We study how much of the proto-area response can be inferred from the bulk von Neumann entropy. In the perturbative holographic code setting considered by CCKLP and Witten, we keep the recovery frame fixed and analyze the response averaged over the GUE at second order in the encoding perturbation. Near the maximally mixed bulk spectrum, the response is fixed by the entropy through quadratic order. For $d_1\geq3$, spectral information first enters at cubic order through the third centered moment, so states with the same entropy can have different responses. We determine the resulting response range on small equal-entropy level sets and obtain its sharp leading $D^{3/2}$ dependence, including the coefficient. The corresponding minimax error for prediction from entropy alone is one half of this range. For $d_1=2$, the entropy instead determines the response exactly.
Comments
v2: Revised version with strengthened results and improved discussion of equal-entropy ambiguity
We study how much of the proto-area response can be inferred from the bulk von Neumann entropy. In the perturbative holographic code setting considered by CCKLP and Witten, we keep the recovery frame fixed and analyze the response averaged over the GUE at second order in the encoding perturbation. Near the maximally mixed bulk spectrum, the response is fixed by the entropy through quadratic order. For d1≥3, spectral information first enters at cubic order through the third centered moment, so states with the same entropy can have different responses. We determine the resulting response range on small equal-entropy level sets and obtain its sharp leading D3/2 dependence, including the coefficient. The corresponding minimax error for prediction from entropy alone is one half of this range. For d1=2, the entropy instead determines the response exactly.
A holographic code assigns a state-dependent entropy remainder to every boundary region, but such remainders need not be the extremal lengths of any one bulk metric. We study this common-source question for the proto-area of a calibrated recovery, the entropy of a boundary region minus that of the matter recovered from it. At zeroth order the reference values must match the lengths of one reference geometry in a fixed renormalization scheme, and this matching also determines the entropy-to-length factor. On a graph with a fixed minimum-cut chamber the data derivatives must lie in the image of the cut-incidence matrix, whose left null vectors are witnesses. On the hyperbolic disk the first variation of renormalized length is the rank-two geodesic X-ray transform, with closed range in an adapted data space, and the geodesic displacement, fixed by a forced Jacobi equation, enters at second order. The resulting normal Hessian gives a gauge-invariant criterion, necessary and sufficient, for a regular pair of response coefficients to come from one metric two-jet. For the constant-curvature family generated by a boundary reparametrization the Hessian has a nonzero rank-four mode, so the image of the length map is curved. For a fixed recovery the proto-area Hessian is minus the Bogoliubov-Kubo-Mori information lost in recovery, which the Petz map decomposes orthogonally at a product reference, and which, on a finite-dimensional space of logical directions with positive minimum retained fraction, is at most a sharp multiple of the retained information. Geometric realization requires this loss to reproduce the normal length Hessian on every interval. Two codes show that exact regional area operators can fail to sew into one graph geometry, and that a matched reference with vanishing first response can fail at second order. All criteria refer to one recovery branch and one extremal-surface branch.
Quantum error-correcting codes provide a powerful framework for emergent spacetime, yet existing subsystem erasure-correcting holographic code models describe only quantum fields on a fixed background: in such codes, the entropic area term is state independent and cannot capture gravitational backreaction. We argue that this limitation is intrinsic to exact subsystem complementary recovery and that incorporating backreaction naturally leads to approximate quantum error correction. We introduce a Ryu-Takayanagi-like entropy decomposition for approximate subsystem erasure-correcting codes, defining bulk matter entropy via optimal recovery and a complementary proto-area entropy as the difference between boundary entropy and recoverable bulk entropy. For a broad class of skewed quantum codes obtained by small nonlocal perturbations of exact codes, the proto-area increases monotonically with bulk entropy, closely aligning with the changes in extremal surface areas in gravitational systems. We identify the origin of this response as a form of tripartite non-local magic in the Choi state of the encoding map, which vanishes in stabilizer codes and controls the leading matter-geometry coupling in approximate subsystem erasure-correcting codes.
In the context of quantum theories of spacetime, one overarching question is how quantum information in the bulk spacetime is encoded holographically in boundary degrees of freedom.It is particularly interesting to understand the correspondence between bulk subregions and boundary subregions in order to address the emergence of locality in the bulk quantum spacetime.For the AdS/CFT correspondence, it is known that this bulk information is encoded redundantly on the boundary in the form of an error-correcting code.Having access only to a subregion of the boundary is as if part of the holographic code has been damaged by noise and rendered inaccessible.In quantum-information science, the problem of recovering information from a damaged code is addressed by the theory of universal recovery channels.We apply and extend this theory to address the problem of relating bulk and boundary subregions in AdS/CFT, focusing on a conjecture known as entanglement wedge reconstruction.Existing work relies on the exact equivalence between bulk and boundary relative entropies, but these are only approximately equal in bulk effective field theory, and in similar situations it is known that predictions from exact entropic equalities can be qualitatively incorrect.We show that the framework of universal recovery channels provides a robust demonstration of the entanglement wedge reconstruction conjecture as well as new physical insights.Most notably, we find that a bulk operator acting in a given boundary region's entanglement wedge can be expressed as the response of the boundary region's modular Hamiltonian to a perturbation of the bulk state in the direction of the bulk operator.This formula can be interpreted as a noncommutative version of Bayes's rule that attempts to undo the noise induced by restricting to only a portion of the boundary.To reach these conclusions, we extend the theory of universal recovery channels to finite-dimensional operator algebras and demonstrate that recovery channels approximately preserve the multiplicative structure of the operator algebra.
