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REVIEW 3 major objections 5 minor 64 references

Entanglement Cost of Erasure Correction in Quantum MDS Codes

T0 review · 3 major / 5 minor · reviewed 2026-08-07 · deepseek-v4-flash

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Pith's one-line read For any [[n,2t−n]]_Q quantum MDS code, correcting one erased node over a star network costs exactly 2t Q-dimensional qudits of entanglement when the replacement node is the hub, and exactly 2t−1 when a helper node is the hub.

desk verdict The achievability is clean and the question is worthwhile, but Lemma 9 is false as stated, so the lower-bound proofs for 2t and 2t-1 do not go through. read the letter →

arxiv 2505.20284 v1 pith:ZKVGDMWJ submitted 2025-05-26 quant-ph

Kaushik Senthoor This is my paper · ORCID
classification quant-ph
keywords quantumerasurecorrectionMDScodesentanglementcostdistributedstoragestarnetworkSchmidtrankLOCCabsolutelymaximallyentangledstates
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

The paper asks how much pre-shared entanglement a distributed quantum storage system must spend to repair a lost node, when the storage uses a quantum MDS code and the repair network is a star. It proves an exact answer: with the minimal t helper nodes, replacing the erased qudit costs 2t qudits if the replacement node is the hub, and 2t−1 qudits if one of the helpers is the hub. Both numbers are tight, because a simple download-and-return protocol—pull the t qudits to a central node, fix the erasure locally, send them back—achieves the bound. The result matters because it fixes the entanglement overhead of a basic distributed-storage operation and covers absolutely maximally entangled states as the n=2t case.

What carries the argument

The machinery is Schmidt-rank monotonicity under LOCC (Lemma 2), applied after a circuit transformation that removes the n − t − 1 uninvolved nodes: the paper applies a unitary $U_T^{{(E)}}$ defined by the code's encoding structure before the protocol and inverts it after, which deletes the non-helper nodes from the rank accounting and yields the tighter bound. The key identities are the Schmidt ranks of the states before and after the protocol: SR(W_j W'_j) = 1 before, $Q^{2}$ after (and SR(cW_n) = Q after), so each edge must carry a resource state with Schmidt rank at least $Q^{2}$ (or Q for the edge to the replacement node in the helper-hub case).

What would settle it

Exhibit a concrete [[n,2t−n]]_Q quantum MDS code (for example [[5,1,3]]_2) and compute SR(W'_1 W'_2) in the state defined in Lemma C.2; if the rank is below $Q^{2}$, Lemma 9's premise fails. Alternatively, construct an explicit erasure-correction protocol on the star network H1 that consumes fewer than 2t qudits of entanglement.

Watch

Extended reading notes

Core claim

The central claim is that the entanglement cost of quantum erasure correction in an [[n, 2t − n]]_Q quantum MDS code over a star network is exactly 2t when the replacement node is the hub and exactly 2t − 1 when a helper node is the hub, where t is the number of helper nodes accessed (the minimum needed, since the code corrects n − t erasures). The lower bound comes from modelling erasure correction as a deterministic LOCC protocol acting on the encoded state together with a resource state of shared maximally entangled pairs, and then applying Schmidt-rank monotonicity: the protocol must supply resource states whose Schmidt ranks are at least $Q^{2}$ through each edge into the hub, and at least Q through the edge to the replacement node in the helper-hub case. The upper bound is a download-and-return protocol that communicates exactly this many qudits. For AME states (n = 2t) the same numbers give the entanglement cost of replacing one lost share.

Load-bearing premise

The entire lower bound rests on the unproved extension of Lemma C.2: that in the decoded state |chi>, the Schmidt rank of any set of primed output qudits W'_J equals $Q^{{|J|}}$, which the paper cites rather than derives for sets larger than one.

Editorial extensions

If this is right

  • With n = 2t (AME states), repairing one lost share costs 2t qudits with the replacement node as hub and 2t − 1 with a helper as hub.
  • The minimal-helper download-and-return strategy is optimal: no protocol using exactly t helpers can correct an erasure on these star networks with less entanglement.
  • When Q = 2^w, the costs translate to 2tw and (2t − 1)w EPR pairs (qubits), giving the physical entanglement budget for qubit-based distributed storage.
  • The gap between the two topologies is exactly one qudit, so choosing a helper as the hub saves precisely one qudit of entanglement per erased node.

