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 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
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 inclusion of gravity alter the thermodynamics of the
entanglement. We give a suggestive argument that the thermodynamics of the
entanglement behaves like the black-hole thermodynamics if the gravitational
effects are included properly. Thus the entanglement entropy passes a
non-trivial check to be the origin of the black-hole entropy.Comment: 40 pages, Latex file, one figur
