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arxiv: 2606.21500 · v1 · pith:IC3FNHC6new · submitted 2026-06-19 · 🪐 quant-ph

Optimal GHZ-State Distribution in LOSR Quantum Networks via Local Decoding from Information Sets

Leonardo Oleynik , Shehbaz Tariq , Symeon Chatzinotas This is my paper

Pith reviewed 2026-06-26 13:42 UTC · model grok-4.3

classification 🪐 quant-ph
keywords GHZ statesLOSR networksmultipartite sourceslocal unitariesinformation setslinear codesentanglement distribution
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The pith

Local unitaries from code decoders convert multipartite sources to GHZ states of fidelity d^{m-M} in LOSR networks when incident edges form information sets.

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

The paper proves that multipartite sources overcome the GHZ-preparation limit of bipartite sources in LOSR networks. By mapping network hyperedges to coordinates of a linear code, it shows that the local decoders turn the shared state into an N-party GHZ state whenever the incident edges at each node form an information set. The resulting fidelity is d^{m-M}, which equals 1/d for the complete (N-1)-uniform hypergraph; the paper further establishes that this value is optimal among local-unitary methods and strictly exceeds the bipartite-source bound. A reader would care because the construction eliminates real-time classical communication and its associated memory and latency costs.

Core claim

By identifying the hyperedges of the network with the coordinates of a linear code, the authors show that whenever the edges incident to each node form an information set, a fixed collection of local unitaries—the local decoders of the code—transforms the source state into an N-party GHZ state with fidelity d^{m-M}, without requiring any classical communication. For the complete (N-1)-uniform hypergraph this fidelity is 1/d and is optimal among all local-unitary strategies.

What carries the argument

The local decoders of the linear code whose coordinates are the network hyperedges; they map the source state to the target GHZ state precisely when the incident hyperedges form an information set.

If this is right

  • Fidelity 1/d is achieved for the complete (N-1)-uniform hypergraph.
  • This value is optimal among all local-unitary strategies.
  • It exceeds the best fidelity obtainable with only bipartite sources, for example 1/2 versus 1/8 in the four-node case.
  • Multipartite sources together with shared randomness suffice to replace real-time classical communication for entanglement distribution.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • Networks obeying the information-set condition could support larger-scale GHZ distribution by removing communication overhead.
  • The same code-decoder construction might apply to other target states or to noisy versions of the sources if the fidelity formula is adjusted accordingly.
  • Small-scale physical implementations of complete hypergraphs could directly measure whether the predicted 1/d fidelity is reached.

Load-bearing premise

The hyperedges incident to each node must form an information set of the linear code identified with the network.

What would settle it

Applying the local decoders in a complete (N-1)-uniform hypergraph and measuring fidelity below 1/d would falsify the optimality claim.

read the original abstract

Distributing multipartite entanglement is a prerequisite for scalable quantum networks. Networks restricted to local operations and shared randomness (LOSR) avoid the quantum-memory and latency costs associated with the real-time classical communication required by LOCC-based networks. However, when only bipartite sources are available, LOSR networks cannot prepare useful GHZ states. In earlier work, we conjectured that multipartite sources overcome this limitation and supported this claim with a single numerical example. In this work, we prove the conjecture for regular and uniform networks of arbitrary size. By identifying the hyperedges of the network with the coordinates of a linear code, we show that whenever the edges incident to each node form an information set, a fixed collection of local unitaries, namely the local decoders of the code, transforms the source state into an (N)-party GHZ state with fidelity (d^{m-M}), without requiring any classical communication. Here, (m) denotes the number of edges incident to each node and (M) the total number of edges in the network. For the complete ((N-1))-uniform hypergraph (K_N^{(N-1)}), this fidelity reduces to (1/d). We further prove that this value is optimal among all local-unitary strategies and exceeds the best fidelity achievable using only bipartite sources. For example, in the four-node case, the optimal fidelity is (1/2), compared with the bipartite bound of (1/8). These results demonstrate that multipartite sources, together with shared randomness, can replace real-time classical communication for entanglement distribution.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

2 major / 0 minor

Summary. The manuscript proves that in regular uniform LOSR quantum networks using multipartite sources, identifying hyperedges with coordinates of a linear code allows local unitary decoders to produce an N-party GHZ state with fidelity d^{m-M} (no classical communication) whenever the hyperedges incident to each node form an information set of the code. For the complete (N-1)-uniform hypergraph K_N^{(N-1)}, the fidelity reduces to 1/d, which is shown optimal among local-unitary strategies and strictly exceeds the best fidelity achievable with only bipartite sources (e.g., 1/2 vs. 1/8 for N=4).

