Quantum networks are of high interest nowadays and a quantum internet has been long envisioned. Network-entanglement adapts the notion of entanglement to the network scenario and network-entangled states are considered to be a resource to overcome the limitations of a given network structure. In this work, we introduce measures of quantum network-entanglement that are well-defined within the general framework of quantum resource theories, which at the same time have a clear operational interpretation characterizing the extra resources necessary to prepare a targeted quantum state within a given network. In particular, we define the network communication cost and the network round complexity, which turn out to be intimately related to graph-theoretic parameters. We also provide methods to estimate these measures by introducing novel witnesses of network-entanglement. We also provide methods to estimate these measures by introducing novel witnesses of network- entanglement. 1 Introduction In recent years, quantum networks have emerged as a highly promising platform for implementing quantum communication and computation tasks. Quantum networks generalize basic primitives of quantum technologies such as quantum repeaters [1, 2] and they rely on the ability to generate and store entanglement at remote interconnected locations. Hence, there is a Julio I. de Vicente: jdvicent@math.uc3m.es Sixia Yu: yusixia@ustc.edu.cn considerable experimental effort addressed at combining entanglement sources and quantum memories so as to meet the above demands (see e.g. To answer this question, rigorous means for the quantification of network-entanglement become indispensable, which can then unveil the potential of different network-entangledstatesforquantumdistributed tasks. On the other hand, by constructing network-entanglement measures, one can address network design problems and find the network topology that minimizes the cost of preparing a given target state within the constraints dictated by a particular problem at hand, especially when the measures can be easily related to graph-theoretic properties of the corresponding hypergraph. In general, we hope that the development of network-entanglement measures might also help to identify tasks where quantum networks overcome classical networks in terms of communication complexity [32, 33], generalizing fundamental primitives such as superdense coding [34]. While the basic condition for entanglement measures is LOCC monotonicity [13], it should be clear from the above discussion that network-entaglement measures should obey LOSR monotonicity [16]. In this work, we introduce the network-entanglement weight , network communication cost and the network round complexity as measures of network- entanglement, which have clear operational interpretations as the minimal average rounds of classical communication within different classes of LOCC protocols that prepare the state from the network. Furthermore, we show that these quantifiers are well-defined from a resource-theoretic perspective by proving that they are LOSR monotonic, convex and subadditive. The network-entanglement measures Ec, Er and Ew are convex, LOSR monotonic and subadditive. The proof of this observation is provided in the App. II. These desirable properties of Ec and Er do not imply, however, that these measures provide a fine grained quantification of network-entanglement. Do these measures depend effectively on the underlying networkG? If so, how do they depend on the size and other properties ofG? In fact, in the App. I, we show thatEw can also be interpreted as the minimal complexity of preparing a state in a network by LOCC if we allow more than one parties to act non-trivially in a round. There exist pure states and networks which do not saturate these upper bounds and, infact, the lower bounds giveninObservation2canbeachieved. Thus, the network does not fix the value of the measures for allG-entangled pure states. The details are given in App. IV. In the above protocol, all the measurements in the procedure of teleportation commute with each other, since they act on different particles. Thus, they can be implemented simultaneously. If there is no restriction on classical communication of the outcomes, then the whole protocol can be finished in one run, i.e., all the nodes only communicate once. 2 the lower bounds onEw(ρ) for ρ =p GHZ +(1− p)1/210 to be a 10-partite state with local dimension 2 for different values ofk. In addition to this, a see-saw method based on semi-definite programming is presented in App. VIII, which provides estimations from above and works well for small networks. 4 Conclusions We have addressed the quantification of network-entanglement and we have provided different such measures. In particular, the network-entanglement weight, the network communication cost and the network round complexity have a clear operational interpretation in this scenario as the cost of preparing network-entangled states by LOCC. This, in turn, can be related to the quality of local quantum memories and to the information flow in the network. This is rather different from the case of standard entanglement theory, where LOCC operations are considered to be free, and stems from the fact that network-entanglement is a resource theory under LOSR. We have also shown that the latter two measures are closely connected to certain graph-theoretic quantities and to the network-entanglement weight, and we have used this to devise methods to estimate them by developing more efficient witnesses. X Other measures The formalism of quantum resource theories [35] can be readily applied to the network-entanglement scenario to obtain many different measures. Particularly, quantifiers based on the trace distance and fidelity have been often used in entanglement theory [13] and elsewhere. Their mathematical structure makes its estimation feasible and, in turn, they can be used to bound other