  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.
