Complementarity and Correlations

We provide an interpretation of entanglement based on classical correlations between measurement outcomes of complementary properties: States that have correlations beyond a certain threshold are entangled. The reverse is not true, however. We also show that, surprisingly, all separable nonclassical states exhibit smaller correlations for complementary observables than some strictly classical states. We use mutual information as a measure of classical correlations, but we conjecture that the first result holds also for other measures (e.g., the Pearson correlation coefficient or the sum of conditional probabilities).

Figure 1

Complementary correlation measurements. Each of two systems is subject to the measurement of one of two observables: either $A$ or $C$ on system 1 and either $B$ or $D$ on system 2. Correlations are evaluated between the results of $A$ and $B$ and between $C$ and $D$ (dashed lines). $A$ and $C$ are complementary on the first system, $B$ and $D$ on the second.

Figure 2

Examples of complementary correlations for different measures of correlation and different families of states. (a) Correlation $I_{AB}+I_{CD}$ plotted as a function of the parameter $p$ for the families of $p$-dependent two-qubit states indicated in the lower right panel. The dotted-line states are always separable and are nonzero discord QQ states for $p\ne 0$, 1; the dashed-line states (Werner states) are entangled for $p>1/3$, whereas the solid-line states are entangled for $p\ne 1/2$. Above the threshold 1 (stars), the states are certainly entangled. (b) Same as previous for $|{\mathcal{C}}_{AB}|+|{\mathcal{C}}_{CD}|$, note that the QQ state (dotted line) is on the conjectured threshold 1 (stars) for this measure of correlation. (c) Same as previous for ${\mathcal{S}}_{AB}+{\mathcal{S}}_{CD}$. Here there are two entanglement boundaries: The states that have a sum larger than 3 or smaller than 1 are conjectured to be entangled. Again, the dotted state coincides with one of the conjectured boundaries. The dashed line and the solid line are superimposed. Here $|\pm ⟩\equiv (|0⟩\pm |1⟩)/\sqrt{2}$ and $|{\mathrm{\Phi }}^{\pm }⟩\equiv (|00⟩\pm |11⟩)/\sqrt{2}$.