We derive a novel chain rule for a family of channel conditional entropies, covering von Neumann and sandwiched Rényi entropies. In the process, we show that these channel conditional entropies are equal to their regularized version, and more generally, additive across tensor products of channels. For the purposes of cryptography, applying our chain rule to sequences of channels yields a new variant of Rényi entropy accumulation, in which we can impose some specific forms of marginal-state constraint on the input states to each individual channel. This generalizes a recently introduced security proof technique that was developed to analyze prepare-and-measure QKD with no limitations on the repetition rate. In particular, our generalization yields ``fully adaptive'' protocols that can in principle update the entropy estimation procedure during the protocol itself, similar to the quantum probability estimation framework.
We derive a novel chain rule for a family of channel conditional entropies, covering von Neumann and sandwiched Rényi entropies. In the process, we show that these channel conditional entropies are equal to their regularized version, and more generally, additive across tensor products of channels. For the purposes of cryptography, applying our chain rule to sequences of channels yields a new variant of Rényi entropy accumulation, in which we can impose some specific forms of marginal-state constraint on the input states to each individual channel. This generalizes a recently introduced security proof technique that was developed to analyze prepare-and-measure QKD with no limitations on the repetition rate. In particular, our generalization yields ``fully adaptive'' protocols that can in principle update the entropy estimation procedure during the protocol itself, similar to the quantum probability estimation framework.
The entropy accumulation theorem—and its subsequent generalized version—is a powerful tool in the security analysis of many device-dependent and device-independent cryptography protocols. However, it has the drawback that the finite-size bounds it yields are not necessarily optimal; furthermore, it relies on the construction of an affine min-tradeoff function, which can often be challenging to construct optimally in practice. In this work, we address both of these challenges simultaneously by deriving a new entropy accumulation bound. Our bound yields significantly better finite-size performance and can be computed as an intuitively interpretable convex optimization, without any specification of affine min-tradeoff functions. Furthermore, it can be applied directly at the level of Rényi entropies if desired, yielding fully-Rényi security proofs. Our proof techniques are based on elaborating on a connection between entropy accumulation and the frameworks of quantum probability estimation or f -weighted Rényi entropies; in the process, we obtain some new results with respect to those frameworks as well. In particular, those findings imply that our bounds apply to prepare-and-measure protocols without the virtual tomography procedures or repetition-rate restrictions previously required for entropy accumulation.
Security proofs in quantum cryptography rely on conditional entropies. In a many-round protocol, their estimation is a challenging task; one must account for the most general attacks by an eavesdropper, including those that are not independently and identically distributed (i.i.d.) across all rounds. Chain rules address this problem by relating the conditional entropy of a structured, but non-i.i.d. process to a sum of entropy contributions from each round. They are a key ingredient in entropy accumulation theorems (EATs), which provide a versatile security proof framework for many protocols in quantum cryptography. Recently, chain rules in the setting of trusted devices have lead to tight i.i.d. reductions at a finite number of rounds, and whether analogous results can be recovered in the device-independent (DI) setting has not been addressed. Surprisingly, we show that a natural tightening of the chain rule of Dupuis et al. [Commun. Math. Phys. 379, 867-913, (2020)] that would answer this question affirmatively cannot hold, highlighting a limitation of the current DI security proof approach. Nonetheless, we show that an intermediate improvement is possible by proving a new chain rule in this setting. Following the framework of Arqand et al. [Phys. Rev. X 15, 041013 (2025)], we use our chain rule to provide a slightly tighter version of the Rényi EAT in certain contexts. In addition, we provide a self-contained framework that unifies existing chain rules and compares their applications, framing our results in a broader context.
The primary entropic measures for quantum states are additive under the tensor product. In the analysis of quantum information processing tasks, the minimum entropy of a set of states, e.g., the minimum output entropy of a channel, often plays a crucial role. A fundamental question in quantum information and cryptography is whether the minimum output entropy remains additive under the tensor product of channels. Here, we establish a general additivity statement for the optimized sandwiched Rényi entropy of quantum channels. For that, we generalize the results of Devetak et al. (Commun Math Phys 266(1):37-63, 2006) to multi-index Schatten norms. As an application, we strengthen the additivity statement of Van Himbeeck and Brown (A tight and general finite-size security proof for quantum key distribution, 2025) thus allowing the analysis of time-adaptive quantum cryptographic protocols. In addition, we establish chain rules for Rényi conditional entropies that are similar to the ones used for the generalized entropy accumulation theorem of Metger et al. (Commun Math Phys 405(11):261, 2024).
Consider a sequential process in which each step outputs a system Ai and updates a side information register E. We prove that if this process satisfies a natural "non-signalling" condition between past outputs and future side information, the min-entropy of the outputs A1,…,An conditioned on the side information E at the end of the process can be bounded from below by a sum of von Neumann entropies associated with the individual steps. This is a generalisation of the entropy accumulation theorem (EAT), which deals with a more restrictive model of side information: there, past side information cannot be updated in subsequent rounds, and newly generated side information has to satisfy a Markov condition. Due to its more general model of side-information, our generalised EAT can be applied more easily and to a broader range of cryptographic protocols. As examples, we give the first multi-round security proof for blind randomness expansion and a simplified analysis of the E91 QKD protocol. The proof of our generalised EAT relies on a new variant of Uhlmann's theorem and new chain rules for the Renyi divergence and entropy, which might be of independent interest.
We present a continuity bound for quantum conditional entropies in terms of fidelity, with no dependence on the dimension of the conditioning system. Our results encompass both the von Neumann entropy and one version of sandwiched Rényi entropy, which is particularly relevant for applications in quantum key distribution (QKD). In such applications, this bound should provide improved results as compared to previous continuity bounds based on trace distance. We also show that the bounds can be sharpened as expected for classical-quantum states, and verify that they hold when the conditioning system has countably infinite dimension. Finally, we present several variants tailored for exponential functions of the Rényi entropies instead, which should be relevant for QKD applications.
Abstract Consider a sequential process in which each step outputs a system Ai A i and updates a side information register E . We prove that if this process satisfies a natural “non-signalling” condition between past outputs and future side information, the min-entropy of the outputs A1,…,An A 1 , ⋯ , A n conditioned on the side information E at the end of the process can be bounded from below by a sum of von Neumann entropies associated with the individual steps. This is a generalisation of the entropy accumulation theorem (EAT) (Dupuis et al. in Commun Math Phys 379: 867–913, 2020), which deals with a more restrictive model of side information: there, past side information cannot be updated in subsequent rounds, and newly generated side information has to satisfy a Markov condition. Due to its more general model of side-information, our generalised EAT can be applied more easily and to a broader range of cryptographic protocols. As examples, we give the first multi-round security proof for blind randomness expansion and a simplified analysis of the E91 QKD protocol. The proof of our generalised EAT relies on a new variant of Uhlmann’s theorem and new chain rules for the Rényi divergence and entropy, which might be of independent interest.