Can the entanglement entropy be the origin of black-hole entropy?

Entanglement entropy is often speculated as a strong candidate for the origin of black-hole entropy. To judge whether or not this speculation is true, 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 black-hole entropy. This 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 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 of entanglement energy with the entanglement entropy. We find that both of these two kinds of thermodynamics show significant differences from black-hole thermodynamics if no gravitational effects are taken into account. These differences are, in particular, highlighted in the context of the third law of thermodynamics. Finally we see how the inclusion of gravity alters the thermodynamics of the entanglement. We give a suggestive argument that the thermodynamics of the entanglement behaves like black-hole thermodynamics if the gravitational effects are included properly. Thus the entanglement entropy passes a nontrivial check to be the origin of black-hole entropy.