Quantum umlaut information

TL;DR

The paper introduces quantum umlaut information $U(A;B)_\rho$, a reversed-entropy correlation measure with a closed-form expression and an operational interpretation as the asymptotic hypothesis-testing exponent. It extends the concept to quantum channels, establishing additivity for classical-quantum channels but not in general, and proves that the regularised channel umlaut information $U^{\infty}(\mathcal N)$ exactly characterises the zero-rate activated non-signalling error exponent, with single-letter bounds via the geometric variant. The work further connects state and channel formulations through Choi matrices and presents Gaussian-state and BS-variant extensions, along with lower bounds via list decoding and an analysis of covariant channels. A key takeaway is the dichotomy between zero-rate exponents and zero-error communication, plus the need for regularisation in general quantum channels, suggesting avenues for quantum sphere-packing-type bounds and further operational interpretations.

Abstract

We study the quantum umlaut information, a correlation measure defined for bipartite quantum states $ρ_{AB}$ as a reversed variant of the quantum mutual information: $U(A;B)_ρ= \min_{σ_B} D(ρ_A\otimes σ_B\|ρ_{AB})$ in terms of the quantum relative entropy $D$. As in the classical case [Girardi et al., arXiv:2503.18910], this definition allows for a closed-form expression and has an operational interpretation as the asymptotic error exponent in the hypothesis testing task of deciding whether a given bipartite state is product or not. We generalise the umlaut information to quantum channels, where it also extends the notion of `oveloh information' [Nuradha et al., arXiv:2404.16101]. We prove that channel umlaut information is additive for classical-quantum channels, while we observe additivity violations for fully quantum channels. Inspired by recent results in entanglement theory, we then show as our main result that the regularised umlaut information constitutes a fundamental measure of the quality of classical information transmission over a quantum channel -- as opposed to the capacity, which quantifies the quantity of information that can be sent. This interpretation applies to coding assisted by activated non-signalling correlations, and the channel umlaut information is in general larger than the corresponding expression for unassisted communication as obtained by Dalai for the classical-quantum case [IEEE Trans. Inf. Theory 59, 8027 (2013)]. Combined with prior works on non-signalling--assisted zero-error channel capacities, our findings imply a dichotomy between the settings of zero-rate error exponents and zero-error communication. While our results are single-letter only for classical-quantum channels, we also give a single-letter bound for fully quantum channels in terms of the `geometric' version of umlaut information.

Figures (5)

• Figure 1: Composite hypothesis testing
• Figure 2: The umlaut information of the channel $\pazocal{N}$ is the largest umlaut information of $\rho^{(\pazocal{N})}_{A'B}$ by varying $\rho_{A'}$.
• Figure 3: Estimation of the umlaut information of $\pazocal{A}_{\gamma,\beta}$ with $(\gamma,\beta)=(0.5,0.001)$ using the family of states $\rho_{A'}(p)$. The umlaut information of the channel is given by the maximum of the function in the plot.
• Figure 4: The effective channel $\pazocal{M}$ given by composition of an $\Omega$-assisted code $\Pi$ with a channel $\pazocal{N}$.
• Figure 5: A pictorial comparison of the information measures and of the error exponents that were discussed in this paper. $\pazocal{N}_{A\to B}$ is a quantum channel; for any $k\geq 1$, $q\in\mathbb{R}^k$ is a vector such that $\sum_{i=1}^kq_i=1$ and $u_2 =(1/2, 1/2)$. The inequalities and identities involving objects represented in blue have to be considered only if $\pazocal{N}$ is a classical-quantum channel.

Theorems & Definitions (74)

• Definition 1: (Quantum umlaut information)
• Lemma 2
• proof
• Definition 3: (Petz--Rényi $\alpha$-umlaut information)
• Lemma 4
• proof
• Lemma 5: (Gibbs variational principle)
• Definition 6: (Umlaut-marginal)
• Proposition 7: (A closed-form expression for the quantum umlaut information)
• proof