Respectfully, your metaphor doesn’t bear any similarity to the way AdS/CFT works. In AdS/CFT there are dual quantum systems with no information loss. A bit more precisely, the correspondence relates the partition functions of quantum gravity in AdS with the partition functions of conformal fields. This implies the existence of a map between the two, which in this case happens to be extremely complex and nonlocal.

Another way of saying it is that the two descriptions are different representations of the same object. There is no projection or information lost in switching between descriptions.