Peter Brown

Variational techniques have been recently developed to find tighter bounds on the von Neumann entropy in a completely device-independent (DI) setting. This, in turn, has led to significantly improved key rates of DI protocols, in both the asymptotic limit as well as in the finite-size regime. In this work, we derive novel variational expressions for Petz-Rényi divergences instead. We also derive two critical applications of this result. First, we show how these variational expressions can be used to further improve the finite-size key rate of DI protocols, by developing a fully-Rényi entropy accumulation theorem that can utilize these expressions for key rate computations. Second, we derive a security condition for DI advantage distillation that is based on the pretty good fidelity. We implement these techniques to derive increased noise tolerances for DIQKD protocols, which surpass the previously known bounds.

We consider the security of Quantum Key Distribution (QKD) protocols consisting of a finite number of rounds. We provide a security proof that is both and provides tight finite-size correction terms. In particular, when expanded in the block length $n$, the rate of randomness generation has the optimal asymptotic rate and optimal leading-order finite-size correction term. The proof is also general, applying to generic randomness generation and QKD protocols that have fully characterized devices and consist of a finite number of rounds.

Abstract In the analysis of quantum information processing tasks, the choice of distance measure between states or channels often plays a crucial role. This submission introduces new quantum Rnyi divergences for states and channels that are based on a convex optimization program involving the matrix geometric mean. These divergences have mathematical and computational properties that make them applicable to a wide variety of problems. We use these Rnyi divergences to obtain semidefinite programming lower bounds on the key rates for device-independent cryptography, and in particular we find a new bound on the minimal detection efficiency required to perform device-independent quantum key distribution without additional noisy preprocessing. Furthermore, we give several applications to quantum Shannon theory, in particular proving that adaptive strategies do not help in the strong converse regime for quantum channel discrimination and obtaining improved bounds for quantum capacities.

Device-independent (DI) cryptography represents the highest level of security, enabling cryptographic primitives to be executed safely on untrusted hardware. Moreover, with successful proof-of-concept demonstrations in randomness expansion, randomness amplification, and quantum key distribution, the field is steadily advancing toward commercial viability. Critical to this continued progression is the development of tighter finite-size security proofs. In this work, we provide a simple method to obtain tighter finite size security proofs for protocols based on the CHSH game which is the nonlocality test used in all of the proof-of-concept experiments. We achieve this by analytically solving key-rate optimization problems based on Rényi entropies, providing a simple method to obtain tighter finite-size key rates.

The amount of cryptographically secure randomness one can extract from a source can be linked to an optimization over conditional sandwiched Rényi entropies. Though these can be difficult to compute in general, we consider a simplified setting in which closed-form expressions can be obtained. More precisely, we consider a quantity we call the maximal intrinsic Rényi randomness and combine it with recent results on privacy amplification to derive simple expressions for the maximal quantity of $\epsilon$-secure randomness that can be extracted from finite uses of some memoryless source (which may be entangled with some eavesdropper). Overall this gives a simple method to compute this operationally relevant quantity and provides a benchmarking tool for the rates of finite size security proofs for randomness generation and quantum key distribution protocols. Along the way, we also prove closed-form expressions for the intrinsic randomness measured with respect to other Rényi entropy families.

We introduce an explicit construction for a key distribution protocol in the Quantum Computational Timelock (QCT) security model, where one assumes that computationally secure encryption may only be broken after a time much longer than the coherence time of available quantum memories. Taking advantage of the QCT assumptions, we build a key distribution protocol called HM-QCT from the Hidden Matching problem for which there exists an exponential gap in one-way communication complexity between classical and quantum strategies.

Variational techniques have been recently developed to find incredibly tight bounds on the von Neumann entropy in a completely device-independent (DI) setting. This, in turn, has led to significantly improved key rates of DI protocols, in both the asymptotic limit as well as in the finite-size regime. In this paper, we discuss two approaches towards applying these variational methods for Petz-Rényi divergences instead. We then show how this can be used to further improve the finite-size key rate of DI protocols, utilizing a fully-Rényi entropy accumulation theorem developed in a partner work. Petz-Rényi divergences can also be applied to study DI advantage distillation, in which two-way communication is used to improve the noise tolerance of quantum key distribution (QKD) protocols. We implement these techniques to derive increased noise tolerances for DIQKD protocols, which surpass all previous known bounds.

Nonlocal tests on multipartite quantum correlations can certify randomness in a device-independent (DI) way. Such correlations admit a rich structure, making the task of choosing an appropriate witness, known as a Bell inequality, difficult. For example, extremal Bell inequalities are tight witnesses of nonlocality, however achieving their maximum violation places constraints on the underlying quantum system, which are often incompatible with optimal randomness generation. As a result we find a trade-off between maximum randomness and Bell violation. Understanding this trade-off for more than two parties has not been explored, and would inform the best way to generate DI randomness in this setting. Moreover, suitable techniques that enable maximum randomness certification for arbitrarily many parties are missing. Here, we study the maximum amount of randomness that can be certified by correlations exhibiting a violation of the Mermin-Ardehali-Belinskii-Klyshko (MABK) inequality. We find that maximum quantum violation and maximum randomness are incompatible for any even number of parties, with incompatibility diminishing as the number of parties grow, and conjecture the precise trade-off. We also show that maximum MABK violation is not necessary for maximum randomness for odd numbers of parties. To obtain our results, we derive new families of Bell inequalities certifying maximum randomness from a new technique for randomness certification, which we call "expanding Bell inequalities". Our technique allows one to take a bipartite Bell expression, known as the seed, and transform it into a multipartite Bell inequality tailored for randomness certification, showing how intuition learned in the bipartite case can find use in more complex scenarios.