Information Theory 3

Abstract: We study the construction of fully universal private channel coding protocols for classical-quantum channels. While earlier schemes achieved universal decoding, they relied on random encoders, preventing complete universality. We close this gap by showing that spectral expansion of a graph associated with a codebook guarantees universal channel resolvability: if the graph has a large spectral gap, the output state induced by the codewords is asymptotically indistinguishable from the target state, independent of the channel. This yields the first deterministic, channel-independent resolvability coding in the quantum regime. Combining this with universal channel coding, we construct a fully universal private coding protocol that achieves standard private information rates, highlighting the role of expander graphs in secure quantum communication.

Abstract: In their seminal work, Bennett et al. [IEEE Trans. Inf. Theory (2002)] showed that, with sufficient shared randomness, one noisy channel can simulate another at a rate equal to the ratio of their capacities. We establish that when coding above this channel interconversion capacity, the exact strong converse exponent is characterized by a simple optimization involving the difference of the corresponding Renyi channel capacities with Holder dual parameters. We extend this result to the entanglement-assisted interconversion of classical-quantum channels, showing that the strong converse exponent is likewise determined by differences of sandwiched Renyi channel capacities. The converse bound is obtained by relaxing to non-signaling assisted codes and applying Holder duality together with the data processing inequality for Renyi divergences. Achievability is proven by concatenating refined channel coding and simulation protocols that go beyond first-order capacities, achieving exponentially small conversion errors.

Abstract: 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.