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Entanglement-Assisted Quantum Locally Recoverable Codes: Bounds, Optimal Constructions, and Achievability

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Pith's one-line read The paper shows that shared entanglement removes the dual-containment barrier that has kept most classical locally recoverable codes out of CSS quantum constructions.

desk verdict Solid EA-qLRC framework with honest bounds and useful hull-dimension criterion, but the purity of the Tamo–Barg and cyclic-pair constructions is asserted, not proved, and the optimality claims lean on it. read the letter →

arxiv 2608.06854 v1 pith:FOTTARKT submitted 2026-08-07 cs.IT math.IT

Vijay Kumar , Ramakrishna Bandi This is my paper · ORCID
classification cs.ITmath.IT MSC 94B0594B1594B6581P70
keywords Entanglement-assistedquantumlocallyrecoverablecodesCSSconstructionhulldimensionLCDcyclicTamo-BargSingleton-likebound
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

This paper establishes that entanglement assistance erases the dual-containment requirement that has kept most classical locally recoverable codes out of quantum CSS constructions. The authors define entanglement-assisted quantum locally recoverable codes (EA-qLRCs) through recovery channels and prove that two stabilizer generators per erased coordinate, one with an X-error and one with a Z-error, whose combined support has size at most r+1, are enough to guarantee local recovery of that coordinate using the surviving r qudits and the shared ebits. From this they derive a Singleton-like bound on distance, dimension, locality, and entanglement consumption, and they characterize exactly when pure CSS-like EA-qLRCs attain it: the constituent classical codes must themselves be optimal LRCs with equal parameters and compatible flooring of k/r, which in the single-code case reduces to a threshold on the hull dimension s=dim(C∩C^⊥). Explicit optimal maximally entangled codes are then built from LCD cyclic LRCs of length dividing q−1 or q+1, while a Tamo-Barg construction is shown to attain the bound only in the regime k≤r where locality is vacuous. The paper closes with two Gilbert-Varshamov-like achievability bounds that hold unconditionally for q>3.

What carries the argument

The load-bearing mechanism is the two-generator local recovery criterion (Theorem 2): if for every coordinate i there are two stabilizer generators—one acting as X on i, one acting as Z on i—whose combined support sits inside a set Γ_i of size at most r+1, then every single-qudit error on i is correctable from Γ_i\{i} together with the decoder's ebits, because the whole single-qudit operator algebra of the erased system is reproduced by operators on the survivors. The paper's named objects are the CSS-like construction from a classical pair (C_1,C_2), the hull dimension s=dim(C∩C^⊥), which fixes entanglement c=n−k−s and sets the optimality threshold s≤(k−1) mod r, the LCD condition C∩C^⊥={0} that yields maximally entangled codes, and cyclic defining-set machinery in which the local checks are residue classes modulo n/(r+1). A locality-preserving monomial equivalence argument is what carries the Gilbert-Varshamov achievability statements to all q>3.

What would settle it

For the Tamo-Barg example with q=7, r=2, ℓ=4 (the claimed pure code [[6,2,3,2;2]]_7), compute every weight-3 codeword of C_TB and check membership in C_TB∩C_TB^⊥; a single such codeword in the hull would give δ>d=3 and falsify the purity claim.

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Extended reading notes

Core claim

On the paper's own terms, the discovery is that locality in entanglement-assisted quantum codes is a purely local support condition, not a global algebraic one: for every coordinate i, parity-check vectors c_1^(i)∈C_1^⊥ and c_2^(i)∈C_2^⊥ with i in both supports and |supp(c_1^(i))∪supp(c_2^(i))|≤r+1 make the CSS-like EA code locally recoverable with locality r, whether or not C_1^⊥⊆C_2. The main performance statement is the converse bound 2δ≤n−κ+c−2⌈κ/r⌉+4, with necessary and sufficient equality conditions in Theorem 6 and the single-code hull version in Theorem 7: an optimal pure CSS-like EA-qLRC from C_1=C_2=C is optimal exactly when C is a classically optimal LRC with hull dimension s≤(k−1) mod r, and when s=0 the code is maximally entangled with c=n−k. The explicit constructions are claimed optimal maximally entangled CSS-like EA-qLRCs from LCD cyclic LRC families of length dividing q±1, with parameters [[n,k,d;n−k]]_q and locality r, while the Tamo-Barg construction is proved to miss optimality for every k>r.

