A major application of quantum communication is the distribution of entangled particles for use in quantum key distribution. Owing to noise in the communication line, quantum key distribution is, in practice, limited to a distance of a few hundred kilometres, and can only be extended to longer distances by use of a quantum repeater, a device that performs entanglement distillation and quantum teleportation. The existence of noisy entangled states that are undistillable but nevertheless useful for quantum key distribution raises the question of the feasibility of a quantum key repeater, which would work beyond the limits of entanglement distillation, hence possibly tolerating higher noise levels than existing protocols. Here we exhibit fundamental limits on such a device in the form of bounds on the rate at which it may extract secure key. As a consequence, we give examples of states suitable for quantum key distribution but unsuitable for the most general quantum key repeater protocol.
We provide a versatile upper bound on the number of maximally entangled qubits, or private bits, shared by two parties via a generic adaptive communication protocol over a quantum network when the use of classical communication is not restricted. Although our result follows the idea of Azuma et al (2016 Nat. Commun. 7 13523) of splitting the network into two parts, our approach relaxes their strong restriction, consisting of the use of a single entanglement measure in the quantification of the maximum amount of entanglement generated by the channels. In particular, in our bound the measure can be chosen on a channel-by-channel basis, in order to make it as tight as possible. This enables us to apply the relative entropy of entanglement, which often gives a state-of-the-art upper bound, on every Choi-simulable channel in the network, even when the other channels do not satisfy this property. We also develop tools to compute, or bound, the max-relative entropy of entanglement for channels that are invariant under phase rotations. In particular, we present an analytical formula for the max-relative entropy of entanglement of the qubit amplitude damping channel. IntroductionWhenever two parties, say Alice and Bob, want to communicate by using a quantum channel, its noise unavoidably limits their communication efficiency . In the limit of many channel uses, their asymptotic optimal performance can be quantified by the channel capacity, which represents the supremum of the number of qubits/bits that can be faithfully transmitted per channel use. Obtaining an exact expression for this quantity is typically far from trivial. Indeed, in addition to the difficulty of studying the asymptotic behaviour of the channel, the value of the capacity also depends on the task Alice and Bob want to perform, as well as on the free resources available to them . Two representative tasks, which will be considered in our paper, involve the generation and distribution of a string of shared private bits (pbits) [2,3] or of maximally entangled states (ebits) . These are known to be fundamental resources for more complex protocols, such as secure classsical communication [5,6], quantum teleportation , and quantum state merging . An example of free resource involves the possibility of exchanging classical information over a public classical channel, such as a telephone line or over the internet. Depending on the restrictions on this, the capacity is said to be assisted by zero, forward, backward, or two-way classical communication . In this paper we will focus on the last option, that is, no restriction will be imposed on the use of classical communication.Although the capacity of a quantum channel is by definition an abstract and theoretical quantity, it is also practically useful in that it can be compared with the performance of known transmission schemes. This comparison could then give an indication on the extent of improvements that could be expected in the future. From this perspective, similar conclusions could be obt...
The ability to distribute entanglement over complex quantum networks is an important step towards a quantum internet. Recently, there has been significant theoretical effort, mainly focusing on the distribution of bipartite entanglement via a simple quantum network composed only of bipartite quantum channels. There are, however, a number of quantum information processing protocols based on multipartite rather than bipartite entanglement. Whereas multipartite entanglement can be distributed by means of a network of such bipartite channels, a more natural way is to use a more general network, that is, a quantum broadcast network including quantum broadcast channels. In this work, we present a general framework for deriving upper bounds on the rates at which GHZ states or multipartite private states can be distributed among a number of different parties over an arbitrary quantum broadcast network. Our upper bounds are written in terms of the multipartite squashed entanglement, corresponding to a generalisation of recently derived bounds [K. Azuma et al., Nat. Commun. 7, 13523 (2016)]. We also discuss how lower bounds can be obtained by combining a generalisation of an aggregated quantum repeater protocol with graph theoretic concepts.
