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Nonarchimedean Holographic Entropy from Networks of Perfect Tensors
High Energy Physics - Theory · Submitted 10 Dec 2018
abstract compiled · 2026-09-23 · no language-model review
The authors' main claim We obtain a $p$-adic version of entropy which obeys a Ryu-Takayanagi like formula for bipartite entanglement of connected or disconnected regions, in both genus-zero and genus-one $p$-adic backgrounds, along with a Bekenstein-Hawking-type formula for black hole entropy.
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The abstract, sentence by sentence
We consider a class of holographic quantum error-correcting codes, built from perfect tensors in network configurations dual to Bruhat-Tits trees and their quotients by Schottky groups corresponding to BTZ black holes. The resulting holographic states can be constructed in the limit of infinite network size. We obtain a $p$-adic version of entropy which obeys a Ryu-Takayanagi like formula for bipartite entanglement of connected or disconnected regions, in both genus-zero and genus-one $p$-adic backgrounds, along with a Bekenstein-Hawking-type formula for black hole entropy. We prove entropy inequalities obeyed by such tensor networks, such as subadditivity, strong subadditivity, and monogamy of mutual information (which is always saturated). In addition, we construct infinite classes of perfect tensors directly from semiclassical states in phase spaces over finite fields, generalizing the CRSS algorithm, and give Hamiltonians exhibiting these as vacua.
background contribution method result interpretation main claim
| # | sentence | role | the authors' own work | states the main conclusion | strength |
|---|---|---|---|---|---|
| s0 | We consider a class of holographic quantum error-correcting codes, built from perfect tensors in network configurations dual to Bruhat-Tits trees and their quotients by Schottky groups corresponding to BTZ black holes. | contribution 0.80 | 0.72 | 0.07 | asserted 0.96 |
| s1 | The resulting holographic states can be constructed in the limit of infinite network size. | method 0.66 | 0.71 | 0.05 | asserted 0.99 |
| s2 | We obtain a $p$-adic version of entropy which obeys a Ryu-Takayanagi like formula for bipartite entanglement of connected or disconnected regions, in both genus-zero and genus-one $p$-adic backgrounds, along with a Bekenstein-Hawking-type formula for black hole entropy. | result 0.98 | 0.94 | 0.66 | asserted 1.00 |
| s3 | We prove entropy inequalities obeyed by such tensor networks, such as subadditivity, strong subadditivity, and monogamy of mutual information (which is always saturated). | result 0.87 | 0.95 | 0.16 | asserted 1.00 |
| s4 | In addition, we construct infinite classes of perfect tensors directly from semiclassical states in phase spaces over finite fields, generalizing the CRSS algorithm, and give Hamiltonians exhibiting these as vacua. | contribution 0.96 | 0.95 | 0.10 | asserted 1.00 |
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The paper
Original abstract
We consider a class of holographic quantum error-correcting codes, built from perfect tensors in network configurations dual to Bruhat-Tits trees and their quotients by Schottky groups corresponding to BTZ black holes. The resulting holographic states can be constructed in the limit of infinite network size. We obtain a $p$-adic version of entropy which obeys a Ryu-Takayanagi like formula for bipartite entanglement of connected or disconnected regions, in both genus-zero and genus-one $p$-adic backgrounds, along with a Bekenstein-Hawking-type formula for black hole entropy. We prove entropy inequalities obeyed by such tensor networks, such as subadditivity, strong subadditivity, and monogamy of mutual information (which is always saturated). In addition, we construct infinite classes of perfect tensors directly from semiclassical states in phase spaces over finite fields, generalizing the CRSS algorithm, and give Hamiltonians exhibiting these as vacua.
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References
How the literature uses it
13 arXiv papers cite this in 18 sentences
Papers citing it · 11 sentences without a confident label
Cites it as background 7 papers
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Several papers [ref] in the $p$-adic AdS/CFT literature have studied theories on the Tate curve[See also [ref] for a study of partition functions on this geometry from a tensor network point of view.] from a bulk perspective and further argued that this geometry furnishes the $p$-adic analog of the Euclidean BTZ black hole, with the parameter $w$ providing the analog of black hole entropy and area.[In light of the related themes of Ref. [ref] and the present paper, it may be useful to note that the former sought to extend the results of Ref. [ref] to $p$-adic AdS/CFT, while the present paper derives inspiration mainly from Refs. [ref] and [ref] .] While Schwarzschild-AdS and thermal AdS furnish distinct bulk geometries at the real place, we argue in this paper that the selfsame Bruhat-Tits quotient that in the literature has been interpreted as a kind of BTZ black hole also provides a bu
Finite Temperature at Finite Places · Thermal CFTs at the finite places · 0.86
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In this section we recall some basic facts and results for $p$-adic holography and Mellin amplitudes and provide relevant definitions and normalizations which will be used throughout.[For recent developments in $p$-adic holography, see Refs. [ref] .] More details can be found in Ref. [ref] (see also Refs. [ref] ).
