Distribution of entanglement is an essential task in quantum information processing and the realization of quantum networks. In our work, we theoretically investigate the scenario where a central source prepares anN-partite entangled state and transmits each entangled subsystem to one ofNreceivers through noisy quantum channels. The receivers are then able to perform local operations assisted by unlimited classical communication to distill target entangled states from the noisy channel output. In this operational context, we define the EPR distribution capacity and the GHZ distribution capacity of a quantum channel as the largest rates at which Einstein-Podolsky-Rosen (EPR) states and Greenberger-Horne-Zeilinger (GHZ) states can be faithfully distributed through the channel, respectively. We establish lower and upper bounds on the EPR distribution capacity by connecting it with the task of assisted entanglement distillation. We also construct an explicit protocol consisting of a combination of a quantum communication code and a classical-post-processing-assisted entanglement generation code, which yields a simple achievable lower bound for generic channels. As applications of these results, we give an exact expression for the EPR distribution capacity over two erasure channels and bounds on the EPR distribution capacity over two generalized amplitude damping channels. We also bound the GHZ distribution capacity, which results in an exact characterization of the GHZ distribution capacity when the most noisy channel is a dephasing channel.

Distribution of entanglement is an essential task in quantum information processing and the realization of quantum networks. In our work, we theoretically investigate the scenario where a central source prepares an N-partite entangled state and transmits each entangled subsystem to one of$N$receivers through noisy quantum channels. The receivers are then able to perform local operations assisted by unlimited classical communication to distill target entangled states from the noisy channel output. In this operational context, we define the EPR distribution capacity and the GHZ distribution capacity of a quantum channel as the largest rates at which Einstein-Podolsky-Rosen (EPR) states and Greenberger-Horne-Zeilinger (GHZ) states can be faithfully distributed through the channel, respectively. We establish lower and upper bounds on the EPR distribution capacity by connecting it with the task of assisted entanglement distillation. We also construct an explicit protocol consisting of a combination of a quantum communication code and a classical-post-processing-assisted entanglement generation code, which yields a simple achievable lower bound for generic channels. As applications of these results, we give an exact expression for the EPR distribution capacity over two erasure channels and bounds on the EPR distribution capacity over two generalized amplitude damping channels. We also bound the GHZ distribution capacity, which results in an exact characterization of the GHZ distribution capacity when the most noisy channel is a dephasing channel.

The Pinsker inequality gives a simple method for lower bounding the relative entropy solely in terms of the total variation distance, which at times is easier to work with. We present a simple method for establishing Pinsker inequalities for$f$-divergences using a version of multivariate Taylor's theorem. By combining this with recent work on bounding$f$-divergences in terms of$\chi^{2}$-divergence, we establish that, under a large class of$f$-divergences, the rate at which finite-dimensional timehomogeneous Markov chains with a fixed, full rank stationary distribution converge to this distribution is upper bounded by the input-dependent contraction coefficient of the$\chi^{2}$-divergence. Given previous work, this bound can be tight, is the fastest it could converge using the theory of contraction coefficents, and is efficient to compute. We further extend these ideas to quantum timehomogeneous Markov chains measured under Petz$f$-divergences albeit without the same guarantees of computational efficiency or being the fastest contraction coefficient.

Distributed source simulation is the task where two (or more) parties share some correlated randomness and use local operations and no communication to convert this into some target correlation. Wyner’s seminal result showed that asymptotically the rate of uniform shared randomness needed for this task is given by a mutual information induced measure, now referred to as Wyner’s common information. This asymptotic result was extended by Hayashi in the quantum setting to separable states, the largest class of states for which this task can be performed to vanishing error. In this work we characterize this task in a near-tight manner in the one-shot setting using the smooth entropy framework. We do this by introducing one-shot operational quantities and correlation measures that characterize them. We establish asymptotic equipartition properties for our correlation measures thereby recovering the previous vanishing-error asymptotic results. In doing so, we consider technical points in one-shot network information theory and provide methods for cardinality bounds in the smooth entropy calculus. We also introduce entangled state versions of the distributed source simulation task and determine bounds in this setting via quantum embezzling. This provides a strong characterization of this network task in the one-shot, quantum regime.

In one-shot and zero-error information theory, the conditional min-entropy is a fundamental tool. It may be expressed as a conic program over the positive semidefinite cone. Recently, Chitambar et al. showed that the same conic program altered to be over the separable cone is a measure of transmitting classical communication over a quantum channel called the ‘communication value.’ In this work, we extend this idea to a broad class of convex cones to induce new families of entropic quantities. We show this methodology has operational relevance by characterizing a generalized notion of communication value and relating a class of cone-restricted entropies to a partial ordering on converting quantum channels via bistochastic preprocessing. We also show regularized smooth versions of these entropic quantities do not in general converge to the von Neumann entropy, which shows tasks characterized by these quantities are not equivalent even in an asymptotic i.i.d. fashion.

