Can we always find a unitary operation connecting qubit states with given eigendecompositions?

Consider the density matrices $rho_0 = |0 rangle langle 0|$ and $rho_1 = |1 rangle langle 1|$. Let ${p_1, p_2}$, and ${p_3, p_4}$ be two probability distributions, that is,
$$0 leq p_1, p_2, p_3, p_4 leq 1$$
$$p_1 + p_2 = 1$$
$$~text{and}~ p_3 + p_4 = 1.$$
These probability distributions refers two mixed states $rho = p_1 rho_0 + p_2 rho_1$ and $rho’ = p_3 rho_0 + p_4 rho_1$. Now I have the following questions:

1. Is there is a unitary matrix $U$ such that $rho’ = U^dagger rho U$ ?

2. How to calculate $U$ when all $p_1, p_2, p_3$ and $p_4$ are known, if $U$ exists?

3. Can we represent $U$ with a quantum circuit with qubits if $U$ exists?