Consider the (asymmetric) quantum discord, defined as (borrowing notation from Eq. 4.13c of Zurek's review): $$\mathcal D(\mathcal S:\mathcal A) = I(\mathcal S:\mathcal A) - \chi(\rho_{\mathcal A}),$$ where $$\chi(\rho_{\cal A}) \equiv \max_{\{\boldsymbol\pi_k\}}(H(\rho_{\mathcal A}) - \sum_k p_k H(\mathcal A|\boldsymbol\pi_k)).$$ Here we're considering bipartite states $\rho\equiv\rho_{\mathcal S\mathcal A}$, denoting with $I(\mathcal S:\mathcal A)$ the standard quantum mutual information, and the maximum is taken with respect to all possible POVMs $\{\boldsymbol\pi_k\}_k$, so that $\sum_k \boldsymbol\pi_k=I$ and $\boldsymbol\pi_k\ge0$. The outcome probabilities read $p_k\equiv \operatorname{tr}(\boldsymbol\pi_k \rho_{\cal SA})$, and $H(\mathcal A|\boldsymbol\pi_k)$ is the entropy of $\mathcal A$ conditioned on having measured $\mathcal S$ and found the $k$-th outcome, meaning that $H(\mathcal A|\boldsymbol\pi_k)$ is the (von Neumann) entropy of the state $\rho_k\equiv \operatorname{tr}_{\cal S}(\boldsymbol\pi_k\rho_{\cal SA})/p_k$.
It is mentioned in the above review (just above (4.12c)) that there are situations where the maximum in the definition of $\chi(\rho_{\cal A})$, or equivalently, the accessible mutual information, is achieved with non-projective POVMs. I'm looking for explicit "simple" examples of this being the case.
In (Modi et al. 2011) the authors mention (section I.3, page 11) that the discord is always optimized by rank-1 extremal POVMs. Most topical reviews (Modi et al. 2011, Modi 2013, Bera et al. 2017) mention the fact that the discord is in general achieved with non-projective POVMs, but I'm failing to actually extract an explicit simple examples from the provided citations. Going through the relevant references given in (Modi 2011), I see that (Chen et al. 2011) work out the discord for two-qubit X states. However, if I understand correctly, this is still a numerical result. (Galve et al. 2011) show that for rank-2 states PVMs are sufficient, but only show numerically that POVMs are more generally needed. (Shi et al. 2011) also analyze the discord of X-states and show that non-projective PVMs are needed, but also in this case I don't know how to extract an explicit example of such a state and associated POVM.
Is there any analytical example of a state where the discord/accessible information (be it the asymmetric or symmetric one) is achieved with non-projective measurements?
