ar X iv : g r - qc / 9 70 10 59 v 1 2 8 Ja n 19 97 YITP - 96 - 36 Can the entanglement entropy be the origin of black - hole entropy ?

Entanglement entropy is often speculated as a strong candidate for the origin of the black-hole entropy. To judge whether this speculation is true or not, it is effective to investigate the whole structure of thermodynamics obtained from the entanglement entropy, rather than just to examine the apparent structure of the entropy alone or to compare it with that of the black hole entropy. It is because entropy acquires a physical significance only when it is related to the energy and the temperature of a system. From this point of view, we construct a ‘thermodynamics of entanglement’ by introducing an entanglement energy and compare it with the black-hole thermodynamics. We consider two possible definitions of entanglement energy. Then we construct two different kinds of thermodynamics by combining each of these different definitions 1 of entanglement energy with the entanglement entropy. We find that both of these two kinds of thermodynamics show significant differences from the black-hole thermodynamics. These differences are in particular highlighted in the context of the third law of thermodynamics. It strongly suggests that the black-hole thermodynamics cannot be fully explained by the concept of entanglement alone.