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Bidirectional holographic codes and sub-AdS locality

Zhao Yang, Patrick Hayden, Xiao-Liang Qi

arXiv:1510.03784v1hep-thcond-mat.str-elquant-ph

TL;DR

Earlier holographic tensor networks left boundary states outside a code subspace without the same interpretation and could not probe sub-AdS locality. The paper constructs a bidirectional holographic code and reports that its highly nonlocal bulk can appear local at sub-AdS scales or in flat space.

  • Problem

    Earlier error-correcting holographic codes applied their mapping only to a boundary subspace, while holographic bulk operators were expected to admit redundant recovery from different boundary regions.

    Source: §1 · p. 3

  • Method

    The paper contracts pluperfect tensors into a bidirectional holographic code mapping the full boundary Hilbert space to a gauge-constrained physical bulk subspace.

    Source: §2.1 · p. 5, §3 · p. 9, §5 · p. 18

  • Results

    At sub-AdS scales or in flat space, almost all low-order bulk configurations can map isometrically, making the bulk appear local until highly complicated correlations are measured.

    Source: §4.3 · p. 17, §5 · p. 18

  • Takeaways & Limitations

    The BHC’s physical bulk is nonlocal, yet its low-energy behavior can resemble local degrees of freedom on bulk sites.

    Source: §5 · p. 18

    Random tensors only weakly approximate pluperfect tensors, and that approximation may not justify substituting them into most arguments without studying error propagation.

    Source: §2.3 · p. 8

Abstract

from arXiv · show

Tensor networks implementing quantum error correcting codes have recently been used to construct toy models of holographic duality explicitly realizing some of the more puzzling features of the AdS/CFT correspondence. These models reproduce the Ryu-Takayanagi entropy formula for boundary intervals, and allow bulk operators to be mapped to the boundary in a redundant fashion. These exactly solvable, explicit models have provided valuable insight but nonetheless suffer from many deficiencies, some of which we attempt to address in this article. We propose a new class of tensor network models that subsume the earlier advances and, in addition, incorporate additional features of holographic duality, including: (1) a holographic interpretation of all boundary states, not just those in a "code" subspace, (2) a set of bulk states playing the role of "classical geometries" which reproduce the Ryu-Takayanagi formula for boundary intervals, (3) a bulk gauge symmetry analogous to diffeomorphism invariance in gravitational theories, (4) emergent bulk locality for sufficiently sparse excitations, and (5) the ability to describe geometry at sub-AdS resolutions or even flat space.

1 Introduction

The introduction presents the bidirectional holographic code (BHC) as a tensor-network construction intended to address limitations of earlier holographic codes while adding features beyond a boundary code subspace.

1 Introduction

  • Earlier constructions addressed bulk-operator recovery only within a boundary subspace, separated RT-state and redundant-operator-mapping networks, and left sub-AdS locality inaccessible.The cited discussion describes these as limitations of prior proposals.Source: §1 · pp. 3–4
  • The BHC maps the full boundary Hilbert space isometrically into a physical bulk subspace defined by gauge symmetry, where no local gauge-invariant operators exist.The introduction characterizes the full bulk theory as intrinsically nonlocal.Source: §1 · p. 4
  • Some bulk states act as classical geometries and correspond to boundary states satisfying the RT formula for single intervals.These geometries have non-positive curvature.Source: §1 · p. 4
  • Low-density excitations can be mapped isometrically so bulk sites appear independent, whereas sufficiently high excitation density reveals the theory’s nonlocality.The introduction compares the sparse-excitation behavior to local quantum field theory.Source: §1 · p. 4
  • The BHC can use sub-AdS-resolution or flat geometries while keeping actual bulk degrees of freedom proportional to boundary degrees of freedom.At sub-AdS resolution, the bulk’s low-energy physics appears local despite the apparent excess of bulk points.Source: §1 · p. 4

2 Definition of the bidirectional holographic code

The bidirectional holographic code is built from pluperfect tensors, whose three conditions support the boundary mapping, classical-geometry states, and isometric encoding of selected excitations. An explicit qutrit construction demonstrates their existence, while large-D random states approximate them only weakly.

