Given a system $A$ and its complement $\bar{A}$, we know that the entanglement entropy is given by $$ S_A = - \text{Tr} ( \rho_A \log \rho_A ), $$ where $\rho_A$ is the reduced density matrix obtained by tracing out the degrees of freedom in $\bar{A}$.
It is possible to geometrize this quantity for a field theory by considering its dual spacetime and calculating the area of a minimal surface anchored on the boundary of the region $A$, as proposed by Ryu-Takayanagi.
WLOG, let us assume that the dual spacetime is Schwarzschild-AdS$_3$, such that the field theory is at finite temperature and minimal surfaces are simply geodesics. It is possible to show that these geodesics start wrapping around the event horizon as the size of the region $A$ increases. However, the length of a geodesic completely wrapped around the horizon corresponds to the Bekenstein-Hawking thermal entropy.
Therefore we know that the entanglement entropy contains information about the thermal state of the system, at least for large enough regions. What if the size of $A$ is such that the corresponding minimal surface does not go deep in the bulk? Is there any way of formalizing the distinction between thermal and quantum entropies for a subsystem $A$ of arbitrary size?