Mario Berta

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.

We study the quantum umlaut information, a correlation measure defined for bipartite quantum states as a reversed variant of the quantum 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 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)]. In the classical unassisted setting, the channel umlaut information has a further operational interpretation as the zero-rate error exponent of list decoding in the large list limit. 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.

Many fundamental quantities in quantum information processing are instances of quantum divergences - functionals on quantum states that satisfy natural axioms grounded in information-theoretic principles. Recently, a new class of divergences - the f-divergences - has gained prominence in quantum information theory and received operational interpretations, while being long established in the classical setting. Furthermore, Frenkel showed that the Umegaki relative entropy is a special case of a quantum f-divergence for the function f(x) = x log x; building on this, Hirche et al. introduced a parameterized family of f-divergences that, in appropriate regimes, recovers the sandwiched and Petz relative entropies as regularizations. Taken together, these results reveal a tight link between the best-understood quantum divergences - the Umegaki, Petz, and sandwiched relative entropies - on a technical level and the general class of f-divergences, thereby strongly motivating a program that connects f-divergences to concrete quantum information tasks as already started by Cheng et al. In this contribution, we develop a variational formulation that approximates general quantum f-divergences to arbitrary precision. These approximations yield (i) efficient evaluation of the quantum relative entropy of channels and already used as the core numerical method in quantum many body physics and (ii) computation of asymptotic key rates in DIQKD in particular in the scenario of two switches in routed Bell scenarios.

We propose a class of randomized quantum algorithms for the task of sampling from matrix functions, without the use of quantum block encodings or any other coherent oracle access to the matrix elements. As such, our use of qubits is purely algorithmic, and no additional qubits are required for quantum data structures. For N times N Hermitian matrices, the space cost is łog(N)+1 qubits and depending on the structure of the matrices, the gate complexity can be comparable to state-of-the-art methods that use quantum data structures of up to size O(N^2), when considering equivalent end-to-end problems. Within our framework, we present a quantum linear system solver that allows one to sample properties of the solution vector, as well as algorithms for sampling properties of ground states and Gibbs states of Hamiltonians. As a concrete application, we combine these sub-routines to present a scheme for calculating Green's functions of quantum many-body systems.