Marco Tomamichel

We introduce an improved one-shot characterisation of randomness extraction against quantum side information (privacy amplification), strengthening known one-shot bounds and providing a unified derivation of the tightest known asymptotic constraints. Our main tool is a new class of smooth conditional entropies defined by lifting classical smooth divergences through measurements. A key role is played by the measured smooth Rényi relative entropy of order 2, which we show to admit an equivalent variational form: it can be understood as allowing for smoothing over not only states, but also non-positive Hermitian operators. Building on this, we establish a tightened leftover hash lemma, significantly improving over all known smooth min-entropy bounds on extractable randomness and recovering the sharpest classical achievability results. We extend these methods to decoupling, the coherent analogue of privacy amplification, obtaining a corresponding improved one-shot bound. Relaxing our smooth entropy bounds leads to one-shot achievability results in terms of measured Rényi divergences, which in the asymptotic i.i.d. limit recover the state-of-the-art error exponents of [Dupuis, IEEE T-IT 69, 7784 (2023)]. We show an approximate optimality of our results by giving a matching one-shot converse bound up to additive logarithmic terms. This yields an optimal second-order asymptotic expansion of privacy amplification under trace distance, establishing a significantly tighter one-shot achievability result than previously shown in [Shen et al., IEEE T-IT 70, 5077 (2024)] and proving its optimality for all hash functions.

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.

In many quantum information tasks, we encounter scenarios where information about two incompatible observables must be retrieved. A natural approach is to perform consecutive measurements, raising a key question: How does the information gained from the first measurement compare to that from both? The consecutive measurement theorem (CMT) provides a general relation between these quantities and has found applications in quantum cryptography. However, its previous formulations are often either too loose or too brittle to yield meaningful bounds. In this work, we first establish a tight CMT and apply it to achieve the best upper bounds on the quantum value of certain nonlocal games and their parallel repetitions to date. We then develop a robust CMT and explore a novel application of CMT to obtain a tighter no-go theorem for quantum oblivious transfer in some regime. These contributions strengthen the theoretical tools for analyzing quantum advantage and have concrete implications for nonlocal games and quantum cryptographic protocols.

In classical information theory, the maximal correlation and \chi^{2}-contraction coefficient establish limits on distributed and sequential processing. Two distinct quantum maximal correlation coefficients have been proposed, but they do not extend all the classical results. Building on work of Petz, we use the family of non-commutative L^{2}(p) spaces that extend the data processing inequality for variance to quantum theory to extend the classical results to quantum theory. We introduce families of quantum maximal correlation coefficients and identify quantum \chi^{2}-divergences as non-commutative generalizations of the variance of the likelihood ratio. We establish a family of maximal correlation coefficients that must all be ordered on a single copy level for an arbitrary number of copies of one state to be able to be converted to a single copy of another target state under local operations. We prove the equivalent characterizations of perfect classical correlation extraction via local operations in quantum theory. We clarify the relationship between maximal correlation and \chi^{2}-contraction coefficients by proving they are the same operator norms evaluated on distinct maps. Then we establish new equivalent conditions to the saturation of the data processing inequality for \chi^{2}-divergences. This implies previous saturation results for the \chi^{2} and sandwiched Rényi divergences. Finally, we establish the quantum maximal correlation coefficients and \chi^{2}-contraction coefficients are often efficiently computable. This results in a generic method for efficiently computing mixing times of time-homogeneous quantum Markov chains with a unique full rank fixed point.

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.

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.

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 introduce and analyse the problem of encoding classical information into different resources of a quantum state. More precisely, we consider a general class of communication scenarios characterised by encoding operations that commute with a unique resource destroying map and leave free states invariant. Our motivating example is given by encoding information into coherences of a quantum system with respect to a fixed basis (with unitaries diagonal in that basis as encodings and the decoherence channel as a resource destroying map), but the generality of the framework allows us to explore applications ranging from super-dense coding to thermodynamics. For any state, we find that the number of messages that can be encoded into it using such operations in a one-shot scenario is upper-bounded in terms of the information spectrum relative entropy between the given state and its version with erased resources. Furthermore, if the resource destroying map is a twirling channel over some unitary group, we find matching one-shot lower-bounds as well. In the asymptotic setting where we encode into many copies of the resource state, our bounds yield an operational interpretation of resource monotones such as the relative entropy of coherence and its corresponding relative entropy variance.

Quantum Key Distribution (QKD), invented by Bennett and Brassard and Ekert can be considered the first application of quantum information science. We focus on the assumption that an arbitrarily large number M of signals can be exchanged between the legitimate parties (Alice and Bob) and subsequently used for the computation of the final key. Here, we apply a novel proof technique to the BB84 QKD protocol and derive almost tight bounds on the minimum value M required to achieve a given level of security. The technique is based on a formulation of the uncertainty relation in terms of smooth entropies. We demonstrate significant improvements of the finite-key rate over existing results.

