6 Relevant Works
- Holographic states emphasizes faithful reproduction of entanglement and correlation properties.
- Holographic codes emphasize exact error-correction.
- HaPPY codes sacrifices exact (become state-dependent) error-correction for algebraic decay.
- Both HaPPY codes and HTN are based on hyperbolic tilings, which are readily generalizable to \(S^2\) boundary. The hyperbolic tiling structure is sufficient to ensure the Ryu-Takayanagi formula.
HaPPY Code
In (Pastawski et al. 2015), the authors defined holographic states and codes based on a regular tiling of hyperbolic space, and perfect tensors (corresponding to maximal absolutely entangled states).
Definition 6.1 (holographic state) A holographic state is a tensor network composed of perfect tensors which cover a geometric manifold with boundary, with interior tensor legs contracted. The open legs are associated with physical degrees of freedom.
Due to the property of perfect tensors, which can be viewed as isometric mappings, we can associate open indices with the bulk tensors and turn the whole TN into an isometric mapping from bulk to boundary degrees of freedom.
Definition 6.2 (holographic code) A holographic code is an isometry from bulk to boundary degrees of freedom, represented by a TN covering some geometric manifold with boundary.
For example, a pentagon tiling of \(\mathbb H^2\) needs to be combined with a rank-\(6\) tensor to form a holographic code. Holographic states are associated with AdS / CFT models by varying the tensor ansatz, while holographic codes provide a whole code isometric mapping of Hilbert spaces.
The merits and shortcomings of the HaPPY code model are:
- Exact RT formula for hyperbolic tilings (Theorem 2, and Appendix B).
- Erasure correction: bulk local observable can be realized by many boundary operators.
- Two-point correlators can be nontrivial for some cases.
Definition 6.3 (wedges and cones) Given a (not necessarily connected) boundary region:
- If \(A\) is connected, its causal wedge is the set of points
reached by applying the greedy algorithm.
- The greedy algorithm starts from \(A\) and progressively add tensors which are “more-than-half connected” to the current set.
- Any bulk local operator on the causal wedge can be reconstructed as a boundary operator supported on \(A\) (Theorem 5).
- Its geometric entanglement wedge is the set of bulk points bounded by \(A\) and the minimal surface with the same boundary as \(A\).
- Its greedy entanglement wedge is defined analogously.
In general, entanglement wedge is larger than the causal cone. - Hypothesis: some operators in the causal cone are reconstructable.
HTN and Holography
Hyper-invariant tensor-networks (Evenbly 2017) is built upon HaPPY codes. It is a holographic state construction which satisfies the following properties:
- Uniform in the holographic bulk (unlike MERA).
- Produces correlations and entanglement compatible with
critical ground states.
- Algebraic decay of correlations: the numerical spectra of descending super-operators are nontrivial.
- Holographic causality (area law): see below.
- Every finite subsystem has a canonical representation, even in the limit of infinite-depth networks.
Possible future work:
- Fully characterize the degrees of freedom of doubly unitary tensors.
Definition 6.4 (various cones) The various definitions of cones about a boundary region \(\mathcal R\) are:
- Causal cone : the set of tensors in the bulk which possibly affect \(\mathrm{tr}_{\bar {\mathcal R}} |\psi\rangle\langle\psi|\).
- Entanglement wedge : the set of bulk tensors bounded by \(\mathcal R\) and \(\gamma_R\), where \(\gamma_R\) is the minimal surface whose boundary matches the boundary of \(\mathcal R\) (and such that maximizes the volume of the enclosure).
Theorem 6.1 (holographic causality) In the limit \(N\to \infty\), the HTN construction satisfies, for simply connected \(\mathcal R\), \(\mathcal R(\mathcal R)\approx \mathcal E(\mathcal R)\).
Proof: Apply the coarse-graining super-operator continuously. By symmetry, choose the bulk “center” to be the place where the shrinking regime transitions to the steady regime.
HTN Codes
HTN Codes (Steinberg, Feld, and Jahn 2023) constructed a way to associate bulk degrees of freedom to HTNs. It yields an model which satisfies algebraic boundary decay, at the cost of having state-dependent, instead of perfect, error-correction (complementary reality).