Reading between the lines

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

  • The Schmidt-rank shortfall flagged in the paper's Lemma C.2, if repaired, would also supply the lower bound for correcting several erased nodes at once, since the same rank argument applies to larger erased sets.
  • The one-qudit saving for helper-hub topologies suggests a design heuristic: in a star network, place the repair computation at an existing storage node rather than the new node.
  • A natural numerical test is to compute the exact entanglement cost for small codes (for example the [[5,1,3]]_2 code) with a brute-force search over protocols; if any protocol beats 2t, the lower-bound premise fails.
  • The proof technique of surrounding the LOCC window with a code unitary and its inverse could be adapted to bound the entanglement cost of distributed syndrome extraction, not just erasure correction.
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Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

3 major / 5 minor

Summary. The paper studies the entanglement cost of correcting one erased node in a distributed quantum storage system based on an [[n, 2t-n]]_Q quantum MDS code, under the assumption that exactly t helper nodes are used and that the communication network is a star. Two topologies are considered: the replacement node as the hub (H1) and a helper node as the hub (H2). The paper claims that the exact entanglement costs are 2t and 2t-1 Q-dimensional qudits respectively. The upper bounds are established by explicit download-and-return protocols, and the lower bounds are derived by modeling erasure correction as an LOCC circuit, applying Schmidt-rank monotonicity, and computing the relevant Schmidt ranks before and after the LOCC protocol. The main technical content is in Lemmas 8 and 9 and in Appendix C, which computes the rank of the primed output subsystems after the protocol.

Significance. If the lower-bound argument is completed, the result is a clean, parameter-free characterization of the entanglement cost of distributed erasure correction for QMDS codes in the minimal-helper regime, with explicit achievability protocols. This is a natural and useful problem for modular quantum computing and quantum data centers, and the LOCC/Schmidt-rank framework is a promising approach. The paper is also careful to separate the deterministic-LOCC modeling assumption and to note that the non-minimal-helper regime remains open. However, the central lower bounds rest on a single rank computation whose proof is currently incomplete; the achievability directions and the overall structure of the argument are sound enough that the issue appears fixable rather than fatal.

major comments (3)
  1. [Appendix C, Lemma 9 proof, Eq. (112)] The proof of Lemma 9 reaches Eq. (112), where it asserts SR(W'_J)_chi = Q^{|J|} and attributes this to Lemma C.2. This is not supported by Lemma C.2 as stated. Lemma C.2 proves only SR(W_j)_chi = Q for individual unprimed qudits j in [n-1] in the state of Eq. (98), which has a different register structure from the state |chi> in Eq. (111). The state in Eq. (111) involves the primed block W'_T, with the last input to U_T fixed to |epsilon>, and it has no R register; for n < 2t the register count also does not match Lemma C.2. The needed statement is about subsets of the primed output qudits, not about single unprimed physical qudits. Since Theorems 2 and 3 use SR(W_j W'_j)_{psi_out} = Q^2, the lower bounds in Eqs. (45) and (57) are not established as written. The authors should provide a direct proof of the singleton case SR(W'_j)_chi = Q (and, if the general-J statement is kept, prove it separately).
  2. [Appendix C, Lemma C.1] Lemma C.1 is false as stated. For an [[5,1,3]]_2 QMDS code, the parameter t equals 3, but a three-qubit subset of a code state has entropy 2 log 2, not 3 log 2, because its complement consists of two qubits, which are maximally mixed by the code distance. The proof of Lemma C.1 infers that any subsystem of W_T is maximally mixed from the fact that W_T as a whole is maximally entangled with the rest; this inference is invalid for a general unitary U_T. The appendix should state only the maximal-mixedness facts that actually follow from the code distance (subsets of size at most d-1 = n-t), and Lemma C.2's proof, which uses the pair case S(W_j W_n)_Phi = 2 log Q, should be reworked to use only those facts.
  3. [Lemma 9 statement and proof, Eqs. (105)-(112)] The statement of Lemma 9 claims SR(W_J W'_J)_{psi_out} >= Q^{|J|+1} for arbitrary J of size up to t-1. The proof of this general claim relies on the same unsupported subset statement SR(W'_J)_chi = Q^{|J|}. Even if the singleton case can be fixed, the general claim is stronger than needed for Theorems 2 and 3, and it is not justified by the distance-2 property of the auxiliary [[t+1,t-1]] code used in the proof, since that property only guarantees maximal mixedness of single qudits. The authors should either prove the general claim with a genuinely new argument or weaken Lemma 9 to the singleton case that is actually used.
minor comments (5)
  1. [Eq. (40)] The definition of H1 lists {cWe, Wj1} twice; the second and subsequent edges should be {cWe, Wj} for j in T, as in the preceding sentence.
  2. [Lemma 5 proof, Eq. (76)] The quantifier in the proof of Lemma 5 contains the undefined expression '2t-d-1'; it should be 'for all (s1, ..., s_{t-1}) in A_Q^{t-1}'.
  3. [Lemma 3 statement] The word 'transfroms' should be 'transforms'.
  4. [Lemma 6 proof, Eqs. (86)-(87)] The equalities SR(WL cWe)_{chi_in} = Q^{t-1} and SR(WL cWe)_{chi_out} = Q^t are stated without derivation; a short explanation using Lemma 4 would improve readability and verifiability.
  5. [Appendix C, Lemma C.2] Lemma C.2 is stated only for the original [[n, 2t-n]] code state, but the application in Lemma 9 requires the analogous statement for the auxiliary [[t+1, t-1]] code and for primed qudits. The appendix should state a version that covers the actual state |chi> in Eq. (111).

