Álvaro Yángüez Bachiller

Quantum information processing is limited, in practice, to efficiently implementable operations. This motivates the study of quantum divergences and entropies that preserve their operational meaning while faithfully capturing computational constraints. In this joint submission, we introduce a mathematical framework for computational quantum information theory from which computational divergences and established computational entropies naturally arise. Within this framework, we define a computational max-divergence and computational measured Rényi divergences. We relate these quantities to efficient hypothesis testing and use them to analyze computational resource theories such as entanglement. We further connect the new computational max-divergence to the previously established computational min-entropy and computational hypothesis testing relative entropy. In particular, we show that the computational max-divergence serves as a parent quantity for the computational min-entropy, thereby endowing the computational max-divergence with an operational meaning. Moreover, we prove that the computational hypothesis testing relative entropy is approximately characterized by a smoothed version of the computational max-divergence. Together, these results extend well-known information-theoretic relationships to the computational setting and unify several operationally-motivated computational quantities.

While one-way functions (OWFs) serve as the minimal assumption for computational cryptography in the classical setting, in quantum cryptography, we have even weaker cryptographic assumptions such as pseudo-random states, and EFI pairs, among others. Moreover, the minimal assumption for computational quantum cryptography remains an open question. Recently, it has been shown that pseudoentanglement is necessary for the existence of quantum cryptography (Goulão and Elkouss 2024), but no cryptographic construction has been built from it. In this work, we study the cryptographic usefulness of quantum pseudoresources —a pair of families of quantum states that exhibit a gap in their resource content yet remain computationally indistinguishable. We show that quantum pseudoresources imply a variant of EFI pairs, which we call EPFI pairs, and that these are equivalent to quantum commitments and thus EFI pairs. Our results suggest that, just as randomness is fundamental to classical cryptography, quantum resources may play a similarly crucial role in the quantum setting. Finally, we focus on the specific case of entanglement, analyzing different definitions of pseudoentanglement and their implications for constructing EPFI pairs. Moreover, we propose a new cryptographic functionality that is intrinsically dependent on entanglement as a resource.