Information theory provides a framework for answering fundamental questions about the optimal performance of many important quantum communication and computational tasks. In many cases, the optimal rates of these tasks can be expressed in terms of regularized formulas that consist of linear combinations of von Neumann entropies optimized over state extensions. However, evaluation of regularized formulas is often intractable, since it involves computing a formula’s value in the limit of infinitely many copies of a state. To find optimized linear entropic functions of quantum states whose regularized versions are tractable to compute, we search for examples which are additive. We use the method of cross2017uniform, which considers bipartite formulas, to identify convex polyhedral cones of tripartite correlation measures which satisfy a stronger a form of additivity called uniform additivity. We rely only on strong subadditivity of the von Neumann entropy and use these cones to prove that three previously established tripartite optimized correlation measures are additive.

Entanglement distillation is a fundamental information processing task whose implementation is key to quantum communication and modular quantum computing. Noise experienced by such communication and computing platforms occurs not only in the form of Pauli noise such as dephasing (sometimes called$T_{2}$) but also non-Pauli noise such as amplitude damping (sometimes called$T_{1}$). We initiate a study of practical and asymptotic distillation over what we call the joint damping-dephasing noise channel. In the practical setting, we propose a distillation scheme that completely isolates away the damping noise. In the asymptotic setting we derive lower bounds on the entanglement sharing capacities including the coherent and reverse coherent information. Like the protocol achieving the reverse coherent information, our scheme uses backward only communication. However for realistic damping noise$(T_{1}\neq 2T_{2})$our strategy can exceed the reverse coherent strategy which is the best known for pure damping. In addition, our companion paper [1] presents evidence showing that the channel displays non-additivity at the 2-letter level.

Understanding quantum channels and the strange behavior of their capacities is a key objective of quantum information theory. Here we study a remarkably simple, low-dimensional, single-parameter family of quantum channels with exotic quantum information-theoretic features. As the simplest example from this family, we focus on a qutrit-to-qutrit channel that is intuitively obtained by hybridizing together a simple degradable channel and a completely useless qubit channel. Such hybridizing makes this channel’s capacities behave in a variety of interesting ways. For instance, the private and classical capacity of this channel coincide and can be explicitly calculated, even though the channel does not belong to any class for which the underlying information quantities are known to be additive. Moreover, the quantum capacity of the channel can be computed explicitly, given a clear and compelling conjecture is true. This “spin alignment conjecture,” which may be of independent interest, is proved in certain special cases and additional numerical evidence for its validity is provided. Finally, we generalize the qutrit channel in two ways, and the resulting channels and their capacities display similarly rich behavior. In the companion paper [1], we further show that the qutrit channel demonstrates superadditivity when transmitting quantum information jointly with a variety of assisting channels, in a manner unknown before.

The capacity of a channel characterizes the maximum rate at which information can be transmitted through the channel asymptotically faithfully. For a channel with multiple senders and a single receiver, computing its sum capacity is possible in theory, but challenging in practice because of the nonconvex optimization involved. To address this challenge, we investigate three topics in our study. In the first part, we study the sum capacity of a family of multiple access channels (MACs) obtained from nonlocal games. For any MAC in this family, we obtain an upper bound on the sum rate that depends only on the properties of the game when allowing assistance from an arbitrary set of correlations between the senders. This approach can be used to prove separations between sum capacities when the senders are allowed to share different sets of correlations, such as classical, quantum or no-signalling correlations. We also construct a specific nonlocal game to show that the approach of bounding the sum capacity by relaxing the nonconvex optimization can give arbitrarily loose bounds. Owing to this result, in the second part, we study algorithms for non-convex optimization of a class of functions we call Lipschitz-like functions. This class includes entropic quantities, and hence these results may be of independent interest in information theory. Subsequently, in the third part, we show that one can use these techniques to compute the sum capacity of an arbitrary two-sender MACs to a fixed additive precision in quasi-polynomial time. We showcase our method by efficiently computing the sum capacity of a family of two-sender MACs for which one of the input alphabets has size two. Furthermore, we demonstrate with an example that our algorithm may compute the sum capacity to a higher precision than using the convex relaxation.

A key objective of quantum information theory is to understand quantum channels and their capacities. Here we study a remarkably simple, low-dimensional, single-parameter family of quantum channels with exotic quantum information-theoretic features. We focus on the simplest example from this family, a qutrit-to-qutrit channel intuitively obtained by hybridizing together a simple degradable channel with a completely useless qubit channel. Such hybridizing makes this channel’s capacities behave in a variety of interesting ways. For instance, the private and classical capacity of this channel coincide and can be explicitly calculated, even though the channel lies outside any previous class with calculable capacities. Moreover, the quantum capacity of the channel can be computed explicitly, given a clear and compelling conjecture is true. This "spin alignment conjecture", which may be of independent interest, is proved in certain special cases and backed numerically in certain other cases. Finally, we generalize the qutrit channel; the resulting channels and their capacities display similarly rich behavior. Our companion paper [22] demonstrates superadditivity when transmitting quantum information jointly across our qutrit channel used with a variety of assisting channels, in a manner unknown before.

Computing the sum capacity of a multiple access channel (MAC) is a non-convex optimization problem. It is therefore common to compute an upper bound on the sum capacity using a convex relaxation. We investigate the performance of such a relaxation by considering a family of MACs obtained from nonlocal games. First, we derive an analytical upper bound on the sum capacity of such MACs, while allowing the senders to share any given set of correlations. Our upper bound depends only on the properties of the game available in practice, thereby providing a way to obtain separations between the sum capacity assisted by different sets of correlations. In particular, we obtain a bound on the sum capacity of the MAC obtained from the magic square game that is tighter than the previously known result. Next, we introduce a game for which the convex relaxation of the sum capacity can be arbitrarily loose, demonstrating the need to find other techniques to compute or bound the sum capacity. We subsequently propose an algorithm that can certifiably compute the sum capacity of any two-sender MAC to a given precision.

We consider various forms of a process, which we call gluing, for combining two or more complementary quantum channel pairs (B,C) to form a composite. One type of gluing combines a perfect channel with a second channel to produce a generalized erasure channel pair (Bg,Cg). We consider two cases in which the second channel is (i) an amplitude-damping, or (ii) a phase-damping qubit channel; (ii) is the dephrasure channel of Leditzky et al. For both (i) and (ii), (Bg,Cg) depends on the damping parameter 0 ≤ p ≤ 1 and a parameter 0 ≤ λ ≤ 1 that characterizes the gluing process. In both cases we study Q(1)(Bg) and Q(1)(Cg), where Q(1)is the channel coherent information, and determine the regions in the (p, λ) plane where each is zero or positive, confirming previous results for (ii). A somewhat surprising result for which we lack any intuitive explanation is that Q(1)(Cg) is zero for λ ≤ 1/2 when p=0, but is strictly positive (though perhaps extremely small) for all values of λ > 0 when p is positive by even the smallest amount. In addition we study the nonadditivity of Q(1)(Bg) for two identical channels in parallel. It occurs in a well-defined region of the (p, λ) plane in case (i). In case (ii) we have extended previous results for the dephrasure channel without, however, identifying the full range of (p, λ) values where nonadditivity occurs. Again, an intuitive explanation is lacking.