Marco Tomamichel

Divergences lie at the core of information-theoretic applications. A recently introduced family of f-divergences, defined via an integral representation, has exhibited remarkable properties --- for instance, for the study of contraction coefficients. However, many familiar properties of their classical analogous have remained elusive. In this work, we develop alternative representations of the quantum f-divergences by leveraging the recently established quantum layer-cake theorem. These new formulations enable us to establish several key properties, including monotonicity and connections to other divergences. As our main application, we show how these representations unify and streamline various proofs in quantum hypothesis testing, yielding tighter achievability bounds through conceptually simple arguments that apply across different error regimes.

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.

Weak coin flipping is a cryptographic primitive in which two mutually distrustful parties generate a shared random bit to agree on a winner via remote communication. While a stand-alone secure weak coin flipping protocol can be constructed from noiseless communication channels, its composability has not been explored. In this work, we demonstrate that no weak coin flipping protocol can be abstracted into a black box resource with composable security. Despite this, we also establish the overall stand-alone security of weak coin flipping protocols under sequential composition.

The task of testing the validity of a hypothesis underlies numerous applications in quantum information theory. The most commonly investigated approach is that of gathering all the available (quantum) data and making a final decision based on a collective measurement. However, such offline strategies are often far from practical, both in the amount of data required as well as in the complexity of the required measurement. In some settings, when the goal is quick detection, offline algorithms are not applicable at all, as they can only make a decision once all samples are received. Sequential methods offer the use of online strategies, where samples are requested on a need-to-know basis, drastically reducing the number of required samples in order to guarantee the, task specific, associated performance criteria. While extensively investigated and applied in the classical setting, we know far less about the optimal performance of such online strategies when quantum data is available. In this joint submission we present major recent progress on sequential methods for the fundamental tasks of quantum state discrimination, channel discrimination and quickest change point detection. In summary, we provide a comprehensive picture of the optimal asymptotic performance of online strategies in these settings under different performance criteria.

We prove that the relative entropy of entanglement is additive when at least one of the two states belongs to some specific class. We show that these classes include bipartite pure, maximally correlated, GHZ, Bell diagonal, isotropic, and generalized Dicke states. Previously, additivity was established only if both states belong to the same class. Moreover, we extend these results to entanglement monotones based on the α-z Rényi relative entropy. Notably, this family of monotones includes also the generalized robustness of entanglement and the geometric measure of entanglement. In addition, we prove that any monotone based on a quantum relative entropy is not additive for general states. We also compute closed-form expressions of the monotones for bipartite pure, Bell diagonal, isotropic, generalized Werner, generalized Dicke, and maximally correlated Bell diagonal states. Our results rely on proving new necessary and sufficient conditions for the optimizer of the monotones based on the α-z R'enyi relative entropy, which allow us to reduce the original optimization problem to a simpler linear one. Even though we focus mostly on entanglement theory, we formulate some of our technical results in a general resource theory framework, and we expect that they could be used to investigate other resource theories.

In many cryptographic tasks, we encounter situations where we would like to retrieve some information about two incompatible observables. A natural strategy to tackle this problem involves consecutive measurements of two observables, raising the critical question: How does the information gained from the first measurement relate to that obtained through both consecutive measurements? A loose relation between these two quantities has been established by the consecutive measurement theorem and is found useful in quantum proofs of knowledge and nonlocal games. In this work, we establish a tight consecutive measurement theorem, and apply our theorem to improve the best-known bounds on the quantum value of CHSH_q(p) games and their parallel repetition. Moreover, we explore a novel application of the consecutive measurement theorem to find tighter trade-off relations for quantum oblivious transfer in most regimes. This advancement enhances the analytical toolkit to study quantum advantage and has direct implications for quantum cryptographic protocols.

Noisy channel is a valuable resource for cryptography. It can be used to build cryptographic primitives like bit commitment and oblivious transfer that are information-theoretically secure between two untrusting parties. Existing studies on this topic focus on the channel that does not change over successive uses. In this work, we study non-independent and identically distributed (non-i.i.d.) channels with constraint on min-entropy. The dishonest player is able to configure the channel at his will under the constraint. We devise a protocol that is complete, hiding, and binding, and give its commitment rate.

We analyse two party non-local games whose predicate requires Alice and Bob to generate matching bits, and their three party extensions where a third player receives all inputs and is required to output a bit that matches that of the original players. We propose a general device independent quantum key distribution protocol for the subset of such non-local games that satisfy a monogamy-of-entanglement property characterised by a gap in the maximum winning probability between the bipartite and tripartite versions of the game. This gap is due to the optimal strategy for two players requiring entanglement, which due to its monogamy property cannot be shared with any additional players. Based solely on the monogamy-of-entanglement property, we provide a simple proof of information theoretic security of our protocol. Lastly, we numerically optimize the finite and asymptotic secret key rates of our protocol using the magic square game as an example, for which we provide a numerical bound on the maximal tripartite quantum winning probability which closely matches the bipartite classical winning probability. Further, we show that our protocol is robust for depolarizing noise up to about $2.2%$, providing the first such bound for general attacks for magic square based quantum key distribution.

The no-cloning theorem is a cornerstone of quantum cryptography. Here we generalize and rederive under weaker assumptions various upper bounds on the maximum achievable fidelity of probabilistic and deterministic cloning machines. Building on ideas by Gisin [Phys.~Lett.~A, 1998], our results hold even for cloning machines that do not obey the laws of quantum mechanics, as long as remote state preparation is possible and the non-signalling principle holds. We apply our general theorem to several subsets of states that are of interest in quantum cryptography.