  Quantum energy teleportation (QET) is the phenomenon in which locally
inaccessible energy is activated as extractable work through collaborative
local operations and classical communication (LOCC) with an entangled partner.
It closely resembles the more well-known quantum information teleportation
(QIT) where quantum information can be sent through an entangled pair with
LOCC. It is tempting to ask how QET is related to QIT. Here we report a first
study of this connection. Despite the apparent similarity, we show that these
two phenomena are not only distinct but moreover are mutually exclusive to each
other. We show a perturbative trade-off relation between their performance in a
thermal entangled chaotic many-body system, in which both QET and QIT are
simultaneously implemented through a traversable wormhole in an emergent
spacetime. To better understand their competition, we study the
finite-dimensional counterpart of two entangled qudits and prove a universal
non-perturbative trade-off bound. It shows that for any teleportation scheme,
the overall performance of QET and QIT together is constrained by the amount of
the entanglement resource. We discuss some explanations of our results.