Recently, Cheng et al generalized quantum state splitting (QSS) to the multipartite setting for applications in quantum network information theory. Furthermore, Cheng and Gao recently noted that unipartite convex splitting reduces to soft covering for classical-quantum states. Soft covering is a common primitive in (classical, network) information theory that does not seem to need an extension to the multipartite setting, e.g. when establishing rates for many party distributed source simulation (DSS). This suggests a gap in our understanding of how these tasks are related. In this work we clarify this relation. We first identify “incoherent” QSS without entanglement assistance with DSS in the bipartite setting. Motivated by this, we identify “coherent” DSS as a special case of multipartite QSS. We then establish a one-shot rate region for multipartite QSS in terms of smooth multipartite mutual information quantities, which in turn implies a one-shot rate region for coherent DSS. Lastly, we establish a strong asymptotic equipartition property for mutual information quantities to establish a strong converse rate region for multipartite QSS and coherent DSS.

Divergences are a fundamental framework for measuring dissimilarity in information theory and statistics. Here, we use the multivariate Taylor's theorem to obtain new integral representations of twice-differentiable Bregman and$\boldsymbol{f}$-divergences in finite dimensions. This results in input-dependent bounds on many$f$-divergences in terms of the$\chi^{2}$-divergence as well as reverse Pinsker inequalities. As an application, we show how this provides upper bounds on the mixing time of irreducible, scrambling Markov chains under a large class of$f$divergences in terms of the input-dependent$\chi^{2}$contraction coefficient.

Distributed source simulation is the task where two (or more) parties share some correlated randomness and use local operations and no communication to convert this into some target correlation. Wyner’s seminal result showed that asymptotically the rate of uniform shared randomness needed for this task is given by a mutual information induced measure, now referred to as Wyner’s common information. In this work we characterize the quantum version of this task in the one-shot setting using the smooth entropy framework and one-shot operational quantities. We further establish asymptotic equipartition properties for our correlation measures. We also introduce entanglement versions of the distributed source simulation task and determine bounds in this setting via quantum embezzling.

There are various ways to quantify the communication capabilities of a quantum channel. In this work we introduce the communication value (cv) of quantum channel, which describes the optimal probability of guessing the channel input from its output. By connecting to prior work on zero-error channel simulation, we show that the cv and its entanglement-assisted variant also offer dual interpretations as the classical communication cost for perfectly simulating different aspects of a channel using non-signaling resources. Our study involves characterizing the communication value as a generalized conditional min-entropy over the cone of separable operators. Using this characterization, we evaluate the cv for all qubit channels and higher-dimensional channels with certain symmetries. We find that the any entanglement-breaking channel has multiplicative cv when used in parallel with any other channel; the same is shown to hold for Pauli channels and partially depolarizing channels. In contrast, the cv is found to be non-multiplicative for a subset of the well-known Werner-Holevo channels. A final component of this work investigates relaxations of the channel cv to other cones such as the set of operators having a positive partial transpose (PPT).

There are various ways to quantify the communication capabilities of a quantum channel. In this work we study the communication value (cv) of channel, which describes the optimal success probability of transmitting a randomly selected classical message over the channel. The cv also offers a dual interpretation as the classical communication cost for zero-error channel simulation using non-signaling resources. We first provide an entropic characterization of the cv as a generalized conditional min-entropy over the cone of separable operators. We evaluate the cv exactly for all qubit channels and the Werner-Holevo family of channels. The latter is shown to have non-multiplicative cv when d > 2. On the other hand, we prove that any pair of qubit channels have multiplicative cv when used in parallel. Even stronger, all entanglement-breaking channels and the partially depolarizing channel are shown to have multiplicative cv when used in parallel with any channel. We then turn to the entanglement-assisted cv and prove that it is equivalent to the conditional min-entropy of the Choi matrix of the channel. Combining with previous work on zero-error channel simulation, this implies that the entanglement-assisted cv is the classical communication cost for perfectly simulating a channel using quantum non-signaling resources. A final component of this work investigates relaxations of the channel cv to other cones such as the set of operators having a positive partial transpose (PPT).

Many information sources are not just sequences of distinguishable symbols but rather have invariances governed by alternative counting paradigms such as permutations, combinations, and partitions. We consider an entire classification of these invariances called the twelvefold way in enumerative combinatorics and develop a method to characterize lossless compression limits. Explicit computations for all twelve settings are carried out for i.i.d. uniform and Bernoulli distributions. Comparisons among settings provide quantitative insight.