Quantum networks are promising venues for quantum information processing. This motivates the study of the entanglement properties of the particular multipartite quantum states that underpin these structures. In particular, it has been recently shown that when the links are noisy two drastically different behaviors can occur regarding the global entanglement properties of the network. While in certain configurations the network displays genuine multipartite entanglement (GME) for any system size provided the noise level is below a certain threshold, in others GME is washed out if the system size is big enough for any fixed non-zero level of noise. However, this difference has only been established considering the two extreme cases of maximally and minimally connected networks (i.e. complete graphs versus trees, respectively). In this article we investigate this question much more in depth and relate this behavior to the growth of several graph theoretic parameters that measure the connectivity of the graph sequence that codifies the structure of the network as the number of parties increases. The strongest conditions are obtained when considering the degree growth. Our main results are that a sufficiently fast degree growth (i.e. Ω ( N ) , where N is the size of the network) is sufficient for asymptotic robustness of GME, while if it is sufficiently slow (i.e. o ( log N ) ) then the network becomes asymptotically biseparable. We also present several explicit constructions related to the optimality of these results.
Quantum networks are promising venues for quantum information processing. This motivates the study of the entanglement properties of the particular multipartite quantum states that underpin these structures. In particular, it has been recently shown that when the links are noisy two drastically different behaviors can occur regarding the global entanglement properties of the network. While in certain configurations the network displays genuine multipartite entanglement (GME) for any system size provided the noise level is below a certain threshold, in others GME is washed out if the system size is big enough for any fixed non-zero level of noise. However, this difference has only been established considering the two extreme cases of maximally and minimally connected networks (i.e. complete graphs versus trees, respectively). In this article we investigate this question much more in depth and relate this behavior to the growth of several graph theoretic parameters that measure the connectivity of the graph sequence that codifies the structure of the network as the number of parties increases. The strongest conditions are obtained when considering the degree growth. Our main results are that a sufficiently fast degree growth (i.e. Ω ( N ) , where N is the size of the network) is sufficient for asymptotic robustness of GME, while if it is sufficiently slow (i.e. o ( log N ) ) then the network becomes asymptotically biseparable. We also present several explicit constructions related to the optimality of these results.
Quantum networks are under current active investigation for the implementation of quantum communication tasks. With this motivation in mind, we study the entanglement properties of the multipartite states underlying these networks. We show that, in sharp contrast to the case of pure states, genuine multipartite entanglement is severely affected by the presence of noise depending on the network topology: the amount of connectivity determines whether genuine multipartite entanglement is robust for any system size or whether it is completely washed out under the slightest form of noise for a sufficiently large number of parties. The impossibility to obtain genuine multipartite entanglement in some networks implies some fundamental limitations for their applications. In addition, the family of states considered in this work proves very useful to find new examples of states with interesting properties. We show this by constructing states of any number of parties that display superactivation of genuine multipartite nonlocality.
The study of entanglement in multipartite quantum states plays a major role in quantum information theory and genuine multipartite entanglement signals one of its strongest forms for applications. However, its characterization for general (mixed) states is a highly nontrivial problem. We introduce a particularly simple subclass of multipartite states, which we term pair-entangled network (PEN) states, as those that can be created by distributing exclusively bipartite entanglement in a connected network. We show that genuine multipartite entanglement in a PEN state depends on both the level of noise and the network topology and, in sharp contrast to the case of pure states, it is not guaranteed by the mere distribution of mixed bipartite entangled states. Our main result is a markedly drastic feature of this phenomenon: the amount of connectivity in the network determines whether genuine multipartite entanglement is robust to noise for any system size or whether it is completely washed out under the slightest form of noise for a sufficiently large number of parties. This latter case implies fundamental limitations for the application of certain networks in realistic scenarios, where the presence of some form of noise is unavoidable. To illustrate the applicability of PEN states to study the complex phenomenology behind multipartite entanglement, we also use them to prove superactivation of genuine multipartite nonlocality for any number of parties.
