This is a preliminary version of a book in progress on the theory of quantum communication. We adopt an information-theoretic perspective throughout and give a comprehensive account of fundamental results in quantum communication theory from the past
decade (and earlier), with an emphasis on the modern one-shot-to-asymptotic approach that underlies much of todays state-of-the-art research in this field. In Part I, we cover mathematical preliminaries and provide a detailed study of quantum mechanics from an information-theoretic perspective. We also provide an extensive and thorough review of the quantum entropy zoo, and we devote an entire chapter to the study of entanglement measures. Equipped with these essential tools, in Part II we study classical communication (with and without entanglement assistance), entanglement distillation, quantum communication, secret key distillation, and private communication. In Part III, we cover the latest developments in feedback-assisted communication tasks, such as quantum and classical feedback-assisted communication, LOCC-assisted quantum communication, and secret key agreement.
We introduce various measures of forward classical communication for bipartite quantum channels. Since a point-to-point channel is a special case of a bipartite channel, the measures reduce to measures of classical communication for point-to-point ch
annels. As it turns out, these reduced measures have been reported in prior work of Wang et al. on bounding the classical capacity of a quantum channel. As applications, we show that the measures are upper bounds on the forward classical capacity of a bipartite channel. The reduced measures are upper bounds on the classical capacity of a point-to-point quantum channel assisted by a classical feedback channel. Some of the various measures can be computed by semi-definite programming.
The security of quantum key distribution has traditionally been analyzed in either the asymptotic or non-asymptotic regimes. In this paper, we provide a bridge between these two regimes, by determining second-order coding rates for key distillation i
n quantum key distribution under collective attacks. Our main result is a formula that characterizes the backoff from the known asymptotic formula for key distillation -- our formula incorporates the reliability and security of the protocol, as well as the mutual information variances to the legitimate receiver and the eavesdropper. In order to determine secure key rates against collective attacks, one should perform a joint optimization of the Holevo information and the Holevo information variance to the eavesdropper. We show how to do so by analyzing several examples, including the six-state, BB84, and continuous-variable quantum key distribution protocols (the last involving Gaussian modulation of coherent states along with heterodyne detection). The technical contributions of this paper include one-shot and second-order analyses of private communication over a compound quantum wiretap channel with fixed marginal and key distillation over a compound quantum wiretap source with fixed marginal. We also establish the second-order asymptotics of the smooth max-relative entropy of quantum states acting on a separable Hilbert space, and we derive a formula for the Holevo information variance of a Gaussian ensemble of Gaussian states.
Before global-scale quantum networks become operational, it is important to consider how to evaluate their performance so that they can be built to achieve the desired performance. We propose two practical figures of merit for the performance of a qu
antum network: the average connection time and the average largest entanglement cluster size. These quantities are based on the generation of elementary links in a quantum network, which is a crucial initial requirement that must be met before any long-range entanglement distribution can be achieved and is inherently probabilistic with current implementations. We obtain bounds on these figures of merit for a particular class of quantum repeater protocols consisting of repeat-until-success elementary link generation followed by joining measurements at intermediate nodes that extend the entanglement range. Our results lead to requirements on quantum memory coherence times, requirements on repeater chain lengths in order to surpass the repeaterless rate limit, and requirements on other aspects of quantum network implementations. These requirements are based solely on the inherently probabilistic nature of elementary link generation in quantum networks, and they apply to networks with arbitrary topology.
Extendibility of bosonic Gaussian states is a key issue in continuous-variable quantum information. We show that a bosonic Gaussian state is $k$-extendible if and only if it has a Gaussian $k$-extension, and we derive a simple semidefinite program, w
hose size scales linearly with the number of local modes, to efficiently decide $k$-extendibility of any given bosonic Gaussian state. When the system to be extended comprises one mode only, we provide a closed-form solution. Implications of these results for the steerability of quantum states and for the extendibility of bosonic Gaussian channels are discussed. We then derive upper bounds on the distance of a $k$-extendible bosonic Gaussian state to the set of all separable states, in terms of trace norm and Renyi relative entropies. These bounds, which can be seen as Gaussian de Finetti theorems, exhibit a universal scaling in the total number of modes, independently of the mean energy of the state. Finally, we establish an upper bound on the entanglement of formation of Gaussian $k$-extendible states, which has no analogue in the finite-dimensional setting.