Load-bearing premise

The claimed optimal cyclic and Tamo-Barg constructions are pure because their minimum-weight classical codewords are assumed to avoid the hull intersection with the other constituent's dual; the paper does not prove this avoidance in general, and the claimed quantum distances and optimality rest on it.

Editorial extensions

If this is right

  • Any classical LRC pair whose duals carry two local checks with joint support at most r+1 per coordinate gives an EA-qLRC with locality r; no dual-containment is required.
  • The Singleton-like bound 2δ≤n−κ+c−2⌈κ/r⌉+4 is universal for CSS-like EA-qLRCs, and equality is characterized by classically optimal constituent codes with equal parameters and ⌈k_1/r⌉=⌈κ/r⌉.
  • In the single-code case, optimality reduces to the hull condition s=dim(C∩C^⊥)≤(k−1) mod r, and LCD codes yield maximally entangled optimal EA-qLRCs with c=n−k.
  • Explicit optimal maximally entangled EA-qLRCs exist from LCD cyclic LRCs of length dividing q−1 or q+1, with parameters [[n,k,d;n−k]]_q and locality r.
  • The Tamo-Barg-based construction attains the Singleton-like bound only in the vacuous regime k≤r, while the Gilbert-Varshamov-like achievability rates hold unconditionally for q>3, leaving q=2,3 open.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • Editorial extension: the same two-generator local-support criterion should transplant to entanglement-assisted codes with (r,ρ)-locality, where ρ local checks per coordinate would need 2ρ generators with combined support at most r+ρ; the paper does not treat this.
  • Editorial extension: since optimality hinges on the hull threshold s≤(k−1) mod r, any systematic family of classical optimal LRCs with hull bounded by that threshold would yield optimal EA-qLRCs; searching beyond LCD codes (s=0) is a natural next step.
  • Editorial extension: the q=2,3 gap might be closed by explicit binary LCD-LRC families rather than monomial equivalence, because the obstruction is an equivalence failure, not a locality obstruction; the explicit cyclic families already cover some binary parameter ranges.
  • Editorial extension: the asymptotic comparison suggests the sphere-packing-like bound dominates only when locality r is large relative to length; testing finite-length crossover points for q=2 with r between 6 and 12 would calibrate the regime boundary.
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Formalized claims in Lean

  1. Claim #1: On the paper's own terms, the discovery is that locality in entanglement-assisted quantum codes is a purely local support condition, not a global algebraic one: for every coordinate i, parity-check vectors c_1^(i)∈C_1^⊥ and c_2^(i)∈C_2^⊥ with i in both supports and |supp(c_1^(i))∪supp(c_2^(i))|≤r+1 make the CSS-like EA code locally recoverable with locality r, whether or not C_1^⊥⊆C_2. The main pe

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

2 major / 5 minor

Summary. The paper develops a framework for entanglement-assisted quantum locally recoverable codes (EA-qLRCs) built from pairs of classical LRCs via a CSS-like construction that does not require dual-containment. It defines EA-qLRCs through recovery channels, proves a sufficient stabilizer-based locality criterion (Theorem 2), derives four explicit converse bounds (Singleton-, Griesmer-, Plotkin-, and sphere-packing-like) together with a Cadambe-Mazumdar-like bound, characterizes equality in the Singleton-like bound for pure codes, constructs families from Tamo-Barg and cyclic codes, provides optimal maximally entangled constructions from LCD cyclic codes, and establishes two Gilbert-Varshamov-like achievability bounds for q>3. It closes with a unified comparison of all bounds in the maximally entangled regime.