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the derivation is parameter-free and rests on external QMDS and LOCC facts; the Appendix C gap is an omitted proof, not a circular reduction.

full rationale

The paper's claimed derivation chain is parameter-free and does not fit any parameter to data. The lower bounds in Theorem 2 and Theorem 3 are obtained by combining the Schmidt-rank monotonicity of Lo-Popescu (Lemma 2), the teleportation resource model, and a claimed rank fact for the output state |ψout> (Lemma 9). Lemma 9's proof in Appendix C contains an explicit derivation gap: Eq. (112) asserts SR(W'_J)_χ = Q^{|J|} 'due to Lemma C.2', but Lemma C.2 (Eqs. 98-99) only establishes SR(W_j)_χ = Q for individual unprimed qudits in a different state |χ>_{RW[n-1]}, not for primed subsets W'_J in the state |χ>_{W[t-1]W'_T} of Eq. (111). This is a correctness or omitted-proof issue, not circularity: Lemma C.2 itself is derived from the external maximal-mixedness property of QMDS code subsystems [39,40] and is not the target result. No load-bearing step is defined in terms of the entanglement cost it purports to derive; the upper bounds are achieved by explicit download-and-return protocols, and no self-citation is load-bearing. Accordingly, no circular step is exhibited and the score is 0.

Assumptions & free parameters 0 free parameters · 6 assumptions · 0 invented entities

The paper introduces no free parameters or new physical entities. The central results rest on standard coding-theoretic facts about QMDS codes and on LOCC entanglement monotonicity, plus an explicit restriction to deterministic protocols. The only notable gap is in the proof of Lemma 9, which is a derivation issue rather than an added axiom.

assumptions (6)
  • domain assumption Quantum MDS codes have the property that any n-t physical qudits of a code state are maximally mixed.
    Invoked in Lemma 4, Lemma C.1, and Theorem 1, based on results in [39] and [40]. This is a proven property of the code family, so it is a domain assumption rather than an ad hoc one.
  • standard math Schmidt rank is monotonically non-increasing under deterministic LOCC.
    The core tool for all lower bounds, stated as Lemma 2 and cited to [44].
  • standard math Nielsen's theorem characterizes bipartite LOCC transformations via majorization.
    Used in the proof of Lemma 3 (Appendix A) as Lemma A.1, cited to [52].
  • standard math The quantum Singleton bound holds for quantum codes.
    Used in the definition of QMDS codes and in Lemma 5's distance argument, cited to [46].
  • domain assumption [[n,2t-n]]_Q quantum MDS codes exist for the range of parameters considered with suitably chosen prime power Q.
    The paper notes such codes do not always exist for Q=2 and works with the general Q-dimensional setting. The results are conditional on code existence for the chosen Q, n, t.
  • domain assumption The erasure correction protocols are restricted to deterministic LOCC protocols.
    Stated in Section II.B; entanglement cost is minimized over this protocol class, so the bounds are for deterministic exact correction.

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Pith. "Pith review of Entanglement Cost of Erasure Correction in Quantum MDS Codes." pith.science (2026). https://pith.science/paper/ZKVGDMWJ

@misc{pith2026250520284,
  author       = {Pith},
  title        = {Pith review of: Entanglement Cost of Erasure Correction in Quantum MDS Codes},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/ZKVGDMWJ}},
  note         = {Machine review of arXiv:2505.20284}
}
read the original abstract

In distributed quantum storage, physical qubits of a code will be stored across the network. When qubits in one of the nodes are lost i.e. when the node is erased, the remaining nodes need to communicate with a new node to replace the lost qubits. Here, we look at the problem of how much entanglement cost is needed to perform such a distributed quantum erasure correction. We focus on distributed quantum storage based on quantum maximum distance separable (MDS) codes. We derive lower bounds on the entanglement cost when the quantum network used for the erasure correction has a star topology. We show that the simple method of downloading the non-erased qudits and performing operations at a single node is optimal when the minimal number of non-erased nodes are accessed. It remains to be seen what the entanglement cost will be when a non-minimal number of non-erased nodes are accessed. The techniques used in this work can be developed further to study the entanglement cost of quantum erasure correction in more general code families and network topologies.

Figures

Figures reproduced from arXiv: 2505.20284 by the authors.

Figure 1
Figure 1. Download-and-return protocol for erasure correction of the [PITH_FULL_IMAGE:figures/full_fig_p002_1.png] view at source ↗
Figure 2
Figure 2. Download-and-return protocol for erasure correction of the [PITH_FULL_IMAGE:figures/full_fig_p003_2.png] view at source ↗
Figure 3
Figure 3. (a) Erasure correction to replace the erased node [PITH_FULL_IMAGE:figures/full_fig_p006_3.png] view at source ↗
Figures from the paper (1 more)
Figure 4
Figure 4. Figure 4: LOCC protocol for quantum erasure correction of node [PITH_FULL_IMAGE:figures/full_fig_p007_4.png]

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Pith tools

Reviewed August 7, 2026 · model on record in the stance chip above.