad Simón Bolívar, AP 89000, Caracas 1080, Venezuela.


Abstract

In this brief erratum, we complete the analysis presented previously in [RMF 64 (2018) 662-670] regarding the quantifiers of the classical correlations and the so-called local available quantum correlations for Bell diagonal states. A correction is introduced in their previous expressions once two cases within the optimizations are included.

Keywords: Quantum correlations; quantum discord; entanglement; Bell diagonal states; Werner states

PACS: 03.65.Ud; 03.67.-a; 03.67.Mn

In Ref. [1], the analytical results of the correlation quantifiers related to the so-called local available quantum correlations (LAQC) [2] for the family of Bell diagonal states [3] were presented. These states are written in the Bloch representation as

ρBD=1414+i=13ciσiσi, (1)

where the coefficients c i ∈ [−1,1] are such that ρ BD is a well-behaved density matrix (i. e. has non-negative eigenvalues) and σ i are the well-known Pauli matrices.

The classical correlations quantifier defined in Ref. [2] can be written in terms of the R ij (θ A A B B ) coefficients that define the optimal computational basis as

C(ρAB)=minθA,ϕAθB,ϕB{i,jRij(θA,ϕA,θB,ϕB)×log2[Rij(θA,ϕA,θB,ϕB)Ri(θA,ϕA)Rj(θB,ϕB)]}. (2)

Since Bell diagonal (BD) states have null local Bloch vector, it is straightforward that they are invariant under subsystem exchange AB. Therefore, only two angles, θ and ϕ, are necessary, and the coefficients R ij (θ,ϕ) are given by

Rijθ,ϕ=141+-1i+jc3+-1i+j12cos2θ2sin2θ2×c1+c2+cos2ϕc1-c2-2c3, (3)

with R 00(θ,ϕ) = R 11(θ,ϕ), R 01(θ,ϕ) = R 10(θ,ϕ), and R i = 1/2.

The minimization in (2) leads to three different cases:

  • I For θ = 0 and ϕ = 0:

R000,0=141+c3R010,0=141-c3. (4)

  • II For θ = π/2 and ϕ = 0:

R00π2,0=141+c1R01π2,0=141-c1. (5)

  • III For θ = π/2 and ϕ = π/2:

R00π2,π2=141+c2,R01π2,π2=141-c2. (6)

Therefore, by defining

cmminc1,c2,c3, (7)

we can write the classical correlations quantifier (2) as

CρBD=1+cm2log21+cm+1-cm2log21-cm. (8)

The above expression is the same as Eq. (33) in [1] but now the minimization achieved for θ = π/2 and ϕ = 0 when c m = |c 1| has been included.

The LAQC quantifier is given by

LρABmax{Φ1,Φ2}IΦ1,Φ2, (9)

where

IΦ1,Φ2=i,jPiA,jB,Φ1,Φ2×log2PiA,jB,Φ1,Φ2PiA,Φ1PjB,Φ2, (10)

with P(i A ,j B ,Φ1 ,Φ2) the probability distributions associated with the complementary basis [4] of ρ AB written in the optimal computational basis, and P(i A ,Φ1) an P(j B ,Φ2) are the corresponding marginal probabilities. Contrary to what is stated in [1], the density matrix of BD states does not remain invariant when written in the optimal computational basis. That is only true for Werner [5] and Werner-like states [6,7].

The density matrix ρ~BD and their corresponding P(i,j,Φ) for each θ and ϕ, with P(0,0,Φ) = P(1,1,Φ), P(0,1,Φ) = P(1,0,Φ), and P(i,Φ) = 1/2, are the following:

  • I) For θ = 0 and ϕ = 0:

ρρ~BD=141+c300c1-c201-c3c1+c200c1+c21-c30c1-c2001+c3 (11)

and

P0,0,Φ=141+c1+c22+c1-c22cos2ΦP1,0,Φ=141-c1+c22-c1-c22cos2Φ., (12)

  • II )For θ=π/2 and ϕ= 0:

ρρ~BD=141+c100c3-c201-c1c3+c200c3+c21-c10c3-c2001+c1 (13)

and

P0,0,Φ=141+c3+c22+c3-c22cos2Φ,P1,0,Φ=141-c3+c22-c3-c22cos2Φ. (14)

  • III) For θ = π/2 and ϕ = π/2:

ρρ~BD=141+c200c3-c101-c2c3+c100c3+c11-c20c3-c1001+c2 (15)

and

P0,0,Φ=141+c3+c12+c3-c12cos2Φ,P1,0,Φ=141-c3+c12-c3-c12cos2Φ. (16)

For each θ and ϕ, Φ depends on |c 1| > |c 2|, |c 2| > |c 3|, or |c 1| > |c 3|, respectively. Therefore, as was done with the classical correlations quantifier (8), defining

cMmaxc1,c2,c3 (17)

allows us to write a general expression for the LAQC quantifier that encompasses all these possibilities:

LρBD=1+cM2log21+cM+1-cM2log21-cM. (18)

As with the classical correlations quantifiers, the above expression is equivalent to the one presented in Eq. (36) of [1]. Nevertheless, this newly defined c M also includes |c 3|. The case of c M = |c 3| arises when the density matrix ρ BD is written in the optimal computational basis with θ = π/2.

Acknowledgments

This work was partially funded by the 2020 BrainGain Venezuela grant awarded to H. Albrecht by the Physics without Frontiers program of the ICTP. The authors would like to thank the support given by the research group GID-30, Teoría de Campos y Óptica Cuántica, at the Universidad Simón Bolívar, Venezuela, as well as to thank D. Mundarain, from the Universidad Católica del Norte, Chile, and M.I. Caicedo and J. Stephany, from Universidad Simón Bolívar, Venezuela, for their comments and suggestions.

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Received: June 01, 2021; Accepted: June 01, 2021

e-mail: albrecht@usb.ve

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