Given two qubit states $\rho_1$ and $\rho_2$. By applying some unitary $U$ we get $\rho = U(\rho_1 \otimes \rho_2)U^\dagger$. Can $\rho$ be entangled if either of $\rho_1$ or $\rho_2$ is a maximally mixed state?


The question is equivalent to the following: can a balanced mixture of (not all separable) states be entangled?

To see the equivalence, observe that if $\rho_2=I/d$ ($d$ is here the dimension of the space) then $$\rho=\frac1 d\sum_{k=1}^d U(\rho_1\otimes|k\rangle\langle k|)U^\dagger,$$ and if we don't put further restrictions on $U$ and $\rho_1$, then the only thing we know about $U(\rho_1\otimes |k\rangle\langle k|)U^\dagger$ is that they are orthogonal states (and if we are to hope the mixture is entangled, the states cannot be all separable).

This is not the case. As a counterexample in the two-qubit scenario, consider $$ \sqrt2|\Phi_+\rangle \equiv |00\rangle + |11\rangle, \qquad \sqrt3|\Psi\rangle \equiv |00\rangle + |10\rangle - |11\rangle. $$ You can then verify that $(|\Phi_+\rangle\!\langle\Phi_+|+|\Psi\rangle\!\langle\Psi|)/2$ is still entangled (using e.g. the PPT criterion).

Translated into the notation of the question, this example would correspond to $\rho_1=|0\rangle\langle 0|$ (or any other pure state), and $U$ such that $U|0,0\rangle=|\Phi_+\rangle$ and $U|0,1\rangle=|\Psi\rangle$.

This paper might be of interest to figure out when mixing pure maximally entangled states can result in an entangled state: (Flores and Galapon 2016).

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