### 2.3 Thermal entropy

Instead of a pure state one could have started with a mixed thermal state at temperature with
density matrix , where is the Hamiltonian of the global system. In this
case the relation (4) is no more valid and the entropy depends on the size of the total system as well as on
the size of each sub-system. By rather general arguments, in the limit of large volume the reduced
density matrix approaches the thermal density matrix. So that in this limit the entanglement
entropy (3) reproduces the thermal entropy. For further references we give here the expression
for the thermal entropy of a massless field residing inside a spatial -volume at temperature
.