Recursion Relations in $p$-adic Mellin Space · Mellin Amplitudes and Pre-Amplitudes · 1.00
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See [ref] for recent progress.
Generalizations of Reflected Entropy and the Holographic Dual · Introduction · 0.90
What it cites · 39 sentences without a confident label
Contests 3 papers
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In fact one is tempted to view the Bruhat–Tits tree itself as a tensor network [ref] , or closely related to a tensor network [ref] ; however these models were unsuccessful in reproducing the RT formula or various expected entropy inequalities in the $p$-adic setting, so the question of whether entanglement entropy results familiar from the usual real holography hold over the $p$-adics has remained open.
Tensor networks, $p$-adic fields, and algebraic curves: arithmetic and the AdS$_3$/CFT$_2$ correspondence · Introduction · contest 0.86
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In fact one is tempted to view the Bruhat–Tits tree itself as a tensor network [ref] , or closely related to a tensor network [ref] ; however these models were unsuccessful in reproducing the RT formula or various expected entropy inequalities in the $p$-adic setting, so the question of whether entanglement entropy results familiar from the usual real holography hold over the $p$-adics has remained open.
Tensor network and ($p$-adic) AdS/CFT · Introduction · contest 0.86
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In fact one is tempted to view the Bruhat–Tits tree itself as a tensor network [ref] , or closely related to a tensor network [ref] ; however these models were unsuccessful in reproducing the RT formula or various expected entropy inequalities in the $p$-adic setting, so the question of whether entanglement entropy results familiar from the usual real holography hold over the $p$-adics has remained open.
Holographic Codes on Bruhat--Tits buildings and Drinfeld Symmetric Spaces · Introduction · contest 0.86
Builds on 1 paper
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Our construction both generalizes the families of perfect tensors used in the construction of holographic codes in [ref] , and gives a physical interpretation of the perfect-tensor condition.
Holographic quantum error-correcting codes: Toy models for the bulk/boundary correspondence · Introduction · 0.93
Uses the method or tool of 2 papers
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Our constructions make use of results of Gurevich and Hadani [ref] , who demonstrated the existence of a “canonical quantization functor” for symplectic vector spaces over finite fields.
Quantization of symplectic vector spaces over finite fields · Perfect Tensors Associated to Semiclassical States · 0.95
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Starting with classical error-correcting codes in the form of Reed–Solomon codes built over projective lines over finite fields [ref] , one may upgrade these to quantum codes by applying the CRSS algorithm [ref] , which we show can be generalized to directly obtain perfect tensors from certain self-orthogonal codes.
Quantum Error Correction and Orthogonal Geometry · Introduction · 0.84
Compares with 6 papers
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The result RTgenus0 is obtained by explicitly computing the reduced density matrix and diagonalizing it.[Some issues associated with infinite dimensional density matrices in the limit when the cutoff $Λ→∞$ are discussed in section [ref].] The entanglement entropy obtained is in direct analogy with the corresponding classic result for the entanglement of a connected interval (of size $|x-y|$ over the real line) with its complement, on a static spatial slice of a massless CFT$_2$ with UV cutoff $ϵ$ [ref] .
Entanglement Entropy and Quantum Field Theory · Results · 0.99
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As remarked earlier, this result is consistent with the minimum-cut rule obeyed by networks of perfect tensors in the case of bipartite entropy of connected regions [ref] , although we do not assume it in our setup.
Holographic quantum error-correcting codes: Toy models for the bulk/boundary correspondence · Results · 0.88
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It is interesting to compare this observation with the result over real CFTs where it was shown that the inequality is saturated for a massless fermion in two dimensions (i.e. mutual information is exactly extensive), but not for instance for the massless scalar [ref] .
Holographic Mutual Information is Monogamous · More entropy inequalities: SSA and MMI · 1.00
Cites as background 53 papers
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Let us briefly recall the setup of $p$-adic AdS/CFT [ref] .
Tensor networks, $p$-adic fields, and algebraic curves: arithmetic and the AdS$_3$/CFT$_2$ correspondence · The Bruhat–Tits tree as a $p$-adic bulk space · 1.00
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Let us briefly recall the setup of $p$-adic AdS/CFT [ref] .
$p$-adic AdS/CFT · The Bruhat–Tits tree as a $p$-adic bulk space · 1.00
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In this section, we give a brief review of the main results of [ref] , which constructed a canonical model of the Heisenberg representation.
Quantization of symplectic vector spaces over finite fields · Functorial quantization · 0.88
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