What do states that are uncorrelated with respect to a given local measurement look like?

Question

Consider a bipartite state $rho$, and let $Pi^Aequiv {Pi^A_a}_a$ and $Pi^Aequiv{Pi^B_b}_b$ be local projective measurements.
Suppose that measuring $rho$ in these bases gives uncorrelated outcomes. More precisely, this means that the probability distribution
$$p(a,b)equiv operatorname{Tr}[(Pi^A_aotimes Pi^B_b)rho],tag A$$
is uncorrelated: $p(a,b)=p_A(a)p_B(b)$ for some $p_A,p_B$.
I should stress that I'm considering this property for a fixed basis here, so this can easily hold regardless of the separability of $rho$.
Given $Pi^A,Pi^B$, is there a way to characterise the set of $rho$ producing uncorrelated measurement outcomes?
For example, if the measurements are rank-1 and diagonal in the computational basis, $Pi^A_a=Pi^B_a=|arangle!langle a|$, then any state of the form
$$sum_{ab} sqrt{p_A(a) p_B(b)} e^{ivarphi_{ab}}|a,brangletag B$$
will give uncorrelated outcomes (assuming of course $p_A,p_B$ are probability distributions).
These are product states for $varphi_{ij}=0$, but not in general.
Nonpure examples include the maximally mixed state (for any measurement basis) and the completely dephased version of any pure state of the form (B).
Is there a way to characterise these states more generally? Ideally, I'm looking for a way to write the general state satisfying the constraints. If this is not possible, some other way to characterise the class more or less directly.

Quantum Mechanic · Answer

Here's another example that works, essentially extending the rank-$1$ argument to arbitrary rank. Suppose all of the projectors within a set are orthogonal to each other. We can write each projector as
$$Pi^A_0=sum_{i=0}^{n_0(A)} |iranglelangle i|,quad Pi^A_1=sum_{i=n_0(A)+1}^{n_0(A)} |iranglelangle i|,quadcdots$$ for some integers $n_0<n_1<cdots$ that depend on the fixed measurement basis and can differ between the two systems $n_i(A)neq n_i(B)$.
We can define a general eigenstate of each projection operator as
$$|phi_a^Arangle=sum_{i=n_a(A)}^{n_{a+1}(A)}phi_i|irangle$$ for any amplitudes $phi_i$ that could depend on $a$ and $A$, and similarly for system $B$.
Then any state of the form
$$rho=sum_k rho_k|Phi_kranglelanglePhi_k|$$ with each component pure state taking the form
$$|Phi_krangle=sum_{a,b}Phi_{a,b} |phi_a^Arangle |phi_b^Brangle$$ will generate the measurement results
$$p(a,b)=sum_k rho_k langlephi_a^A|Pi^A_a|phi_a^Aranglelanglephi_b^B|Pi^B_b|phi_b^Brangle.$$ Each of $rho_k$, $langlephi_a^A|Pi^A_a|phi_a^Arangle$, and $langlephi_b^B|Pi^B_b|phi_b^Brangle$ is positive, so I suspect $p(a,b)$ to only be uncorrelated when there is a single nonzero $rho_k$. This means that one could only consider pure states that satisfy some strange condition that looks reminiscient of mixed-state separability. Thoughts?

What do states that are uncorrelated with respect to a given local measurement look like?

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