Analytical progress on symmetric geometric discord: Measurement-based upper bounds

Quantum correlations may be measured by means of the distance from the state to the subclass of states $\Omega$ having well-defined classical properties. In particular, a geometric measure of asymmetric discord [Dakić et al., Phys. Rev. Lett. 105, 190502 (2010)] was recently defined as the Hilbert-Schmidt distance from a given two-qubit state to the closest classical-quantum (CQ) correlated state. We analyze a geometric measure of symmetric discord defined as the Hilbert-Schmidt distance from a given state to the closest classical-classical (CC) correlated state. The optimal member of $\Omega$ is just a specially measured original state for both the CQ and CC discords. This implies that this measure is equal to the quantum deficit of postmeasurement purity. We discuss some general relations between the CC discords and explain why an analytical formula for the CC discord, contrary to the CQ discord, can hardly be found even for a general two-qubit state. Instead of such an exact formula, we find simple analytical-measurement-based upper bounds for the CC discord which, as we show, are tight and faithful in the case of two qubits and may serve as independent indicators of two-party quantum correlations. In particular, we propose an adaptive upper bound, which corresponds to the optimal states induced by single-party measurements: optimal measurement on one of the parties determines an optimal measurement on the other party. We discuss how to refine the adaptive upper bound by nonoptimal single-party measurements and by an iterative procedure which usually rapidly converges to the CC discord. We also raise the question of optimality of the symmetric measurements realizing the CC discord on symmetric states and give a partial answer for the qubit case.

Figure 1

Venn-type diagram showing the sets of the CC (${\Omega }_S={\Omega }_{AB}$), CQ (${\Omega }_A$), and QC (${\Omega }_B$) states together with the closest states ${\sigma }_i^{*}$ ($i=S,A,B$) according to the CC ($D_S$), CQ ($D_A$), and QC ($D_B$) geometric discords, respectively. States ${\sigma }_{S^{\prime }}$ and ${\sigma }_{S^{\prime \prime }}$, which correspond to the adaptive upper bound $D_S^{\left(\mathrm{aub}\right)}$, are the closest CC states for ${\sigma }_A^{*}$ and ${\sigma }_B^{*}$, respectively. This is an intuitive graph, but more precisely, the point ${\sigma }_A^{*}$ (${\sigma }_B^{*}$) should be on the line between $\rho$ and ${\sigma }_{S^{\prime }}$ (${\sigma }_{S^{\prime \prime }}$).

Figure 2

Geometric discords and their tight upper bounds for the state $\rho (p,\phi =\pi /2)$, given by Eq. (46), as a function of the parameter $p$: The CC discord, $D_S$ [solid (blue) curve], and CQ and QC discords, $D_A=D_B$ [dotted (magenta) curve], together with the adaptive upper bound, $D_S^{\left(\mathrm{aub}\right)}$ [dashed (red) curve], and unoptimized nonadaptive upper bounds, $D_{S11}^{\left(\mathrm{nub}\right)}=D_{S22}^{\left(\mathrm{nub}\right)}$ [dot-dashed (green) curve]. We note that the optimized upper bound ${\tilde{D}}_S^{\left(\mathrm{aub}\right)}=D_S$ and $D_S^{\left(\mathrm{aub}\right)}={\tilde{D}}_{S12}^{\left(\mathrm{nub}\right)}={\tilde{D}}_{S21}^{\left(\mathrm{nub}\right)}$ for any $p$. All the upper bounds, except ${\tilde{D}}_S^{\left(\mathrm{aub}\right)}$, are discontinuous at $p=1/2$, while the corresponding vertical lines are added for clarity only.

Figure 3

Same as Fig. 2, but for the state $\rho (p,\phi )$ as a function of phase $\phi$ for fixed $p=2/3$.