Quantum networks form the backbone of long-distance quantum information processing. Genuine multipartite entanglement (GME) serves as a key indicator of network performance and overall state quality. However, the widely used methods for certifying GME suffer from a major drawback that they either detect only a limited range of states or are applicable only to systems with a small number of parties. To overcome these limitations, we propose a family of sub-symmetric witnesses (SSWs), which are tractable both theoretically and experimentally. Analytically, we establish a connection between SSWs and the cut space of graph theory, enabling several powerful detection criteria tailored to practical quantum networks. Numerically, we show that the optimal detection can be formulated as a linear program, offering a significant efficiency advantage over the semidefinite programs commonly employed in quantum certification. Experimentally, SSWs can be evaluated via local measurements, with resource requirements independent of the local dimension in general, and even independent of the overall network size in many practical networks.
Characterizing large noisy multiparty quantum states using genuine multipartite entanglement is a challenging task. In this paper, we calculate lower bounds of genuine multipartite entanglement localized over a chosen multiparty subsystem of multiqubit stabilizer states in the noiseless and noisy scenarios. In the absence of noise, adopting a graph-based technique, we perform the calculation for arbitrary graph states as representatives of the stabilizer states and show that the graph operations required for the calculation have a polynomial scaling with the system size. As demonstrations, we compute the localized genuine multipartite entanglement over subsystems of large graphs having linear, ladder, and square structures. We also extend the calculation for graph states subjected to single-qubit Markovian or non-Markovian Pauli noise on all qubits and demonstrate, for a specific lower bound of the localizable genuine multipartite entanglement corresponding to a specific Pauli measurement setup, the existence of a critical noise strength beyond which all of the postmeasured states are biseparable. The calculation is also useful for arbitrary large stabilizer states under noise due to the local unitary connection between stabilizer states and graph states. We demonstrate this by considering a toric code defined on a square lattice and computing a lower bound of localizable genuine multipartite entanglement over a nontrivial loop of the code. Similar to the graph states, we show the existence of the critical noise strength in this case also and discuss its interesting features.
Graph states are entangled states that are essential for quantum information processing. As experimental advances enable the realization of large-scale graph states, efficient fidelity estimation methods are crucial for assessing their robustness against noise. However, calculations of exact fidelity become intractable for large systems due to the exponential growth in the number of stabilizers. In this work, we show that the fidelity between any ideal graph state and its noisy counterpart under IID Pauli noise can be mapped to the partition function of a classical spin system, enabling efficient computation via statistical mechanical techniques. Using this approach, we analyze the fidelity for regular graph states under depolarizing noise and uncover the emergence of phase transitions in fidelity between the pure-state regime and the noise-dominated regime. Specifically, in 2D, phase transitions occur only when the degree satisfies d ≥ 6 , while in 3D they already appear at d ≥ 5 . However, for graph states with excessively high degree, such as fully connected graphs, the phase transition disappears. Robustness of graph states against noise is thus determined by their connectivity and spatial dimensionality. Graph states with lower degree and/or dimensionality, which exhibit a smooth crossover, demonstrate greater robustness, while highly connected or higher-dimensional graph states are more fragile. Extreme connectivity, as the fully connected graph state possesses, restores robustness. Furthermore, we show that the fidelity can be rewritten in the form of the partition function of a constraint-percolation problem. Within this picture, we discuss the qualitative difference between 2D regular graph states with d = 6 and d = 5 regarding the presence or absence of a phase transition, as well as the suppressed critical behavior of fully connected graph states.
Quantum network states are multipartite states built from distributing pairwise entanglement among parties and underpin the paradigm of quantum networks for quantum information processing. In this work we introduce the problem of partial distillability in noisy quantum networks. This corresponds to the possibility of distilling by local operations and classical communication an arbitrary pure state for an arbitrary subset of parties starting from a single copy of a quantum network state of an increasing number of parties that share noisy (i.e. mixed) bipartite entangled states. Here, distillation means that the target state is obtained with fidelity as close to 1 as desired as the size of the network increases when the pairwise entangled links support a constant amount of noise. While we prove an obstruction to multipartite distillation protocols after teleportation with channels with constant noise, we show that partial distillability is indeed possible if certain well-established graph-theoretic parameters that measure the connectivity in the network grow fast enough with its size. We give necessary as well as sufficient conditions for partial distillability in terms of these parameters and we moreover provide explicit constructions of networks with partial distillability and a relatively slow connectivity growth.