Significance. If the construction claims are fully justified, this is a substantial contribution to quantum locally recoverable coding: it opens classical LRC families to quantum local recovery via entanglement assistance, supplies explicit finite-length and asymptotic bounds, and provides several optimal or near-optimal families. The stabilizer criterion of Theorem 2, the subcode-puncturing reduction in Lemma 5 and Theorem 3, and the LCD-based maximally entangled constructions of Section VII are coherent and appear sound. The Gilbert-Varshamov-type achievability results for q>3, obtained through monomial equivalence to LCD codes, are a useful and non-obvious extension. The principal unresolved point is the unproved purity of the non-LCD Tamo-Barg and general cyclic-pair constructions, which affects the reported parameters and the optimality boundary claimed in Sections VI and VII.

major comments (2)
  1. [Section VI, Theorems 9-12 and Remarks 11-12] The purity of the Tamo-Barg and general cyclic-pair constructions is asserted but not proved. Theorem 1 defines the EA distance as δ = min{wt(C1\(C1∩C2⊥)), wt(C2\(C1⊥∩C2))}; equality δ = min{d1,d2} requires that all minimum-weight codewords of each constituent code avoid the corresponding hull intersection. Proposition 2 computes only the hull dimension s=|S∩S⊥|, and the proof of Theorem 11 computes only κ and c from Lemma 8; neither verifies the required weight condition. Consequently, the parameters in Theorem 9 ([[q−1, 1, ≥q−ℓ, r; ...]]), the distance min{d1,d2} in Theorems 11-12, the reported [[36,11,14;12]] code in Example 2, and the non-optimality conclusions of Remarks 11-12 all rest on an unverified hypothesis. If the actual δ is strictly larger than min{d1,d2}, equality in (12) could occur outside the claimed k≤r regime, and the optimality boundary would change. The authors should either prove that minimum-weight codewords of the constituent codes avoid the relevant hull intersections or restate the theorems with δ in place of min{d1,d2} and adjust the optimality discussion accordingly.
  2. [Section IV, Eq. (16) and Eq. (23)] The asymptotic form of the sphere-packing-like bound is derived by fixing τ while n→∞, but the maximum in (16) is taken over τ ranging up to Θ(n). The true leading-order behavior requires optimizing over α = τ/n, and the resulting expression depends on r. Using only fixed τ yields a valid but strictly weaker upper bound; the claimed r-independence in Remark 7 and the tightness comparison in Remark 8 and Figures 7-8 are therefore not established for the actual bound (16). The authors should provide the optimized asymptotic form or explicitly label (23) as a non-optimized relaxation and qualify the hierarchy and tightness claims accordingly.
minor comments (5)
  1. [Notation, throughout] The notation [m]† = {0,...,m} and [m] = {1,...,m} is introduced in Section II, but later sections use [ℓ−1], [n−1], and similar sets inconsistently; for example, Proposition 2 writes S∩S⊥ using [ℓ−1] where [ℓ−1]† appears intended. Please standardize the interval notation.
  2. [Theorem 13] Theorem 13 states parameters [[n,k,d,n−k]]_q; the standard notation is [[n,k,d;n−k]]_q. Also, since the theorem is in the LRC section, it should explicitly state that the underlying classical code C is assumed to have locality r.
  3. [Abstract and Section IV] The abstract says the converse bounds are derived for 'pure CSS-like EA-qLRCs,' but Theorem 3 and Corollaries 2-4 do not require purity and are stated for any CSS-like EA-qLRC from Proposition 1. The wording should be adjusted to avoid suggesting that purity is needed for the bounds.
  4. [Section VI, Theorem 3 statement] Theorem 3 says the code is 'constructed from Proposition 1 using two classical [n,k_i,d_i]_q codes C_i of locality r.' Proposition 1 requires, for each coordinate, a pair of checks whose union support has size at most r+1, which is stronger than each C_i having classical locality r. Please state the exact hypothesis needed or clarify that the classical locality of C_i is a consequence of the Proposition 1 condition.
  5. [Example 2 and Remark 12] In Example 2, the minimum distances d1 and d2 are initially obtained as lower bounds from Lemma 7; they become exact only after applying the classical LRC Singleton bound. Please make that two-step justification explicit at the point where d1=14 and d2=20 are stated as exact.

