We introduce entanglement purification protocols for d-level systems (qudits) with improved efficiency as compared to previous protocols. While we focus on protocols for bipartite systems, we also propose generalizations to multi-partite qudit systems. The schemes we introduce include recurrence protocols that operate on two copies, as well as hashing protocols that operate on large ensembles. We analyze properties of the protocols with respect to minimal required fidelity and yield, and study their performance in the presence of noise and imperfections. We determine error thresholds and study the dependence on local dimension. We find that our schemes do not only outperform previous approaches, but also show an improved robustness and better efficiency with increasing dimension. While error thresholds for different system sizes are not directly comparable, our results nevertheless suggest that quantum information processing using qudits, in particular for long-distance quantum communication, may offer an advantage over approaches based on qubit systems.

We consider entanglement-based quantum networks where information is stored in a delocalized way within regions or the whole network. This offers a natural protection against failure of network nodes, loss and decoherence, and has built-in security features. Quantum information is transmitted within the network by performing local measurements on individual nodes only. Information can be localized within regions or at a specific node by collaborative actions using only entanglement within a region, or sometimes even without entanglement. We discuss several examples based on error correction stabilizer codes, Dicke states and correlation space encodings. We show how to design fully functional networks using encoded states or correlation space resources. the approach we follow here the failure of one (or several) individual nodes may have only a very limited influence, thereby minimizing the influence of network node failures. (ii) With respect to security of the stored information, the accessible information per node is bounded and can be made arbitrarily small. This implies that multiple nodes need to cooperate in order to access the information, while it is protected against malicious parties. In this context it is also interesting to note that the entanglement shared between an individual party and the rest of the network can be small [33], significantly less than one ebit. (iii) Since information is no longer stored in its bare form, one has to think of encoding, decoding and processing of information. Ideally, this should be done by local operations on individual nodes only, however it may also require some (restricted) amount of shared entanglement. In fact we find that processing of information, in particular transport among an entanglement-based network, is always possible using local operations only [34,35].We consider two different scenarios: (a) storage networks, where quantum information is stored in a distributed way among all or multiple nodes, and (b) generic networks with full functionality, including transport, that are comprised of different connected regions. Each region consists of multiple network nodes and corresponds to a single logical qubit. Regarding (a), we analyze several kind of encodings, where the logical basis states are given by codewords of error correction stabilizer codes, Dicke states [36][37][38], or resource states that have been discussed in the context of quantum computation in correlation space [33][34][35]. The usage of error correction codes for storage is well known and has been widely discussed. Such an approach offers protection against noise or loss, however requires active error correction. When used in a distributed scenario as we consider here, entanglement or non-local operations are required to detect and correct errors. Dicke state encodings in contrast have passive, built in protection features. Even without active error correction, quantum information is only slightly disturbed by loss, decoherence and node failures of a restricted amount of parties. Furtherm...

We show how to make quantum networks, both standard and entanglement-based, genuine quantum by providing them with the possibility of handling superposed tasks and superposed addressing. This extension of their functionality relies on a quantum control register, which specifies not only the task of the network, but also the corresponding weights in a coherently superposed fashion. Although adding coherent control to classical tasks, such as sending or measuring—or not doing so—is in general impossible, we introduce protocols that are able to mimick this behavior under certain conditions. We achieve this by always performing the classical task, either on the desired state or a properly chosen dummy state. We provide several examples, and show that externally controlling quantum superposition of tasks offers additional possibilities and advantages over usually considered single functionality. For instance, superpositions of different target state configurations shared among different nodes of the network can be prepared, or quantum information can be sent among a superposition of different paths or to different destinations.

The topology of classical networks is determined by physical links between nodes, and after a network request the links are used to establish the desired connections. Quantum networks offer the possibility to generate different kinds of entanglement prior to network requests, which can substitute links and allow one to fulfill multiple network requests with the same resource state. We utilize this to design entanglement-based quantum networks tailored to their desired functionality, independent of the underlying physical structure. The kind of entanglement to be stored is chosen to fulfill all desired network requests (i.e. parallel bipartite or multipartite communications between specific nodes chosen from some finite set), but in such a way that the storage requirement is minimized. This can be accomplished by using multipartite entangled states shared between network nodes that can be transformed by local operations to different target states. We introduce a clustering algorithm to identify connected clusters in the network for a given desired functionality, i.e. the required network topology of the entanglement-based network, and a merging algorithm that constructs multipartite entangled resource states with reduced memory requirement to fulfill all desired network requests. This leads to a significant reduction in required time and resources, and provides a powerful tool to design quantum networks that is unique to entanglement-based networks.

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