Background
Let's say I have a gas of $N$ particles where I cannot distinguish between the particles at a temperature $T$.
Its density matrix is given by $\rho$. Note, if my friend happens to measure the energy of all $N$ particles for large $N$ the system can still be described by the density matrix $\rho$.
To see this we can write the density matrix in the energy eigenbasis:
$$ \rho =\sum_{i=0} p_i |E_i \rangle \langle E_i |$$
where $p_i$ is likelihood of eigenstate of the $i$'th energy eigenstate and $|E_i \rangle$ is the energy eigenket. The measurement of energy effectively takes $|E_j \rangle \to |E_j \rangle $ due to orthogonality. Think of $p_i$ as the number of particles with energy $E_i$ out of $N$ particles.
My friend however can use the wavefunction $|\psi \rangle$ to describe the system.
Claim: Both me and my friend will agree on the quantity $\rho |\psi \rangle$ and its time evolution:
$$ i \hbar \frac{\partial \rho | \psi \rangle}{\partial t} = H \rho |\psi \rangle$$
(Use the product rule to prove the above).
Question
In the thought experiment of Wigner's friend is $\rho |\psi \rangle$ an invariant where $|\psi\rangle$ is what Wigner's friend measures and $\rho$ is the pure state Wigner uses to describe the system?