Circularity Check

0 steps flagged · score 2.0 of 10

No significant circularity: the bounds are derived from external classical LRC bounds and the EA-CSS theorem; the only self-citation is a non-load-bearing construction ingredient.

full rationale

The paper's derivation chain is self-contained: Theorem 2 gives a sufficient stabilizer criterion for local recoverability from two local generators; Proposition 1 translates this to parity-check vectors; Theorem 3 reduces the EA-qLRC parameters to classical LRC bounds via an explicit puncturing/subcode argument (Lemma 5), and Corollary 2 and Theorem 5 are direct consequences of the classical Griesmer/Plotkin/sphere-packing results, with no fitted parameter renamed as a prediction. The optimality criterion (Theorem 6) follows algebraically from the classical Singleton-like bound (1) and the dimension identity, so it is not defined in terms of its own conclusion. The only self-citation is [33] (Rajput–Bhaintwal–Bandi) used in Section VII-A and Table II as a source of cyclic LCD-LRC families; this is a construction ingredient for additional examples and is not load-bearing for the main converse bounds or the GV achievability results. Separately, the pure parameters claimed in Theorems 9–12 are not fully justified: Theorem 11's proof computes only dimension and entanglement, while Theorem 1 gives distance as min of the two set differences and equality with min{d1,d2} is asserted, not proved; this is an unproven-support/correctness gap, not a circular reduction, so it does not raise the circularity score. Accordingly, no step reduces by construction to its own input.

Assumptions & free parameters 0 free parameters · 8 assumptions · 0 invented entities

The paper's central bounds rest on prior classical LRC bounds and the EA-CSS construction, which are external and not re-derived. The main unstated load-bearing assumption is the purity of several constructed families, which is asserted rather than proved.

assumptions (8)
  • standard math EA-CSS construction from arbitrary pairs of classical linear codes (Theorem 1, based on Galindo et al. [26])
    The whole framework builds on this prior theorem for EAQECCs; the paper does not re-prove it.
  • standard math Classical LRC distance bounds: Singleton-like [1], CM bound [2], Griesmer-like, Plotkin-like, sphere-packing-like [3],[15]
    Used to derive the corresponding EA-qLRC bounds in Theorem 3, Corollary 2, Theorem 5, and Corollary 3.
  • standard math Stabilizer formalism and Knill-Laflamme conditions for entanglement-assisted codes
    Used in Definition 2, Theorem 2, and the recovery-channel construction; accepted background.
  • standard math Carlet et al. monomial equivalence to LCD for q>3 [35]
    Critical for the achievability bounds; every linear code over F_q, q>3, is monomially equivalent to an LCD code.
  • standard math Classical cyclic LRC-LCD families from Rajput et al. [33] and Chen et al. [9]
    Rows 1-8 of Table II pull these existing optimal classical families into the EA-qLRC setting.
  • domain assumption Noiseless pre-shared entanglement and noise only on the encoder's n physical qudits
    Standard EAQEC assumption stated in Section II.C; the recovery channels in Definition 2 depend on it.
  • ad hoc to paper Purity of Tamo-Barg and general cyclic-pair CSS-like EA-qLRCs: the minimum-weight codewords of each constituent code lie outside the relevant hull intersection
    Theorems 9-12 assert pure codes, but Theorem 1 only gives δ as the minimum of set-difference weights, which can exceed min{d1,d2}; no proof of equality is supplied.
  • domain assumption Convention that 1≤r≤k and locality is vacuous when r≥k
    Used in Remark 11 and Table II notes to classify the Tamo-Barg boundary as degenerate.

