Statistical correlations that can be generated across the nodes in a quantum network depend crucially on its topology.However, this topological information might not be known a priori, or it may need to be verified.In this paper, we propose an efficient protocol for distinguishing and inferring the topology of a quantum network.We leverage entropic quantities-namely, the von Neumann entropy and the measured mutual information-as well as measurement covariance to uniquely characterize the topology.We show that the entropic quantities are sufficient to distinguish two networks that prepare GHZ states.Moreover, if qubit measurements are available, both entropic quantities and covariance can be used to infer the network topology without state-preparation assumptions.We show that the protocol can be entirely robust to noise and can be implemented via quantum variational optimization.Numerical experiments on both classical simulators and quantum hardware show that covariance is generally more reliable for accurately and efficiently inferring the topology, whereas entropy-based methods are often better at identifying the absence of entanglement in the low-shot regime.
Statistical correlations that can be generated across the nodes in a quantum network depend crucially on its topology.However, this topological information might not be known a priori, or it may need to be verified.In this paper, we propose an efficient protocol for distinguishing and inferring the topology of a quantum network.We leverage entropic quantities-namely, the von Neumann entropy and the measured mutual information-as well as measurement covariance to uniquely characterize the topology.We show that the entropic quantities are sufficient to distinguish two networks that prepare GHZ states.Moreover, if qubit measurements are available, both entropic quantities and covariance can be used to infer the network topology without state-preparation assumptions.We show that the protocol can be entirely robust to noise and can be implemented via quantum variational optimization.Numerical experiments on both classical simulators and quantum hardware show that covariance is generally more reliable for accurately and efficiently inferring the topology, whereas entropy-based methods are often better at identifying the absence of entanglement in the low-shot regime.
Paramount for performances of quantum network applications are the structure and quality of distributed entanglement. Here we propose a scalable and efficient approach to reveal the topological information of unknown quantum networks, and quantify entanglement simultaneously. The scheme exploits entropic uncertainty, an operationally meaningful measure of correlation, by performing only two local measurements on each qubit. Moreover, when measurement outcomes in each node are collectively evaluated, integrating uncertainty and mutual information enables a direct count of the number of bipartite sources between any two nodes. This surpasses what is possible via applying either approach solely. Moreover, quantum error mitigation techniques including probabilistic error cancellation (PEC) and virtual distillation (VD), which have been widely applied to suppress biases in single expectation value, are further incorporated to mitigate errors in entropic quantities. We find that PEC successfully removes deviations in correlation estimations. Meanwhile, VD extends the depolarizing noise strength that allows for valid bipartite entanglement certification from 8.8% to 26.4%, thus substantially enhancing robustness against bias-inducing noise in practical situations. The proposal is applicable to a broad variety of platforms and helps to spur future studies toward harnessing the advantages of quantum networks.
Distributed quantum information in networks is paramount for global secure quantum communication. Moreover, it finds applications as a resource for relevant tasks, such as clock synchronization, magnetic field sensing, and blind quantum computation. For quantum network analysis and benchmarking of implementations, however, it is crucial to characterize the topology of networks in a way that reveals the nodes between which entanglement can be reliably distributed. Here, we demonstrate an efficient scheme for this topology certification. Our scheme allows for distinguishing, in a scalable manner, different networks consisting of bipartite and multipartite entanglement sources. It can be applied to semi-device-independent scenarios also, where the measurement devices and network nodes are not well characterized and trusted. We experimentally demonstrate our approach by certifying the topology of different six-qubit networks generated with polarized photons, employing active feed-forward and time multiplexing. Our methods can be used for general simultaneous tests of multiple hypotheses with few measurements, being useful for other certification scenarios in quantum technologies.
Topological order represents a profound state of matter characterized by exotic properties robust against local perturbations, such as fractionalized excitations and ground state degeneracy dependent on system topology. While extensively studied in closed quantum systems, the detection and characterization of topological order in open quantum systems, which inevitably interact with their environment, remain a significant challenge. Environmental coupling typically introduces decoherence and dissipation, potentially obscuring or even destroying fragile quantum correlations. This paper investigates the concept of "entropic fingerprints" as a robust diagnostic tool for identifying topological order in systems subjected to environmental influences. We explore how various entanglement measures, particularly the topological entanglement entropy and related information-theoretic quantities, can be adapted and reinterpreted in the context of non-equilibrium steady states or dynamically evolving open systems. Through a theoretical framework employing Lindblad master equations and tensor network methods, we demonstrate that universal signatures of topological order, though potentially modified, can persist in the steady-state entanglement properties of open systems. Our findings suggest that specific patterns in the bipartite entanglement entropy, conditional entropies, and mutual information, robust against a finite range of environmental parameters, can serve as reliable indicators of underlying topological phases. This research not only offers a pathway for experimentally probing topological materials in realistic, noisy environments but also deepens our fundamental understanding of how quantum information is preserved or degraded under open system dynamics.
The practical utility of a quantum network depends on its ability to establish entanglement between arbitrary node pairs with quality sufficient to execute entanglement enabled tasks. This capability can be assessed globally, through aggregate performance over all node pairs, as well as locally, at the level of individual nodes. Since entanglement-based connections form a layer above the underlying physical topology, quantum connectivity is not adequately captured by classical topological connectivity metrics. To enable characterisation of the quantum connectivity at the level of the network (or its subnetworks), we introduce the quantum connectivity measure (QCM), which quantifies the average connection quality between pairs of network nodes. Further, we describe two quantities, the quantum-connected fraction (QCF) and the quantum clustering coefficient (QCC), naturally derived from the QCM, which capture important features of the functional connectivity of the quantum network at the level of the network and an individual node, respectively. These metrics of quantum connectivity depend crucially on the entanglement distribution protocol and the quantum network parameters in addition to its physical topology. We demonstrate the crucial distinction between topological and quantum connectivity, showing that even a fully connected graph can be functionally disconnected for quantum tasks if average network edge-concurrence falls below a critical threshold. These quantum connectivity metrics thus provide important tools for the design, optimization, and benchmarking of future quantum networks.
Quantum networks play a major role in long-distance communication, quantum cryptography, clock synchronization, and distributed quantum computing. Generally, these protocols involve many independent sources sharing entanglement among distant parties that, upon measuring their systems, generate correlations across the network. The question of which correlations a given quantum network can give rise to remains almost uncharted. Here we show that constraints on the observable covariances, previously derived for the classical case, also hold for quantum networks. The network topology yields tests that can be cast as semidefinite programs, thus allowing for the efficient characterization of the correlations in a wide class of quantum networks, as well as systematic derivations of device-independent and experimentally testable witnesses. We obtain such semidefinite tests for fixed measurement settings, as well as parties that independently choose among collections of measurement settings. The applicability of the method is demonstrated for various networks, and compared with previous approaches.
Entanglement percolation aims at generating maximal entanglement between any two nodes of a quantum network by utilizing strategies based solely on local operations and classical communication between the nodes. As it happens in classical percolation theory, the topology of the network is crucial, but also the entanglement shared between the nodes of the network. In a network of identically partially entangled states, the network topology determines the minimum entanglement needed for percolation. In this work, we generalize the protocol to scenarios where the initial entanglement shared between any two nodes of the network is not the same but has some randomness. In such cases, we find that for classical entanglement percolation, only the average initial entanglement is relevant. In contrast, the quantum entanglement percolation protocol (within the q-swap framework) degrades under these more realistic conditions as the width of the distribution increases, suggesting that Random CEP may become the optimal LOCC strategy in sufficiently heterogeneous quantum networks.