  Based on the gedanken experiment for black hole complementarity in the
Schwarzschild black hole, we calculate the energy required to duplicate
information in the BTZ black hole under the assumption of absorbing boundary
condition and its dual solution of the black string, respectively, in order to
justify the validity of the no-cloning theorem in quantum mechanics. For the
BTZ black hole, the required energy for the duplication of information can be
made fairly small, whereas for the black string it exceeds the total mass of
the black string, although they are related to each other under the dual
transformation. So, the duplication of information might be possible in the BTZ
black hole in contrast to the case of the black string, so that the no-cloning
theorem could be violated for the former case. To save the duplication of
information for the BTZ black hole, we perform an improved gedanken experiment
by using the local thermodynamic quantities near the horizon rather than those
defined at infinity, and show that the no-cloning theorem could be made valid
even in the BTZ black hole. We also discuss how this local treatment for the
no-cloning theorem can be applied to the black string as well as the
Schwarzschild black hole innocuously.
