Abstract
Quantum error correcting codes with finite-dimensional Hilbert spaces have yielded new insights on bulk reconstruction in AdS/CFT. In this paper, we give an explicit construction of a quantum error correcting code where the code and physical Hilbert spaces are infinite-dimensional. We define a von Neumann algebra of type II$_1$ acting on the code Hilbert space and show how it is mapped to a von Neumann algebra of type II$_1$ acting on the physical Hilbert space. This toy model demonstrates the equivalence of entanglement wedge reconstruction and the exact equality of bulk and boundary relative entropies in infinite-dimensional Hilbert spaces.
Highlights
The study of entanglement entropy has utilized results in the mathematical field of operator algebras [1,2,3]
Since AdS/CFT implies that information in the bulk is encoded redundantly in the boundary, quantum error correction is a natural framework in which to elucidate the connection between holographic quantum field theories and their gravity duals [5,6,7,8]
In order to study a more realistic toy model where boundary subregions are characterized by infinite-dimensional von Neumann algebras, we should consider quantum error correcting codes defined on infinite-dimensional Hilbert spaces
Summary
The study of entanglement entropy has utilized results in the mathematical field of operator algebras [1,2,3]. For any jΨi; jΦi ∈ Hcode with jΨi cyclic and separating with respect to Mcode; SΨjΦðMcodeÞ 1⁄4 SuΨjuΦðMphysÞ; and SΨjΦðM0codeÞ 1⁄4 SuΨjuΦðM0physÞ; where SΨjΦðMÞ is the relative entropy: Tensor networks with a finite number of nodes have been used to construct QECC for finite-dimensional Hilbert spaces, which have yielded physical insights into holography [7,9] One such example is the HAPPY code which demonstrates the kinematics of entanglement wedge reconstruction [17]. (iii) Using the code and physical Hilbert spaces and an isometry relating them, we define von Neumann algebras Mcode and Mphys. It follows that our tensor network model satisfies both statements in Theorem 1.1. We first review some preliminary facts about the three qutrit code
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