Repeated Extraction of Scrambled Quantum Data: Sustainability of the Hayden-Preskill Type Protocols

We introduce and study the problem of scrambler hacking, which is the procedure of quantum information extraction from and installation on a quantum scrambler given only partial access. This problem necessarily emerges from a central topic in contemporary physics - information recovery from systems undergoing scrambling dynamics, such as the Hayden-Preskill protocol in black hole studies - because one must replace quantum data with another when extracting it due to the no-cloning theorem. For large scramblers, we supply analytical formulas for the optimal hacking fidelity, a quantitative measure of the effectiveness of scrambler hacking with limited access. In the two-user scenario where Bob attempts to hack Alice's data, we find that the optimal fidelity converges to $64/(9\pi^2)\approx0.72$ with increasing Bob's hacking space relative to Alice's user space. We applied our results to the black hole information problem and showed that the limited hacking fidelity implies the reflectivity decay of a black hole as an information mirror, which questions the solvability of the black hole information paradox through the Hayden-Preskill type protocol.