Significance. If the central construction holds, the result is significant: it supplies an explicit coding-theoretic method to replace real-time classical communication with shared randomness and multipartite sources for GHZ distribution, together with an optimality proof for the complete-hypergraph case that improves on the bipartite bound. The use of information sets and local decoders of linear codes is a concrete, falsifiable advance over the prior numerical conjecture.

major comments (2)
  1. [Abstract] Abstract: the statement that the result is proved 'for regular and uniform networks of arbitrary size' is not supported by the actual theorem, which is conditioned on the existence of a code identification making every incident set an information set. The manuscript supplies no general argument that such an identification exists for every regular uniform network, only the 'whenever' qualifier plus the special case of K_N^{(N-1)}.
  2. [Main theorem statement] The paragraph beginning 'By identifying the hyperedges of the network with the coordinates of a linear code': the fidelity guarantee d^{m-M} and the local-decoder construction are load-bearing on the information-set condition. When the condition fails, no alternative construction or fidelity bound is provided; the paper therefore does not establish the claimed result for general regular uniform networks.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading and for identifying the need to align the abstract and theorem phrasing more precisely with the conditional nature of the result. We address the comments below and will revise the manuscript to improve clarity.

read point-by-point responses

  1. Referee: [Abstract] Abstract: the statement that the result is proved 'for regular and uniform networks of arbitrary size' is not supported by the actual theorem, which is conditioned on the existence of a code identification making every incident set an information set. The manuscript supplies no general argument that such an identification exists for every regular uniform network, only the 'whenever' qualifier plus the special case of K_N^{(N-1)}.

    Authors: We agree that the abstract phrasing 'prove the conjecture for regular and uniform networks of arbitrary size' can be read as claiming an unconditional result for all such networks. The theorem is explicitly conditional on the existence of a code identification for which incident sets are information sets, and no general existence argument is supplied beyond the complete-hypergraph case. We will revise the abstract to state that the construction and fidelity bound apply to regular and uniform networks admitting such an identification (which includes the complete case of primary interest), thereby removing any overstatement while preserving the technical claims. revision: yes

  2. Referee: [Main theorem statement] The paragraph beginning 'By identifying the hyperedges of the network with the coordinates of a linear code': the fidelity guarantee d^{m-M} and the local-decoder construction are load-bearing on the information-set condition. When the condition fails, no alternative construction or fidelity bound is provided; the paper therefore does not establish the claimed result for general regular uniform networks.

    Authors: The main theorem paragraph already qualifies the fidelity guarantee and decoder construction with the 'whenever' information-set condition; the result is not asserted to hold without it. No alternative construction or bound is given when the condition fails because that lies outside the paper's scope. The contribution is the explicit coding-theoretic method that works for arbitrary network size under the stated hypothesis, together with the optimality proof for the complete hypergraph. We will add a short clarifying remark after the theorem statement noting the conditional character and that no general bound is claimed when the information-set property does not hold. revision: partial

Circularity Check

0 steps flagged

No circularity: derivation applies standard linear-code decoding to a modeling identification that is not self-referential

full rationale

The paper models networks by mapping hyperedges to coordinates of a linear code and then invokes the standard fact that local decoders on information sets recover the logical information. The fidelity expression d^{m-M} follows directly from the dimension of the code and the definition of information sets; it is not obtained by fitting parameters inside the paper or by redefining the target GHZ state in terms of the decoders. The earlier conjecture is cited only as motivation; the present argument supplies an explicit construction that holds precisely when the incident sets are information sets, without circular reduction. The optimality claim for the complete hypergraph likewise rests on comparing the derived fidelity against the known bipartite-source bound, not on any self-citation chain.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The central claim rests on the existence of linear codes whose coordinate sets satisfy the information-set property for the chosen hypergraph; no new physical constants or fitted parameters are introduced beyond the standard dimension d of the local Hilbert space.

axioms (2)
  • standard math Standard postulates of quantum mechanics (states as density operators, local unitaries, fidelity as overlap).
    Invoked implicitly throughout the abstract when discussing source states and GHZ fidelity.
  • domain assumption Existence of linear codes over alphabet size d whose incident-edge sets are information sets.
    The construction presupposes such codes exist for the regular uniform hypergraphs considered; this is a standard fact in coding theory but must hold for the specific network topology.

pith-pipeline@v0.9.1-grok · 5824 in / 1587 out tokens · 16839 ms · 2026-06-26T13:42:54.164734+00:00 · methodology

discussion (0)

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Reference graph

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