2.1 Building block: pluperfect tensors

  • Pluperfect tensors have four in-plane indices and a bulk index; their three conditions enable the all-boundary-state mapping, RT-satisfying geometry states, and isometric maps for selected operators.For n = 2, the bulk index has dimension D^4 and the in-plane indices each have dimension D.Source: §2.1 · p. 5

2.2 Explicit construction of a pluperfect tensor

  • A numerical search constructs a qutrit example in which nine orthogonal states satisfy Conditions 2 and 3, then form part of an 81-dimensional basis for the bulk index.The construction begins with a four-leg perfect tensor and selects local-unitary-related states.Source: §2.2 · pp. 6–7

2.3 Pluperfect tensors as idealized random states

  • In the large-D limit, pluperfect tensors idealize D^2 Haar-random orthogonal states in a D^4-dimensional space, but this weak approximation may not support direct substitution in the paper’s arguments.The authors identify error propagation in random tensor networks as an open question.Source: §2.3 · p. 8
  • Condition 3 makes every single-site observable’s matrix elements vanish between distinct pluperfect states, an exact strengthening of the off-diagonal ETH condition.The states therefore behave like idealized quantum-chaotic eigenstates with respect to single-site operators.Source: §2.3 · p. 8

3 Gauge invariance and physical operators

Contracting pluperfect tensors defines an isometric map, with physical bulk states given by its image and invariant under a generalized gauge symmetry. This symmetry acts on links, leaves no nontrivial boundary image for interior operators, and is compared to diffeomorphism invariance.

3 Gauge invariance and physical operators

  • After normalization, contracting internal lines defines an isometry M, and the physical bulk Hilbert space is the image of that map.The physical-state condition is expressed through membership in this image.Source: §3 · p. 9, Figure 3 · p. 9
  • SU(D) transformations on contracted links define a gauge symmetry that can be pushed to bulk operators, with each site’s transformation depending on its four attached links.This link-based symmetry differs from ordinary gauge theories, which have gauge-invariant Wilson loops; here there are no gauge-invariant local bulk operators.Source: §3 · pp. 9–10
  • Any operator acting only on interior bulk sites maps to a boundary operator proportional to the identity.The projected operator commutes with arbitrary SU(D) transformations on boundary legs.Source: §3 · p. 10
  • The gauge redundancy relates multiple tensor-network representations of boundary states and is presented as an analog of diffeomorphism invariance, potentially with a larger symmetry.The bulk Hilbert space corresponds to tensor-network states on the given graph, while entangled bulk states correspond to their linear superpositions.Source: §3 · pp. 10–11

4 Emergent locality in the “low-energy subspace"

The section constructs classical-geometry states with RT entropies and defines bulk operators relative to those states, while locality emerges only for sufficiently sparse excitations. It characterizes which bulk configurations admit isometric boundary mappings and how that allowance scales on a square lattice.

4.1 Classical geometry states

  • Classical-geometry product states in flat or negatively curved networks satisfy the RT formula for singly connected boundary regions through arrow-defined isometries across geodesics.The arrow construction requires non-positive curvature; a loop-free assignment gives an isometry from the geodesic to the boundary region and its complement.Source: §4.1 · p. 12, Figure 5 · p. 13
  • The RT construction does not generically establish the formula for multiple boundary intervals, and superpositions of classical-geometry product states generically fail to satisfy it.Arrow drawing is sufficient but not necessary for isometries, so the limitation does not rule out RT-satisfying networks in particular cases.Source: §4.1 · p. 13

4.2 Bulk “local” operators and the causal wedge

  • Bulk operators become nontrivially representable on the boundary when defined relative to a classical vacuum with other bulk sites fixed in that vacuum.The resulting isometric encodings can support different boundary regions; the causal wedge comprises sites admitting such an isometry and is generally smaller than the entanglement wedge.Source: §4.2 · p. 14, Figure 6 · p. 15
  • For multiple geodesics homologous to a boundary region, the smallest-enclosed-site geodesic is conjectured to bound its causal wedge, but this is not rigorously proved.Operators on sites enclosed by a larger candidate geodesic but not the smaller one cannot be reconstructed isometrically from that boundary region.Source: §4.2 · pp. 14–15

4.3 Kinematics of the bulk holographic theory

  • An allowed n-site configuration is one whose bulk sites map isometrically to the boundary, preserving their operator commutation relations while other sites remain in the ground state.Arrow drawings provide a lower bound on the number of allowed configurations; configurations with more than half the boundary-site count cannot map isometrically.Source: §4.3 · pp. 15–17
  • On an L × L square lattice, the allowed-configuration fraction decreases from 1 at small n toward 0 as n approaches 2L, with disallowed configurations rare until n is comparable to L.The counting is a lower bound, so the actual allowed fraction for a specific BHC can be higher; the backreaction threshold grows with L at least as fast as √L.Source: Figure 8 · p. 16, §4.3 · pp. 17–18
  • As L grows, any fixed finite excitation count lies below the threshold at sufficiently large size, so almost all such bulk n-point functions look like those of a local QFT.The paper associates n ∼ √L with the scale where backreaction becomes significant and n = 2L with black-hole formation.Source: §4.3 · pp. 17–18

5 Conclusion and discussion

The bidirectional holographic code (BHC) maps the entire boundary Hilbert space unitarily to a gauge-constrained bulk, with classical bulk states reproducing the RT formula for single boundary regions. It also yields emergent locality for sufficiently sparse excitations, while leaving important questions about bulk dynamics and other geometries open.