Quantum networks will allow to implement communication tasks beyond the reach of their classical counterparts. A pressing and necessary issue for the design of quantum network protocols is the quantification of the rates at which these tasks can be performed. Here, we propose a simple recipe that yields efficiently computable lower and upper bounds for network capacities. For this we make use of the max-flow min-cut theorem and its generalization to multi-commodity flows to obtain linear programs (LPs). We exemplify our recipe deriving the LPs for bipartite settings, settings where multi-pairs of users obtain entanglement in parallel as well as multipartite settings, covering almost all known situations. We also make use of a generalization of the concept of paths between user pairs in a network to Steiner trees spanning the group of users wishing to establish GHZ states. Contents 26A. Proof of Lemma 2 26 B. Proof of Lemma 3 27 C. Proof of Lemma 4 28 arXiv:1809.03120v2 [quant-ph] 4 Jan 2019 total/worst case network and P total/worst case networkas the maximum rate achievable by means of an adaptive protocol, as well as (3) the maximization over a weighted sum of rates over all user pairs. From [13,14], we can obtain upper bounds on the private capacities that involve (1) a minimization over multicuts, i.e. sets of edges the removal of which connects all pairs, or (2) the so-called minimum cut ratio . Using the respective results of [17,19], we can, up to a factor of order O(log r), upper bound the capacities for both cases (1) and (2) by a maximization over concurrent multi-commodity flows, i.e. flows between several user pairs that can be achieved in parallel. In the first case we maximize the sum of flows for all user pairs, whereas in the second case we maximize the worst case flow that is guaranteed for every pair. Both multi-commodity flow maximizations can be cast into LPs. Applying the aggregated repeater protocol  to multiple user pairs we also obtain lower bounds in terms of the maximum concurrent multi-commodity flows, providing us with the following efficiently computable bounds: f total/worst case Q ↔ ≤ Q total/worst case network ≤ P total/worst case network ≤ O(log r)f total/worst case Esq/E
Quantum networks will enable the implementation of communication tasks with qualitative advantages with respect to the communication networks known today. While it is expected that the first demonstrations of small scale quantum networks will take place in the near term, many challenges remain to scale them. To compare different solutions, optimize over parameter space, and inform experiments, it is necessary to evaluate the performance of concrete quantum network scenarios. Here, the authors review the state-of-the-art of tools for evaluating the performance of quantum networks. The authors present them from three different angles: information-theoretic benchmarks, analytical tools, and simulation.
Bipartite quantum interactions have applications in a number of different areas of quantum physics, reaching from fundamental areas such as quantum thermodynamics and the theory of quantum measurements to other applications such as quantum computers, quantum key distribution, and other information processing protocols. A particular aspect of the study of bipartite interactions is concerned with the entanglement that can be created from such interactions. In this paper, we present our work on two basic building blocks of bipartite quantum protocols, namely, the generation of maximally entangled states and secret key via bipartite quantum interactions. In particular, we provide a non-trivial, efficiently computable upper bound on the positive-partial-transposeassisted (PPT-assisted) quantum capacity of a bipartite quantum interaction. In addition, we provide an upper bound on the secret-key-agreement capacity of a bipartite quantum interaction assisted by local operations and classical communication (LOCC). As an application, we introduce a cryptographic protocol for the read-out of a digital memory device that is secure against a passive eavesdropper.
A bipartite quantum interaction corresponds to the most general quantum interaction that can occur between two quantum systems. In this work, we determine bounds on the capacities of bipartite interactions for entanglement generation and secret key agreement. Our upper bound on the entanglement generation capacity of a bipartite quantum interaction is given by a quantity that we introduce here, called the bidirectional max-Rains information. Our upper bound on the secret-key-agreement capacity of a bipartite quantum interaction is given by a related quantity introduced here also, called the bidirectional max-relative entropy of entanglement. We also derive tighter upper bounds on the capacities of bipartite interactions obeying certain symmetries. Observing that quantum reading is a particular kind of bipartite quantum interaction, we leverage our bounds from the bidirectional setting to deliver bounds on the capacity of a task that we introduce, called private reading of a wiretap memory cell. Given a set of point-to-point quantum wiretap channels, the goal of private reading is for an encoder to form codewords from these channels, in order to establish secret key with a party who controls one input and one output of the channels, while a passive eavesdropper has access to one output of the channels. We derive both lower and upper bounds on the private reading capacities of a wiretap memory cell. We then extend these results to determine achievable rates for the generation of entanglement between two distant parties who have coherent access to a controlled point-to-point channel, which is a particular kind of bipartite interaction.
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