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

Quantum Computing Asked on May 26, 2021

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.

One 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?

Answered by Quantum Mechanic on May 26, 2021

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