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arXiv:2208.12100 · npj Quantum Information

Quantum LOSR networks cannot generate graph states with high fidelity

Yi-Xuan Wang, Zhen‐Peng Xu, Otfried Gühne

Published Jan 22, 2024• 9 citations
arXivHTMLPDF

Abstract

Abstract Quantum networks lead to novel notions of locality and correlations and an important problem concerns the question of which quantum states can be experimentally prepared with a given network structure and devices and which not. We prove that all multi-qubit graph states arising from a connected graph cannot originate from any quantum network with bipartite sources, as long as feed-forward and quantum memories are not available. Moreover, the fidelity of a multi-qubit graph state and any network state cannot exceed 9/10. Similar results can also be established for a large class of multi-qudit graph states.

DOI
10.1038/s41534-024-00806-z

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Abstract Quantum networks lead to novel notions of locality and correlations and an important problem concerns the question of which quantum states can be experimentally prepared with a given network structure and devices and which not. We prove that all multi-qubit graph states arising from a connected graph cannot originate from any quantum network with bipartite sources, as long as feed-forward and quantum memories are not available. Moreover, the fidelity of a multi-qubit graph state and any network state cannot exceed 9/10. Similar results can also be established for a large class of multi-qudit graph states.
Abstract In research concerning quantum networks, it is often assumed that the parties can classically communicate with each other. However, classical communication might introduce a substantial delay to the network, especially if it is large. As the latency of a network is one of its most important characteristics, it is interesting to consider quantum networks in which parties cannot communicate classically and ask what limitations this assumption imposes on the possibility of preparing multipartite states in such networks. We show that graph states of an arbitrary prime local dimension known for their numerous applications in quantum information cannot be generated in a quantum network in which parties are connected via sources of bipartite quantum states and the classical communication is replaced by some pre-shared classical correlations. We then generalise our result to arbitrary quantum states that are sufficiently close to graph states.
The study of quantum nonlocal correlations in networks (QNCNs), beyond the standard Bell theory with a single source, involves simulating large-scale quantum communications. The detectability of QNCNs depends on the input sources and selected measurements, and more essentially on the network structures. In this paper, we prove that same-size acyclic networks with different independence degrees can generate distinct sets of QNCNs in the device-independent scenario. We then distinguish two acyclic networks with different independence degrees by comparing the QNCNs generated from them. Furthermore, we show quantum demonstrations of the aforementioned notions in two-qubit-source networks. We investigate the quantitative influence of different network topologies on the detection of QNCNs in acyclic networks with Werner states as sources. Finally, we study how to maintain high independence degrees in networks formed by connecting two networks. This makes the correlations on the newly formed quantum network more likely to be nonlocal. Our research provides an alternative perspective for the well-known graph classification problem, and establishes an intrinsic connection between the graph theory and the foundation of quantum mechanics.
Quantum networks with bipartite resources and shared randomness present the simplest infrastructure for implementing a future quantum internet. Here, we shall investigate which kinds of entanglement can or cannot be generated from this kind of quantum network by examining their fidelity with different graph states. On the one hand, based on a standard form of graph states under local complementation and a fine-grained uncertainty relation between two projections, we establish upper bounds of fidelity that improve upon previous results by at least 25% as the dimension of local systems tends to infinity. On the other hand, in the triangle network, we propose efficient protocols to generate genuine multipartite entangled states from the network, providing significant nontrivial lower bounds of fidelity with high-dimensional GHZ states. Quantum networks with bipartite resources and shared randomness are important for the future quantum internet, yet the types of entanglement they can generate remain unclear. The authors establish improved upper bounds on fidelity for graph states and propose efficient protocols for generating genuine multipartite entangled states, enhancing quantum network capabilities.
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 distribution of multipartite entanglement is a key requirement for scalable quantum networks. While LOCC networks are theoretically universal, their reliance on quantum memories limits practical deployment. LOSR networks, which avoid real-time classical communication, offer a more feasible alternative but are fundamentally constrained when limited to bipartite sources. We conjecture that these limitations can be overcome with higher-dimensional entangled resources. Focusing on a square LOSR network with tripartite sources, we demonstrate a fidelity of1/21 / 21/2with GHZ states, surpassing the bound established in the literature. This result provides a concrete example of quantum advantage in LOSR networks and opens new directions for entanglement distribution under realistic constraints.
We investigate the properties of different levels of entanglement in graph states which correspond to connected graphs. Combining the operational definition of graph states and the postulates of entanglement measures, we prove that in connected graph states of NNN qubits there is no genuine kkk-qubit entanglement, 2≤k≤N−12\leq k\leq N-12≤k≤N−1, among every kkk qubits. These results about connected graph states naturally lead to the definition of fully multi-qubit entangled states. We also find that the connected graph states of four qubits is one but not the only one class of fully four-qubit entangled states.