Computational Notions of Quantum Min-Entropy

Abstract

We initiate the study of computational entropy in the quantum setting. We investigate to what extent the classical notions of computational entropy generalize to the quantum setting, and whether quantum analogues of classical theorems hold. Our main results are as follows.(1) The classical Leakage Chain Rule for pseudoentropy can be extended to the case that the leakage information is quantum (while the source remains classical). Specifically, if the source has pseudoentropy at least k, then it has pseudoentropy at least k − ℓ conditioned on an ℓqubit leakage. (2) As an application of the Leakage Chain Rule, we construct the first quantum leakage-resilient stream-cipher in the bounded-quantum-storage model, assuming the existence of a quantum-secure pseudorandom generator. (3) We show that the general form of the classical Dense Model Theorem (interpreted as the equivalence between two definitions of pseudo-relativemin-entropy) does not extend to quantum states. Along the way, we develop quantum analogues of some classical techniques (e.g., the Leakage Simulation Lemma, which is proven by a Nonuniform Min-Max Theorem or Boosting). On the other hand, we also identify some classical techniques (e.g., Gap Amplification) that do not work in the quantum setting. Moreover, we introduce a variety of notions that combine quantum information and quantum complexity, and this raises several directions for future work.

Security in Quantum Cryptography

“…Other works such as Chen et al (2017) bound adversaries by circuit sizes. It is not clear how to model that in a finite, composable framework, and is important open work.…”

Interactive Leakage Chain Rule for Quantum Min-entropy

“…The separable case is proved by Desrosiers and Dupuis [10], and the general case is proved by Winkler et al [11], both of which are motivated by cryptographic applications. Furthermore, a computational version of quantum leakage chain rule is explored in [12] with applications in quantum leakage-resilient cryptography.…”

Pseudorandom States, Non-Cloning Theorems and Quantum Money

“…This is a reasonable definition that has been seriously considered in the literature (e.g. [11,12]). Although this definition may be applicable in certain situations, it does not seem to grasp the quantum nature of the problem as purely classical distributions 1 may also satisfy the definition.…”