I have a question arising from von Neumann's C*-algebra formulation of quantum mechanics. In it, a state is a $\mathbb{C}$-linear functional on a C*-algebra $A$

$\rho:A\rightarrow\mathbb{C}$

satisfying the following conditions:

• $\rho$ is positive: for every $a\in A$, we have $ \rho(a^*a) \geq 0 \in \mathbb{R}$;

• $\rho$ is normalized: $\rho(\mathbf{1}) = 1$.

By the Gelfand-Naimark-Segal (GNS) construction, for each such state, there is a *-representation $\pi_{\rho}$ of $A$ in a Hilbert space $\mathscr{H}_{\rho}$ and a unit vector $\Omega\in\mathscr{H}_{\rho}$ such that $$\rho(a)=\langle\pi_\rho(a)\Omega,\Omega\rangle$$ for every $a\in A$.

Question: what conditions do two states $\rho$, $\rho'$ have to satisfy such that the associated representations be equivalent ?