Validity of black hole complementarity in the BTZ black hole

TL;DR

This paper investigates the validity of black hole complementarity and the no-cloning theorem in the BTZ black hole, proposing a local thermodynamic approach to reconcile potential violations and comparing with black string and Schwarzschild black holes.

Contribution

It introduces a local thermodynamic method to uphold the no-cloning theorem in the BTZ black hole, addressing potential violations suggested by energy calculations.

Findings

The required energy for information duplication in the BTZ black hole can be small.

In contrast, for the black string, the energy exceeds the total mass, preventing duplication.

A local thermodynamic approach can restore the validity of the no-cloning theorem.

Abstract

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.