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

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.

• The paper introduces a constructive protocol using linear code information sets for local decoding in LOSR networks to distribute GHZ states.
• It leverages hypergraph network structures and fixed local unitaries to achieve fidelities (e.g., 1/2 in a four-node example) that surpass bipartite limitations.
• Optimality under local unitary operations is proven, offering significant implications for low-latency and scalable quantum network designs.

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

Introduction

The paper "Optimal GHZ-State Distribution in LOSR Quantum Networks via Local Decoding from Information Sets" (2606.21500) rigorously studies the distribution of multipartite entangled states—specifically GHZ states—in quantum networks restricted to local operations and shared randomness (LOSR). The central contribution is an analytic, constructive protocol for GHZ-state preparation utilizing multipartite sources, exploiting a deep connection between hypergraph network structure and linear coding theory. The work settles a previously conjectured quantum advantage for multipartite-source networks, establishes optimality under local unitary operations, and precisely characterizes the conditions under which LOSR networks outperform the bipartite bound.

LOSR Networks and Entanglement Distribution

Quantum networks (QNs) are foundational for distributed quantum information processing, demanding scalable entanglement distribution. Networks governed by LOSR protocols mitigate quantum memory and latency costs that afflict LOCC-based approaches by relying exclusively on local operations and pre-shared randomness, dispensing with real-time classical communication. Bipartite-source LOSR networks have been theoretically proven incapable of preparing GHZ states for $N > 3$ nodes beyond the classical product benchmark, restricting their utility in scalable scenarios. This limitation motivates the exploration of multipartite entangled sources as a means to circumvent the bipartite-bound obstruction.

Information-Set Decoding Construction

The protocol proposes associating the hyperedges of an $m$-regular, $k$-uniform hypergraph network with coordinates of a linear code over $\mathbb{F}_d$. If every node's incident edges constitute an information set in the code, each node applies a fixed local decoder (corresponding to a permutation unitary) to its registers. This yields a product unitary transformation $U_{\mathrm{IS}} = \bigotimes_{i \in V} U_i$, where each $U_i$ permutes the local computational basis according to the information set's decoding. All operations are local and predetermined, thus preserving the LOSR constraints.

Given $M$ hyperedges in total and $m$ edges per node, the fidelity attained with the target $N$-partite GHZ state, $\ket{GHZ{N}{d^m}$, is analytically given by $m$0.

Optimality and Quantum Advantage

For the complete hypergraph $m$1 (i.e., $m$2-uniform network over $m$3 nodes), the protocol achieves fidelity $m$4. The authors provide a rigorous upper bound, proving that this fidelity is optimal under all local unitary strategies, while general local channels may potentially surpass this bound. The quantum advantage is precisely characterized: multipartite sources provide a strictly higher fidelity than any bipartite-source LOSR network exactly when $m$5, i.e., the total number of hyperedges is less than twice the number of edges per node.

Figure 1: Numerical local-unitary benchmark for $m$6. (a) Information-set fidelity $m$7 (log scale) vs. source arity $m$8 for $m$9, (b) optimization history; the dashed line denotes the analytical optimum $k$0.

Numerical Benchmarking

The local unitary optimization in the product-unitary manifold corroborates the analytic results. Benchmarks for $k$1 demonstrate that, for boundary networks $k$2, the information-set decoder attains the analytical optimum $k$3, with convergence becoming slower as $k$4 increases (due to exponential growth in local Hilbert dimensions). For networks with $k$5, numerical local unitary search consistently retrieves the constructed fidelity, further validating the protocol albeit not certifying global optimality beyond the analytically proven cases.

Explicit Circuit Construction and the Four-Node Example

The explicit four-node network ($k$6) is detailed, employing a binary $k$7 single-parity-check code to coordinate the decoders. The constructed local decoder circuits map raw source labels to a unified message ordering $k$8, realizing the GHZ state with fidelity $k$9, thus exceeding the bipartite benchmark $\mathbb{F}_d$0 by a factor of four. The protocol utilizes only fixed local unitaries (swap and cnot gates), requiring no classical communication at runtime.

Implications and Prospects

The results have significant practical implications for low-latency quantum networks: multipartite-source LOSR networks can distribute multipartite entanglement at higher fidelities, overcoming operational and scalability bottlenecks inherent in LOCC-based designs. The constructive mapping from quantum network hypergraph topology to coding-theory information sets formalizes a resource-theoretic equivalence, establishing that shared randomness combined with multipartite entangled sources can substitute for real-time classical communication. Analytically characterizing optimality sharpens the delineation of operational boundaries in quantum network design.

Future research directions include exploring fidelity dependence on source arity ($\mathbb{F}_d$4) for fixed $\mathbb{F}_d$5, optimizing regular hypergraph architectures for entanglement distribution, targeting alternative multipartite entangled states (e.g., cluster, W states), and extending optimality proofs to general local channels and source state variations. Hybrid network architectures combining CC and SR may further optimize scalability and latency trade-offs.

Conclusion

The paper delivers an analytic, constructive, and optimal protocol for multipartite entanglement distribution in LOSR quantum networks with multipartite sources, settling long-standing conjectures regarding GHZ-state preparation. By precisely mapping quantum network structure to linear code information sets, the protocol achieves the maximal possible fidelity under local unitary operations, fundamentally altering the capabilities of LOSR networks for scalable quantum information distribution. The work opens a clear path for future theoretical developments and practical implementations in quantum network engineering.