Computational Notions of Quantum Min-Entropy

TL;DR

This paper explores how classical notions of computational entropy extend to the quantum setting, establishing new theorems, constructing quantum leakage-resilient cryptography, and identifying limitations of classical techniques in quantum contexts.

Contribution

It extends classical leakage chain rules to quantum leakage, constructs a quantum leakage-resilient stream cipher, and analyzes the limitations of classical theorems in quantum information theory.

Findings

Quantum leakage chain rule for pseudoentropy proven

First quantum leakage-resilient stream cipher constructed

Classical Dense Model Theorem does not extend to quantum states

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 $k-\ell$ conditioned on an $\ell$-qubit 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-relative-min-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 Non-uniform 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.