Approximate locality, black hole complementarity and overlapping qubits

How can we reconcile the apparent cloning of information in an evaporating black hole with the principles of unitary quantum mechanics? The authors investigate whether the assumption of exact tensor factorization in Hilbert space is too restrictive, proposing that "overlapping" degrees of freedom—which only approximately commute—can resolve the tension between the black hole interior and Hawking radiation without violating fundamental quantum laws.

Drawing on the construction of overlapping qubits by Chao, Reichardt, Sutherland, and Vidick (CRSV), the authors define a toy model where quasi-local degrees of freedom are embedded within a fundamental Hilbert space. By using the Johnson-Lindenstrauss lemma, they construct operators that satisfy standard commutation relations up to a small error term (epsilon). This allows them to represent the interior and the radiation as two different, yet related, representations of the same underlying algebraic structure. They then analyze how this overlap affects entropy calculations and the resulting Page curve.

• Resolution of No-Cloning: Because the interior and radiation operators are different representations of the same fundamental algebra, they do not require an exact factorization of the Hilbert space. This prevents the violation of the no-cloning theorem, as the "cloning" is an artifact of assuming independent tensor factors where none exist.
• Page Curve Recovery: The authors demonstrate that by explicitly accounting for the operator overlaps in the entropy calculation, the system exhibits a Page curve, consistent with the expected unitary evolution of an evaporating black hole.
• The model shows that a large number of overlapping qubits can be compressed into a significantly smaller Hilbert space, providing a potential information-theoretic mechanism for the holographic principle.