Entanglement is not a lower bound for geometric discord

We show that partial transposition of any $2\otimes n$ state can have at most $(n-1)$ number of negative eigenvalues. This extends a decade old result of the $2\otimes 2$ case by Sanpera et al. [Phys. Rev. A 58, 826 (1998)]. We then apply this result to critically assess an important conjecture recently made by Girolami and Adesso [Phys. Rev. A 84, 052110 (2011)], namely, the (normalized) geometric discord should always be lower bounded by squared negativity. This conjecture has strengthened the common belief that measures of generic quantum correlations should be more than those of entanglement. Our analysis shows that unfortunately this is not the case and we give several counterexamples to this conjecture. All the examples considered here are in finite dimensions. Surprisingly, there are counterexamples in $2\otimes n$ for any $n>2$. Coincidentally, it appears that the $4\otimes 4$ Werner state, when seen as a $2\otimes 8$ dimensional state, also violates the conjecture. This result contributes significantly to the negative side of the current ongoing debates on the defining notion of geometric discord as a good measure of generic correlations.

Figure 1

A successful simulation of the quantity $\mathcal{D}-\mathcal{N}{}^2$ (dimensionless) for ${10}^5$ randomly generated $2\otimes 3$ states using the code given in the Supplemental Material [23]. The red solid line represents the line $\mathcal{D}=\mathcal{N}{}^2$. Note that in this simulation we found only one counterexample to the conjecture.

Figure 2

A simulation of the quantity $\mathcal{D}-\mathcal{N}{}^2$ (dimensionless) for ${10}^5$ randomly generated $2\otimes 4$ states. The red solid line represents the line $\mathcal{D}=\mathcal{N}{}^2$. The number of states violating the conjecture increases very fast with $n$.

Figure 3

Geometric discord ($\mathcal{D}$, shown by solid blue line) and squared negativity ($\mathcal{N}{}^2$, shown by dashed red line) for the $2\otimes 8$ dimensional Werner states. We note that for all $z\in [-1,-8/13)$, the state violates the conjecture. All quantities plotted are dimensionless.