The following submission constitutes a guide and an introduction to a collection of articles submitted as a Ph.D. dissertation at the University of Gdańsk. In the dissertation, we study the fundamental limitations within the selected quantum and supra-quantum cryptographic scenarios in the form of upper bounds on the achievable key rates. We investigate various security paradigms, bipartite and multipartite settings, as well as single-shot and asymptotic regimes. Our studies, however, extend beyond the derivations of the upper bounds on the secret key rates in the mentioned scenarios. In particular, we propose a novel type of rerouting attack on the quantum Internet for which we find a countermeasure and benchmark its efficiency. Furthermore, we propose several upper bounds on the performance of quantum (key) repeaters settings. We derive a lower bound on the secret key agreement capacity of a quantum network, which we tighten in an important case of a bidirectional quantum network. The squashed nonlocality derived here as an upper bound on the secret key rate is a novel non-faithful measure of nonlocality. Furthermore, the notion of the non-signaling complete extension arising from the complete extension postulate as a counterpart of purification of a quantum state allows us to study analogies between non-signaling and quantum key distribution scenarios.
The following submission constitutes a guide and an introduction to a collection of articles submitted as a Ph.D. dissertation at the University of Gdańsk. In the dissertation, we study the fundamental limitations within the selected quantum and supra-quantum cryptographic scenarios in the form of upper bounds on the achievable key rates. We investigate various security paradigms, bipartite and multipartite settings, as well as single-shot and asymptotic regimes. Our studies, however, extend beyond the derivations of the upper bounds on the secret key rates in the mentioned scenarios. In particular, we propose a novel type of rerouting attack on the quantum Internet for which we find a countermeasure and benchmark its efficiency. Furthermore, we propose several upper bounds on the performance of quantum (key) repeaters settings. We derive a lower bound on the secret key agreement capacity of a quantum network, which we tighten in an important case of a bidirectional quantum network. The squashed nonlocality derived here as an upper bound on the secret key rate is a novel non-faithful measure of nonlocality. Furthermore, the notion of the non-signaling complete extension arising from the complete extension postulate as a counterpart of purification of a quantum state allows us to study analogies between non-signaling and quantum key distribution scenarios.
The aim of this thesis is to understand the fundamental limitations on secret key distillation in various settings of quantum key distribution. We first consider quantum steering, which is a resource for one-sided device-independent quantum key distribution. We introduce a conditional mutual information based quantifier for quantum steering, which we call intrinsic steerability. Next, we consider quantum non-locality, which is a resource for device-independent quantum key distribution. In this context, we introduce a quantifier, intrinsic non-locality, which is a monotone in the resource theory of Bell non-locality. Both these quantities are inspired by intrinsic information and squashed entanglement and are based on conditional mutual information. The idea behind these quantifiers is to suppress the correlations that can be explained by a local hidden variable or by an inaccessible quantum system, thus quantifying the remaining intrinsic correlations. We then prove various properties of these two monotones, which includes the following: monotonicity under free operations, additivity under tensor product of objects, convexity, and faithfulness, among others. Next, we prove that intrinsic steerability is an upper bound on the secret-key-agreement capacity of an assemblage, and intrinsic non-locality is an upper bound on the secret-key-agreement capacity of a quantum probability distribution. Thus we prove that these quantities are upper bounds on the achievable key rates in one-sided device-independent and device-independent quantum key distribution protocols. We also calculate these bounds for certain honest devices. The study of these upper bounds is instrumental in understanding the limitations of protocols that can be designed for various settings. These upper bounds inform us that, even if one considers the best possible protocol, there is no possibility of exceeding the upper bounds on key rates without a quantum repeater. The upper bounds introduced in this thesis are an important step for initiating this line of research in one-sided device-independent and in device-independent quantum key distribution.
Abstract 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 (Azuma et al , (2016), Nat. Commun. 7 13523). We also discuss how lower bounds can be obtained by combining a generalisation of an aggregated quantum repeater protocol with graph theoretic concepts.
Distribution and distillation of entanglement over quantum networks is a basic task for Quantum Internet applications. A fundamental question is then to determine the ultimate performance of entanglement distribution over a given network. Although this question has been extensively explored for bipartite entanglement-distribution scenarios, less is known about multipartite entanglement distribution. Here we establish the fundamental limit of distributing multipartite entanglement, in the form of GHZ states, over a quantum network. In particular, we determine the multipartite entanglement distribution capacity of a quantum network, in which the nodes are connected through lossy bosonic quantum channels. This setting corresponds to a practical quantum network consisting of optical links. The result is also applicable to the distribution of multipartite secret key, known as common key, for both a fully quantum network and trusted-node based quantum key distribution network. Our results set a general benchmark for designing a network topology and network quantum repeaters (or key relay in trusted nodes) to realize efficient GHZ state/common key distribution in both fully quantum and trusted-node-based networks. We show an example of how to overcome this limit by introducing a network quantum repeater. Our result follows from an upper bound on distillable GHZ entanglement introduced here, called the "recursive-cut-and-merge" bound, which constitutes major progress on a longstanding fundamental problem in multipartite entanglement theory. This bound allows for determining the distillable GHZ entanglement for a class of states consisting of products of bipartite pure states.
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 Letter, 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 nontrivial, efficiently computable upper bound on the positive-partial-transpose-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. As an application, we introduce a cryptographic protocol for the readout of a digital memory device that is secure against a passive eavesdropper.
The quantum internet holds promise for accomplishing quantum teleportation and unconditionally secure communication freely between arbitrary clients all over the globe, as well as the simulation of quantum many-body systems. For such a quantum internet protocol, a general fundamental upper bound on the obtainable entanglement or secret key has been derived [K. Azuma, A. Mizutani, and H.-K. Lo, Nat. Commun. 7, 13523 (2016)]. Here we consider its converse problem. In particular, we present a universal protocol constructible from any given quantum network, which is based on running quantum repeater schemes in parallel over the network. For arbitrary lossy optical channel networks, our protocol has no scaling gap with the upper bound, even based on existing quantum repeater schemes. In an asymptotic limit, our protocol works as an optimal entanglement or secret-key distribution over any quantum network composed of practical channels such as erasure channels, dephasing channels, bosonic quantum amplifier channels, and lossy optical channels.
Upper bounds on the secret-key-agreement capacity of a quantum channel serve as a way to assess the performance of practical quantum-key-distribution protocols conducted over that channel. In particular, if a protocol employs a quantum repeater, achieving secret-key rates exceeding these upper bounds is evidence of having a working quantum repeater. In this paper, we extend a recent advance [Liuzzo-Scorpo et al., Phys. Rev. Lett. 119, 120503 (2017)] in the theory of the teleportation simulation of single-mode phase-insensitive Gaussian channels such that it now applies to the relative entropy of entanglement measure. As a consequence of this extension, we find tighter upper bounds on the nonasymptotic secret-key-agreement capacity of the lossy thermal bosonic channel than were previously known. The lossy thermal bosonic channel serves as a more realistic model of communication than the pure-loss bosonic channel, because it can model the effects of eavesdropper tampering and imperfect detectors. An implication of our result is that the previously known upper bounds on the secret-key-agreement capacity of the thermal channel are too pessimistic for the practical finite-size regime in which the channel is used a finite number of times, and so it should now be somewhat easier to witness a working quantum repeater when using secret-key-agreement capacity upper bounds as a benchmark.