Let $\tau_1$ and $\tau_2$ be two normal states on a von Neumann algebra $M$. Denote $\sigma_t^{\tau_i}(i=1, 2)$ by the modular automorphism group associated with $\tau_i$. If $\sigma_t^{\tau_1}(x)=\sigma_t^{\tau_2}(x)$ for any $x\in M$, can we conclude that $\tau_1=\tau_2$?
What about the case when $M$ is a factor?
If $M$ is finite but it is not a factor, you can construct two different traces. As the modular group of a trace is trivial, you get two distinct normal states with the same modular group.
If I remember correctly, the condition $\sigma^\varphi_t=\sigma^\psi_t$ is equivalent to the existence of invertible positive $h\eta M_\psi\cap M_\varphi$ with $\psi=h\,\varphi$. What I don't remember is whether the centralizer can be non-trivial in a factor.