The generalized amplitude damping channel (GADC) is one of the sources of noise in superconducting-circuit-based quantum computing. It can be viewed as the qubit analogue of the bosonic thermal channel, and it thus can be used to model lossy processe
s in the presence of background noise for low-temperature systems. In this work, we provide an information-theoretic study of the GADC. We first determine the parameter range for which the GADC is entanglement breaking and the range for which it is anti-degradable. We then establish several upper bounds on its classical, quantum, and private capacities. These bounds are based on data-processing inequalities and the uniform continuity of information-theoretic quantities, as well as other techniques. Our upper bounds on the quantum capacity of the GADC are tighter than the known upper bound reported recently in [Rosati et al., Nat. Commun. 9, 4339 (2018)] for the entire parameter range of the GADC, thus reducing the gap between the lower and upper bounds. We also establish upper bounds on the two-way assisted quantum and private capacities of the GADC. These bounds are based on the squashed entanglement, and they are established by constructing particular squashing channels. We compare these bounds with the max-Rains information bound, the mutual information bound, and another bound based on approximate covariance. For all capacities considered, we find that a large variety of techniques are useful in establishing bounds.
Entanglement distribution is a prerequisite for several important quantum information processing and computing tasks, such as quantum teleportation, quantum key distribution, and distributed quantum computing. In this work, we focus on two-dimensiona
l quantum networks based on optical quantum technologies using dual-rail photonic qubits for the building of a fail-safe quantum internet. We lay out a quantum network architecture for entanglement distribution between distant parties using a Bravais lattice topology, with the technological constraint that quantum repeaters equipped with quantum memories are not easily accessible. We provide a robust protocol for simultaneous entanglement distribution between two distant groups of parties on this network. We also discuss a memory-based quantum network architecture that can be implemented on networks with an arbitrary topology. We examine networks with bow-tie lattice and Archimedean lattice topologies and use percolation theory to quantify the robustness of the networks. In particular, we provide figures of merit on the loss parameter of the optical medium that depend only on the topology of the network and quantify the robustness of the network against intermittent photon loss and intermittent failure of nodes. These figures of merit can be used to compare the robustness of different network topologies in order to determine the best topology in a given real-world scenario, which is critical in the realization of the quantum internet.
It is well known in the realm of quantum mechanics and information theory that the entropy is non-decreasing for the class of unital physical processes. However, in general, the entropy does not exhibit monotonic behavior. This has restricted the use
of entropy change in characterizing evolution processes. Recently, a lower bound on the entropy change was provided in the work of Buscemi, Das, and Wilde~[Phys.~Rev.~A~93(6),~062314~(2016)]. We explore the limit that this bound places on the physical evolution of a quantum system and discuss how these limits can be used as witnesses to characterize quantum dynamics. In particular, we derive a lower limit on the rate of entropy change for memoryless quantum dynamics, and we argue that it provides a witness of non-unitality. This limit on the rate of entropy change leads to definitions of several witnesses for testing memory effects in quantum dynamics. Furthermore, from the aforementioned lower bound on entropy change, we obtain a measure of non-unitarity for unital evolutions.
Bound secret information is classical information that contains secrecy but from which secrecy cannot be extracted. The existence of bound secrecy has been conjectured but is currently unproven, and in this work we provide analytical and numerical ev
idence for its existence. Specifically, we consider two-way post-processing protocols in prepare-and-measure quantum key distribution based on the well-known six-state signal states. In terms of the quantum bit-error rate $Q$ of the classical data, such protocols currently exist for $Q<frac{5-sqrt{5}}{10}approx 27.6%$. On the other hand, for $Qgeqfrac{1}{3}$ no such protocol can exist as the observed data is compatible with an intercept-resend attack. This leaves the interesting question of whether successful protocols exist in the interval $frac{5-sqrt{5}}{10}leq Q<frac{1}{3}$. Previous work has shown that a necessary condition for the existence of two-way post-processing protocols for distilling secret key is breaking the symmetric extendability of the underlying quantum state shared by Alice and Bob. Using this result, it has been proven that symmetric extendability can be broken up to the $27.6%$ lower bound using the advantage distillation protocol. In this work, we first show that to break symmetric extendability it is sufficient to consider a generalized form of advantage distillation consisting of one round of post-selection by Bob on a block of his data. We then provide evidence that such generalized protocols cannot break symmetric extendability beyond $27.6%$. We thus have evidence to believe that $27.6%$ is an upper bound on two-way post-processing and that the interval $frac{5-sqrt{5}}{10}leq Q<frac{1}{3}$ is a domain of bound secrecy.
