The operational meaning of min- and max-entropy

TL;DR

This paper establishes a direct operational interpretation of min- and max-entropy in quantum information, linking them to entanglement, guessing probabilities, and security in key distribution.

Contribution

It provides a novel operational meaning for min- and max-entropy, connecting them to entanglement measures and security tasks in quantum information theory.

Findings

Min-entropy relates to maximum overlap with maximally entangled states.

Max-entropy relates to fidelity with a completely mixed product state.

Guessing probability bounds secret key extractability.

Abstract

We show that the conditional min-entropy Hmin(A|B) of a bipartite state rho_AB is directly related to the maximum achievable overlap with a maximally entangled state if only local actions on the B-part of rho_AB are allowed. In the special case where A is classical, this overlap corresponds to the probability of guessing A given B. In a similar vein, we connect the conditional max-entropy Hmax(A|B) to the maximum fidelity of rho_AB with a product state that is completely mixed on A. In the case where A is classical, this corresponds to the security of A when used as a secret key in the presence of an adversary holding B. Because min- and max-entropies are known to characterize information-processing tasks such as randomness extraction and state merging, our results establish a direct connection between these tasks and basic operational problems. For example, they imply that the (logarithm of the) probability of guessing A given B is a lower bound on the number of uniform secret bits that can be extracted from A relative to an adversary holding B.