| Summary: | 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; 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.
|
|---|