Felix Leditzky

In this work we improve the quantum communication rates of various quantum channels of interest using permutation-invariant quantum codes. We focus in particular on parametrized families of quantum channels and aim to improve bounds on their quantum capacity threshold, defined as the lowest noise level at which the quantum capacity of the channel family vanishes. These thresholds are important quantities as they mark the noise level up to which faithful quantum communication is theoretically possible. Our method exploits the fact that independent and identically distributed quantum channels preserve any permutation symmetry present at the input. The resulting symmetric output states can be described succinctly using the representation theory of the symmetric and general linear groups, which we use to derive an efficient algorithm for computing the channel coherent information of a permutation-invariant code. Our approach allows us to evaluate coherent information values for a large number of channel copies, e.g., at least 100 channel copies for qubit channels. We apply this method to various physically relevant channel models, including general Pauli channels, the dephrasure channel, the generalized amplitude damping channel, and the damping-dephasing channel. For each channel family we obtain improved lower bounds on their quantum capacities. For example, for the 2-Pauli and BB84 channel families we significantly improve the best known quantum capacity thresholds derived in [Fern, Whaley 2008]. These threshold improvements are achieved using a repetition code-like input state with non-orthogonal code states, which we further analyze in our representation-theoretic framework.

Quantum teleportation is a quantum communication primitive that allows a long-distance quantum channel to be built using pre-shared entanglement and one-way classical communication. However, the quality of the established channel crucially depends on the quality of the pre-shared entanglement. In this work, we revisit the problem of using noisy entanglement for the task of teleportation. We first show how this problem can be rephrased as a state discrimination problem. In this picture, a quantitative duality between teleportation and dense coding emerges in which every Alice-to-Bob teleportation protocol can be repurposed as a Bob-to-Alice dense coding protocol, and the quality of each protocol can be measured by the success probability in the same state discrimination problem. One of our main results provides a complete characterization of the states that offer no advantage in one-way teleportation protocols over classical states, thereby offering a new and intriguing perspective on the long-standing open problem of identifying such states. This also yields a new proof of the known fact that bound entangled states cannot exceed the classical teleportation threshold. Moreover, our established duality between teleportation and dense coding can be used to show that the exact same states are unable to provide a non-classical advantage for dense coding as well. We also discuss the duality from a communication capacity point of view, deriving upper and lower bounds on the accessible information of a dense coding protocol in terms of the fidelity of its associated teleportation protocol. A corollary of this discussion is a simple proof of the previously established fact that bound entangled states do not provide any advantage in dense coding.

Communication networks have multiple users, each sending and receiving messages. A multiple access channel (MAC) models multiple senders transmitting to a single receiver, such as the uplink from many mobile phones to a single base station. The optimal performance of a MAC is quantified by a capacity region of simultaneously achievable communication rates. We study the two-sender classical MAC, the simplest and best-understood network, and find a surprising richness in both a classical and quantum context. First, we find that quantum entanglement shared between senders can substantially boost the capacity of a classical MAC. Second, we find that optimal performance of a MAC with bounded-size inputs may require unbounded amounts of entanglement. Third, determining whether a perfect communication rate is achievable using finite-dimensional entanglement is undecidable. Finally, we show that evaluating the capacity region of a two-sender classical MAC is in fact NP-hard.