Quantum minimal surfaces from quantum error correction

We show that complementary state-specific reconstruction of logical (bulk) operators is equivalent to the existence of a quantum minimal surface prescription for physical (boundary) entropies. This significantly generalizes both sides of an equivalence previously shown by Harlow [1]; in particular, we do not require the entanglement wedge to be the same for all states in the code space. In developing this theorem, we construct an emergent bulk geometry for general quantum codes, defining areas associated to arbitrary logical subsystems, and argue that this definition is functionally unique. We also formalize a definition of bulk reconstruction that we call state-specific product unitary reconstruction. This definition captures the quantum error correction (QEC) properties present in holographic codes and has potential independent interest as a very broad generalization of QEC; it includes most traditional versions of QEC as special cases. Our results extend to approximate codes, and even to the non-isometric codes that seem to describe the interior of a black hole at late times.

Automated summary

We demonstrate the equivalence between complementary state-specific reconstruction of logical operators and the existence of a quantum minimal surface prescription for physical entropies. We also introduce the concept of emergent bulk geometry for general quantum codes, and formalize a definition of bulk reconstruction called state-specific product unitary reconstruction.