We derive simple and unified closed-form expressions for projections with respect to fidelity (equivalently, Bures and purified distances) onto several sets of interest. These include projecting bipartite PSD matrices onto the set of PSD matrices with a fixed marginal, and projecting ensembles of PSD matrices onto the set of PSD decompositions of a given matrix, with important special cases given by projections onto the set of quantum channels (via the Choi isomorphism) and onto the set of POVMs. We introduce 'prior-channel decompositions' of completely positive (CP) maps, which uniquely decompose any CP map into a prior PSD matrix and a quantum channel. This decomposition generalizes the Choi--Jamiołkowski isomorphism by establishing a bijective correspondence between arbitrary bipartite PSD matrices and channel--state pairs, and we show that it arises naturally from the fidelity projections developed here. As applications, we show that the 'pretty good measurement' is the fidelity projection of an ensemble onto the set of POVMs, and that the Petz recovery map is the projection of a CP map—constructed from a prior state and a forward channel—onto the set of reverse quantum channels, which recasts the well-known identification of the Petz map with quantum Bayes’ rule in information-geometric terms. Our results also provide an information-geometric underpinning of the Leifer--Spekkens quantum state over time [Leifer and Spekkens, Phys. Rev. A 88, 052130 (2013)].

We offer the first operational interpretation of the α-z relative entropies, a measure of distinguishability between two quantum states introduced by Jakšić et al. and Audenaert and Datta. We show that these relative entropies appear when formulating conditions for large-sample or catalytic relative majorization of pairs of flat states and certain generalizations of them. Indeed, we show that such transformations exist if and only if all the α-z relative entropies of the two pairs are ordered. In this setting, the α and z parameters are truly independent from each other. These results also yield an expression for the optimal rate of converting one flat state pair into another. Our methods use real-algebraic techniques involving preordered semirings and certain monotone homomorphisms and derivations on them.

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.

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.

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.

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.

We give a strengthening as well as a generalization of an inequality for the quantum conditional mutual information of a tripartite quantum state recently proved by Fawzi and Renner, connecting it with the ability to reconstruct the state from its bipartite reductions. We provide three alternative and simplified proofs ranging from quantum state redistribution via duality of semidefinite programming to elementary properties of pinching maps and the operator logarithm.

In the study of Markovian processes, one of the principal achievements is the equivalence between the Phi-Sobolev inequalities and an exponential decrease of the Phi-entropies. In this work, we develop a framework of Markov semigroups on matrix-valued functions and generalize the above equivalence to the exponential decay of matrix Phi-entropies. This result also specializes to spectral gap inequalities and modified logarithmic Sobolev inequalities in the random matrix setting. To establish the main result, we define a non-commutative generalization of the carre du champ operator, and prove a de Bruijn's identity for matrix-valued functions. The proposed Markov semigroups acting on matrix-valued functions have immediate applications in the characterization of the dynamical evolution of quantum ensembles. We consider two special cases of quantum unital channels, namely, the depolarizing channel and the phase-damping channel. In the former, since there exists a unique equilibrium state, we show that the matrix Phi-entropy of the resulting quantum ensemble decays exponentially as time goes on. Consequently, we obtain a stronger notion of monotonicity of the Holevo quantity - the Holevo quantity of the quantum ensemble decays exponentially in time and the convergence rate is determined by the modified log-Sobolev inequalities. However, in the latter, the matrix Phi-entropy of the quantum ensemble that undergoes the phase-damping Markovian evolution generally will not decay exponentially. This is because there are multiple equilibrium states for such a channel. Finally, we also consider examples of statistical mixing of Markov semigroups on matrix-valued functions. We can explicitly calculate the convergence rate of a Markovian jump process defined on Boolean hypercubes, and provide upper bounds of the mixing time on these types of examples.

Quantum key distribution (QKD) is a prime example of the interdisciplinary nature of quantum cryptography and the first application of quantum science that matured into the realm of engineering and commercial development. While the security of the generated key is intuitively guaranteed by the laws of quantum mechanics, a precise analysis of the security properties requires tools from both classical cryptography and information theory. This is particularly relevant when investigating QKD in a practical setting where the available resources are finite. Here, we employ recent results in classical information theory to show that information reconciliation imposes fundamental limitations on the amount of secret key that can be extracted in the finite key regime. In particular, we find that an often used approximation for the information leakage during information reconciliation is flawed and propose an improvement.