Can a mixed state (density matrix) in CFT be conformal?

Question

A vacuum in conformal theory can be defined as a state where correlation functions of primary operators take a particularly simple form. Another definition is being annihilated by all local Virasoro generators $L_n$ with $n>-2$. Can a mixed state with similar properties exist (except for $\rho=|0\rangle\langle0|$ of course)? In particular, can one find an example of a density matrix $\rho$ for which some basis of local fields $V_\Delta$ has the usual form of correlators? For example for two-point functions this should be something like
$$Tr\left(\rho V_{\Delta}(z_1)V_{\Delta}(z_2)\right)\propto \frac1{(z_1-z_2)^{2\Delta}}$$
and similarly for 3-point functions etc.  Potentially the definitions of the primary operators and/or their dimensions might be different from the original theory. Any hints and pointers are welcome!