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Pith. "Pith review of Entanglement-Assisted Quantum Locally Recoverable Codes: Bounds, Optimal Constructions, and Achievability." pith.science (2026). https://pith.science/paper/FOTTARKT

@misc{pith2026260806854,
  author       = {Pith},
  title        = {Pith review of: Entanglement-Assisted Quantum Locally Recoverable Codes: Bounds, Optimal Constructions, and Achievability},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/FOTTARKT}},
  note         = {Machine review of arXiv:2608.06854}
}
abstract

This paper studies entanglement-assisted quantum locally recoverable codes (EA-qLRCs) built via a CSS-like stabilizer construction from pairs of classical locally recoverable codes (cLRCs), without requiring dual-containment. We define such codes through local recovery channels, give a sufficient stabilizer criterion for the construction, and derive Singleton-, Griesmer-, Plotkin-, and sphere-packing-like converse bounds on the parameters of the resulting pure CSS-like EA-qLRCs, along with a Cadambe--Mazumdar-like bound that, as in the classical case, lacks a closed form, plus a comparison of their relative tightness across finite-length and asymptotic regimes. We give necessary and sufficient conditions for a pure CSS-like EA-qLRC to attain the Singleton-like bound with equality; for the single-code case $\mathcal{C}_1=\mathcal{C}_2=\mathcal{C}$, this reduces to a simple condition on the hull dimension $s=\dim(\mathcal{C}\cap\mathcal{C}^\perp)$, which also fixes the entanglement count via $c=n-k-s$. We present CSS-like EA-qLRC constructions from classical LRC families---Tamo--Barg and cyclic codes---and characterize when these attain the Singleton-like bound, showing the cyclic families yield optimal codes while the Tamo--Barg construction, though valid, attains the bound only in the degenerate regime $k \le r$, where locality is vacuous. We complement these constructions with two Gilbert--Varshamov-like achievability bounds, via a classical parity-check augmentation and a sharper concatenated-code construction, and show both hold unconditionally for field size $q>3$ via a monomial-equivalence argument. Finally, we unify all bounds---converse and achievability alike---under a common maximally entangled regime, giving a single comparison of the achievable and forbidden rate--distance--locality region for CSS-like EA-qLRCs.

Figures

Figures reproduced from arXiv: 2608.06854 by the authors.

Figure 1
Figure 1. An entanglement-assisted quantum distributed storage system. A quantum state [PITH_FULL_IMAGE:figures/full_fig_p006_1.png] view at source ↗
Figure 2
Figure 2. Schematic of the recovery channel Ri of Theorem 2: surviving qudits Γi \ {i} and the decoder’s ebits HD enter Ri, which outputs the reconstructed set Γi; the ebit register is consumed, matching Ri : D(HΓi\{i} ⊗ HD) → D(HΓi ). The channel never has access to qudit i itself, and its explicit gate decomposition is not addressed in this work. Theorem 2 provides a sufficient stabilizer criterion for EA-qLRC, which enable… view at source ↗
Figure 3
Figure 3. Comparison of the bounds with q = 64, δ = 3, r = 3 & c = 1 [PITH_FULL_IMAGE:figures/full_fig_p012_3.png] view at source ↗
Figures from the paper (5 more)
Figure 5
Figure 5. Figure 5: Comparison of the bounds with q = 2, δ = 22, r = 3 & c = 1 [PITH_FULL_IMAGE:figures/full_fig_p013_5.png]
Figure 7
Figure 7. Figure 7: Comparison of the four asymptotic bounds at [PITH_FULL_IMAGE:figures/full_fig_p015_7.png]
Figure 8
Figure 8. Figure 8: Net-rate analogue of Figure 7: the crossing window [PITH_FULL_IMAGE:figures/full_fig_p015_8.png]
Figure 9
Figure 9. Figure 9: The sharpened achievability bound RLRC(∆, r) of Theorem 19, plotted against the standard Gilbert–Varshamov rate RGV(∆) = 1 − Hq(∆), for r = 2, 4, 8, 16 over F4 and F8. As r increases, RLRC converges rapidly toward RGV, consistent with the exponential decay of the local…
Figure 10
Figure 10. Figure 10: The shaded red region is provably impossible, bounded below by the pointwise minimum of the four converse bounds (at [PITH_FULL_IMAGE:figures/full_fig_p027_10.png]

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Pith tools

Reviewed August 10, 2026 · model on record in the stance chip above.