Universal chain rules from entropic triangle inequalities

TL;DR

This paper establishes a universal chain rule for the smooth min-entropy using entropic triangle inequalities, enabling better bounds on multipartite quantum systems and relaxing conditions for entropy accumulation.

Contribution

It introduces a universal chain rule for the smooth min-entropy applicable to all system sizes and proves an approximate entropy accumulation theorem with relaxed assumptions.

Findings

Proves a universal chain rule for smooth min-entropy.

Derives a similar relation for smooth max-entropy.

Provides an approximate entropy accumulation theorem with relaxed conditions.

Abstract

The von Neumann entropy of an $n$-partite system $A_1^n$ given a system $B$ can be written as the sum of the von Neumann entropies of the individual subsystems $A_k$ given $A_1^{k-1}$ and $B$. While it is known that such a chain rule does not hold for the smooth min-entropy, we prove a counterpart of this for a variant of the smooth min-entropy, which is equal to the conventional smooth min-entropy up to a constant. This enables us to lower bound the smooth min-entropy of an $n$-partite system in terms of, roughly speaking, equally strong entropies of the individual subsystems. We call this a universal chain rule for the smooth min-entropy, since it is applicable for all values of $n$. Using duality, we also derive a similar relation for the smooth max-entropy. Our proof utilises the entropic triangle inequalities for analysing approximation chains. Additionally, we also prove an approximate version of the entropy accumulation theorem, which significantly relaxes the conditions required on the state to bound its smooth min-entropy. In particular, it does not require the state to be produced through a sequential process like previous entropy accumulation type bounds. In our upcoming companion paper, we use it to prove the security of parallel device independent quantum key distribution.