Code Properties from Holographic Geometries

Abstract

Almheiri, Dong, and Harlow [J. High Energy Phys. 04 (2015) 163.] proposed a highly illuminating connection between the AdS=CFT holographic correspondence and operator algebra quantum error correction (OAQEC). Here, we explore this connection further. We derive some general results about OAQEC, as well as results that apply specifically to quantum codes that admit a holographic interpretation. We introduce a new quantity called price, which characterizes the support of a protected logical system, and find constraints on the price and the distance for logical subalgebras of quantum codes. We show that holographic codes defined on bulk manifolds with asymptotically negative curvature exhibit uberholography, meaning that a bulk logical algebra can be supported on a boundary region with a fractal structure. We argue that, for holographic codes defined on bulk manifolds with asymptotically flat or positive curvature, the boundary physics must be highly nonlocal, an observation with potential implications for black holes and for quantum gravity in AdS space at distance scales that are small compared to the AdS curvature radius.

Information flow in black hole evaporation

“…obtained for the CFT vacuum in [4]. Thus, we find that the region B ∞ of the bath needed, with QM L , to recover the black hole interior exhibits uberholography -it has zero total length.…”

“…Of course, the process of systematically removing intermediate intervals from the bath region can be continued, cutting out smaller and smaller subregions, as discussed in section 3.1.4. Repeating this process ad infinitum, following [4], we produce a fractal structure from which the interior of the black hole can still be reconstructed. It is interesting that the (Minkowski) dimension characterizing this fractal matches that found for the CFT vacuum in [4].…”

“…In fact, by an iterative process where, at each step, a hole is punched into each connected region in this bath region, this total length can be reduced arbitrarily close to zero. This procedure was designated 'überholography', where a bulk region is encoded in a subset of the boundary with lower (fractal) dimension than the dimension of the boundary [4].…”

Discrete gravity on random tensor network and holographic Rényi entropy

“…Given a boundary region A containing a number of open links of the tensor network, the entanglement entropy of the tensor network in each of the above approaches reproduces an analog of the RT formula (See e.g. [37][38][39][40][41][42][43][44][45][46] for some more recent developments)…”

Approximate Quantum Markov Chains

“…To summarize, one-dimensional systems that satisfy the locality assumption (5.110) can be efficiently represented by a finite sequence of recovery maps given by Theorem 5.5. Theorem 5.5 has been successfully applied in other areas such as high energy physics [43,45,117], solid state physics [29,160,179], quantum error correction [66,118], quantum information theory [7,22,33,90,97,101], and foundations of quantum mechanics [109].…”