5 Conclusion and discussion

  • The BHC maps the entire boundary Hilbert space unitarily to a bulk whose physical Hilbert space is selected by a gauge-invariance condition.This condition is unlike ordinary gauge symmetry because the physical Hilbert space has no local operators; classical bulk product states can nevertheless reproduce the RT formula for single boundary regions.Source: §5 · p. 18
  • Bulk operators can be defined at some sites when the remainder is in a classical state, and a single-site operator can map to multiple boundary regions.Bulk sites that map isometrically to a boundary region form that region’s causal wedge.Source: §5 · p. 18
  • For a flat L × L lattice, below a critical number of sites n_c ∝ L, almost all n-site configurations map isometrically to the boundary, yielding emergent locality even at sub-AdS scales or in flat space.The bulk theory remains highly nonlocal, but its nonlocality is difficult to observe without highly complicated correlation functions.Source: §5 · p. 18
  • The mapped bulk Hamiltonian is generically nonlocal, and preserving locality in low-energy dynamics requires special boundary Hamiltonians that remain unidentified.The BHC map itself does not directly describe the dynamics of either theory.Source: §5 · pp. 18–19
  • The physical interpretation of positive-curvature geometries and boundaryless geometries such as de Sitter, the continuum limit, and boundary states with global symmetries remain open questions.For positive curvature, classical-geometry boundary states generically do not satisfy the RT formula; for de Sitter, the BHC naively defines a single bulk state.Source: §5 · p. 19

A Pluperfect tensor search for stabilizer codes

This section expresses stabilizer codes as vectors over integers modulo D and translates the pluperfection conditions into linear-algebraic tests. It describes how to search for operators defining pluperfect tensors by checking those conditions.

A Pluperfect tensor search for stabilizer codes

  • Each k-qudit stabilizer operator is encoded by a 2k-dimensional vector over integers from 0 to D − 1, with entries recording the powers of X_D and Z_D on each qudit.The section uses this vector representation to translate pluperfection conditions into linear algebra.Source: A Pluperfect tensor search for… · p. 19
  • Choosing prime D makes Z_D a finite field, where stabilizer commutation can be expressed as a linear-algebraic relation.A qudit stabilizer code consists of commuting operators built from D-dimensional on-site Pauli matrices X_D and Z_D.Source: A Pluperfect tensor search for… · pp. 19–20
  • Starting from a four-leg perfect tensor, applying products of unitary operators on individual legs makes Condition 2 hold for the states |nm⟩ = A^nB^m|0⟩.The section represents A and B as products of Pauli matrices associated with vectors over Z_D.Source: A Pluperfect tensor search for… · p. 20
  • Condition 3 requires nonidentity operator products to fail to commute with at least one stabilizer, which becomes a full-rank 4 × 4 matrix test over Z_D for each leg.The test includes the vectors for A, B, and the single-site X or Z operator on that leg.Source: A Pluperfect tensor search for… · p. 20
  • Randomly generating vectors A and B and verifying the resulting conditions provides a straightforward way to find operators defining pluperfect tensors.The corresponding operators are products of Pauli matrices.Source: A Pluperfect tensor search for… · p. 20

B Dynamic programming algorithm bounding the number of allowed configurations

The section bounds allowed bulk-excitation configurations by characterizing when bulk excitations can be mapped isometrically to the boundary. On rectangles, iterative boundary expansion and recurrence relations reduce the count to smaller lattices.

B Dynamic programming algorithm bounding the number of allowed configurations

  • An isometry exists exactly when curves can be pushed to the boundary through moves that include vertices, expand at excitation points, never shrink elsewhere, and sweep over every excitation.The stated conditions apply to general graphs.Source: B Dynamic programming algorithm… · p. 21
  • On rectangles, an edge expands by adding an adjacent vertex and then including vertices in its row or column until the curve is rectangular again.The curve length must not decrease, and an edge with multiple adjacent bulk excitations cannot be expanded by an allowed move.Source: B Dynamic programming algorithm… · p. 21, Figure 9 · p. 21
  • A rectangle is allowed if one boundary has at most one excitation and removing it leaves an allowed configuration on the smaller rectangle.This criterion allows the rectangle to be reduced iteratively until its boundaries shrink to a point.Source: B Dynamic programming algorithm… · pp. 21–22
  • The count A(n, x, y) is computed from smaller rectangles by recurrence relations, with inclusion-exclusion correcting configurations counted through multiple boundary removals.The terms F1 through F4 account for configurations reducible in one through four ways.Source: B Dynamic programming algorithm… · p. 22
  • All configurations with up to 3 excitation points are allowed, and the recurrence relations and initial conditions determine A(n, x, y) for any n, x, and y.The initial conditions set A(n, x, y) to zero for negative arguments and specify values for zero excitations and 1 × i rectangles.Source: B Dynamic programming algorithm… · pp. 22–23
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