Hierarchy of Efficiently Computable and Faithful Lower Bounds to Quantum Discord

Quantum discord expresses a fundamental nonclassicality of correlations that is more general than entanglement, but that, in its standard definition, is not easily evaluated. We derive a hierarchy of computationally efficient lower bounds to the standard quantum discord. Every nontrivial element of the hierarchy constitutes by itself a valid discordlike measure, based on a fundamental feature of quantum correlations: their lack of shareability. Our approach emphasizes how the difference between entanglement and discord depends on whether shareability is intended as a static property or as a dynamical process.

Figure 1

Symmetric local broadcasting. A Bose-$k$-symmetric broadcasting channel ${\mathrm{\Lambda }}_{B\to B_{+}^k}$ has output in the fully symmetric subspace of $B^k=B_1{B}_2\dots B_k$. How well ${\rho }_{A{B}_1}={\mathrm{Tr}}_{\setminus A{B}_1}({\mathrm{\Lambda }}_{B\to B_{+}^k}[{\rho }_{AB}])$ can approximate ${\rho }_{AB}$ depends on the classicality of correlations in ${\rho }_{AB}$.

Figure 2

A hierarchy of lower bounds to quantum discord. Let ${\rho }_{AB}(\theta )=\frac{1}{2}{|0⟩⟨0|}_A\otimes {|{\psi }_0(\theta )⟩⟨{\psi }_0(\theta )|}_B+\frac{1}{2}{|1⟩⟨1|}_A\otimes {|{\psi }_1(\theta )⟩⟨{\psi }_1(\theta )|}_B$, with $|{\psi }_a(\theta )⟩=\cos (\theta /2)|0⟩+(-1{)}^a\sin (\theta /2)|1⟩$, $a=0$, 1, for $\theta \in [0,\pi /2]$. From bottom to top, we plot $D_{F,{\mathcal{L}}^{{\mathrm{Sym}}_{+}(k)}}$ for $k=2$, 3, 4, 5 (dashed lines), $D_{F,{\mathcal{L}}^{\mathrm{EB}}}$ (line with circles), as calculated via SDP, and the discord proper $D$ (on $B$) (solid line) as calculated in [94, 95]. Any element in the hierarchy certifies quantitatively that the state ${\rho }_{AB}(\theta )$ is classical on $B$ only for $\theta =0$, $\pi /2$.