Optimal measurements to access classical correlations of two-qubit states

We analyze the optimal measurements to access classical correlations in arbitrary two-qubit states. Two-qubit states can be transformed into the canonical forms via local unitary operations. For the canonical forms, we investigate the probability distribution of the optimal measurements. The probability distribution of the optimal measurements is found to be centralized in the vicinity of a specific von Neumann measurement, which we call the maximal-correlation-direction measurement (MCDM). We prove that, for the states with zero discord and maximally mixed marginals, the MCDM is the optimal measurement. Furthermore, we give an upper bound of quantum discord based on the MCDM, and investigate its performance for approximating the quantum discord.

Figure 1

Probability distribution of the optimal measurements for the 100 000 random states according to the Hilbert-Schmidt measure [52] in the canonical form. Here, we discretized the domain of $\theta$ and $\varphi$ into 100 sections.

Figure 2

MCDM-based discord and quantum discord for the states $\rho (q)=(1-q)|{\psi }_0\rangle \langle {\psi }_0|+q|{\psi }_1\rangle \langle {\psi }_1|$, where $|{\psi }_0\rangle$ and $|{\psi }_1\rangle$ are given by Eq. (21).

Figure 3

MCDM-based discord $\tilde{\mathcal{D}}$ versus $\mathcal{D}$ for 100 000 random-density matrices according to the Hilbert-Schmidt measure [52]. The variance $\mathrm{avg}[(\tilde{\mathcal{D}}-\mathcal{D}){}^2]\simeq 3.7433\times {10}^{-5}$, where $\mathrm{avg}[x]$ means the average of $x$ over all the random density matrices.