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

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 ,ϕ _A ,θ _B ,ϕ _B ) coefficients that define the optimal computational basis as

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

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:

Therefore, by defining

we can write the classical correlations quantifier (2) as

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

where

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:

and

and

and

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

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

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.