In the model of approximate holographic codes introduced by Cao, Cheng, Karthikeyan, Li, and Preskill and analyzed perturbatively by Witten, the error in entanglement wedge reconstruction is smaller than the gravitational backreaction by a factor 1/d_2^2 at second order in the skewing parameter, where log d_2 plays the role of the area term. It was recently observed numerically that this hierarchy fails at fourth order for non-degenerate bulk spectra, through a coefficient growing as d_2^4. In the present work a closed form for this coefficient is proved: writing p = (p_1, ..., p_d1) for the Schmidt spectrum of the bulk state, the leading fourth-order term of the mean reconstruction error is epsilon^4 sigma_W^4 alpha(p) d_2^4, where alpha(p) = (1/2) d_1^2 (sum_i 1/p_i - d_1^2) coincides with (1/2) d_1^4 chi^2(u || p), the Neyman chi-squared divergence of the bulk spectrum from the uniform distribution u = (1/d_1, ..., 1/d_1). The mechanism is identified exactly: the ensemble mean of the second-order state perturbation is a decoherence drift of magnitude proportional to d_2^2 directed toward maximal mixing, and alpha is the Kubo-Mori quadratic form of this drift. The spectral-degeneracy protection of the hierarchy observed numerically is thereby explained as the equality case of Jensen's inequality. The formula, conjectured from five calibration spectra, was subsequently confirmed by a zero-parameter blind prediction at a previously unmeasured configuration (d_1 = 4, a new spectrum), which it matched to 0.08 per cent (-0.13 sigma). This is a corrected version of the preprint DOI 10.13140/RG.2.2.14852.69768. Three presentational defects are corrected and no result, proof, or measured value changes. First, the figure committed in advance for the blind test was 601,378, whereas the closed form proved in the paper gives 15552 alpha = 601,344 exactly; the committed figure corresponds to a rounded alpha = 38.6689 and exceeds the exact one by 34, a relative excess of 5.7e-5. Both figures are now stated with the deviation of the measurement from each, and the test is passed against either. Second, the fourth calibration spectrum was printed as (0.453, 0.333, 0.213), which does not sum to one; it is now given as the exact spectrum (34, 25, 16)/75 that was simulated, for which the stated alpha = 4.0202 is correct. Third, the blind test is described as zero-parameter rather than registered: the prediction uses no fitted quantity of any kind, but it was not deposited with a third party, so no external timestamp is claimed. The corrections are listed in the "Note on this version" on the first page. Simulation scripts and data: Zenodo DOI 10.5281/zenodo.21223051.
In this Dissertation, different aspects of holographic entanglement entropy are explored. In quantum information theory, entanglement is a useful resource needed to perform quantum operations. Entanglement entropy quantifies this resource and even if computable in quantum field theories, it is quite hard to calculate explicitly. However, in the context of gauge/gravity duality, (holographic) entanglement entropy was proposed by Ryu and Takayanagi (RT) to be captured by the area of a minimal surface in Anti DeSitter (AdS) space, which is extremely simple to compute. We begin by using the holographic dictionary to derive the RT formula. This is done by first considering smooth bulk geometries which are dual to n copies of the boundary geometry glued in a particular way. The action of these geometries computes the Renyi entropies S_n for any integer n. We give a prescription to analytically continue the action to non-integer n and argue that, as n goes to 1,S_1, the entanglement entropy, is given by the area of a minimal surface, reproducing the RT conjecture. That is, we reduced the RT proposal to the equality between bulk and boundary partition functions, a standard entry in the dictionary. We then generalize the previous formalism to account for corrections due to bulk quantum fields (1/N corrections in the boundary). This allows us to derive a new formula: boundary entanglement entropy is given by the area of the minimal surface plus bulk entanglement entropy. This extends RT beyond the planar limit and we present several predictions of the proposal. Our exploration of holographic entanglement entropy continues by considering different states: by comparing the quantum corrected entropy for nearby states we obtain the modular hamiltonian operator in bulk perturbation theory. From this expression of the modular hamiltonian, we derive that relative entropy is bulk relative entropy and that the modular flow is the bulk modular flow.
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.