2.3 Thermal entropy

Instead of a pure state one could have started with a mixed thermal state at temperature T with density matrix −1 ρ0(A, B ) = e −T H(A,B), where H (A, B ) is the Hamiltonian of the global system. In this case the relation (4View Equation) 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 (3View Equation) reproduces the thermal entropy. For further references we give here the expression
Sthermal = -d--Γ (d-)ζ(d) Td−1Vd− 1 (7 ) πd∕2 2
for the thermal entropy of a massless field residing inside a spatial (d − 1)-volume Vd−1 at temperature T.
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