Umlaut information

We study the umlaut information, a correlation measure defined for pairs of random variables as a reversed variant of the mutual information. We show that it has an operational interpretation as the asymptotic error exponent in the hypothesis testing task of deciding whether a given joint probability distribution is product or not. We discuss the extension of the umlaut information to channels, which we prove to be additive, and we show that it has a twofold operational interpretation: as the zero-rate error exponent of non-signalling--assisted coding on the one hand, and as the zero-rate error exponent of list decoding in the large list limit on the other. The umlaut information can be extended to bipartite quantum states and quantum channels. We prove that channel umlaut information is additive for classical-quantum channels, while we observe additivity violations for fully quantum channels. We show that, in the fully quantum setting, the regularisation of the channel umlaut information matches the zero-rate error exponent of non-signalling--assisted coding. While our results are single-letter only for classical-quantum channels, we also give a single-letter upper bound for fully quantum channels. Based on: https://arxiv.org/abs/2503.18910, https://arxiv.org/abs/2503.21479