Abstract
Recent work showed holographic error correcting codes to have simple universal features at O(1/G). In particular, states of fixed Ryu-Takayanagi (RT) area in such codes are associated with flat entanglement spectra indicating maximal entanglement between appropriate subspaces. We extend such results to one-loop order (O(1) corrections) by controlling both higher-derivative corrections to the bulk effective action and dynamical quantum fluctuations below the cutoff. This result clarifies the relation between the bulk path integral and the quantum code, and implies that i) simple tensor network models of holography continue to match the behavior of holographic CFTs beyond leading order in G, ii) the relation between bulk and boundary modular Hamiltonians derived by Jafferis, Lewkowycz, Maldacena, and Suh holds as an operator equation on the code subspace and not just in code-subspace expectation values, and iii) the code subspace is invariant under an appropriate notion of modular flow. A final corollary requires interesting cancelations to occur in the bulk renormalization-group flow of holographic quantum codes. Intermediate technical results include showing the Lewkowycz-Maldacena computation of RT entropy to take the form of a Hamilton-Jacobi variation of the action with respect to boundary conditions, corresponding results for higher-derivative actions, and generalizations to allow RT surfaces with finite conical angles.
Highlights
Point here is that codes with complementary recovery are characterized by their pattern of entanglement between appropriate factors of HCFT, and at leading order in G refs. [20, 21] showed this pattern to be the same in each Hφ up to unitary transformations
This result clarifies the relation between the bulk path integral and the quantum code, and implies that i) simple tensor network models of holography continue to match the behavior of holographic CFTs beyond leading order in G, ii) the relation between bulk and boundary modular Hamiltonians derived by Jafferis, Lewkowycz, Maldacena, and Suh holds as an operator equation on the code subspace and not just in code-subspace expectation values, and iii) the code subspace is invariant under an appropriate notion of modular flow
A corollary is confirmation of the above-mentioned conjecture that dynamical IR quantum fluctuations merely determine which state in the code subspace arises from a given path integral and that properties of the code itself are determined by treating the cutoff-scale effective action as a classical variational principle
Summary
It is useful to briefly review the role of quantum error correcting codes in holography. The bulk calculation deriving (2.13) relies on properties of variational principles for higher-derivative actions that we will establish in section 3 below These properties imply that with fixed geometric entropy, a purely classical (saddle-point) calculation of Renyi entropies would again give a flat entanglement spectrum for the |χα Rα2 R2α state, and that the associated density matrix on Rα2 would be a projector onto a subspace of dimension set by the associated saddle-point von Neumann entropy (i.e., by the geometric entropy). This is the prediction of the classical effective action conjecture, and we see that it agrees precisely with the results for the spectrum of |χα Rα2 R2α described above.
Sign-in/Register to view highlights & summary Talk to us
Join us for a 30 min session where you can share your feedback and ask us any queries you have
Schedule a callDisclaimer: All third-party content on this website/platform is and will remain the property of their respective owners and is provided on "as is" basis without any warranties, express or implied. Use of third-party content does not indicate any affiliation, sponsorship with or endorsement by them. Any references to third-party content is to identify the corresponding services and shall be considered fair use under The CopyrightLaw.
