Entropic Inequalities for a Class of Quantum Secret Sharing States

TL;DR

This paper investigates entropic inequalities in quantum secret sharing states, demonstrating partial monotonicity of von Neumann entropy for certain authorized and unauthorized sets derived from monotone span programs.

Contribution

It introduces a class of quantum secret sharing states where entropy exhibits monotonicity properties, bridging a gap between quantum and classical entropy behaviors.

Findings

Entropy of unauthorized sets is partially monotonic.

Entropy of authorized sets is nonincreasing.

Provides a framework for entropic inequalities in quantum secret sharing.

Abstract

It is well-known that von Neumann entropy is nonmonotonic unlike Shannon entropy (which is monotonically nondecreasing). Consequently, it is difficult to relate the entropies of the subsystems of a given quantum state. In this paper, we show that if we consider quantum secret sharing states arising from a class of monotone span programs, then we can partially recover the monotonicity of entropy for the so-called unauthorized sets. Furthermore, we can show for these quantum states the entropy of the authorized sets is monotonically nonincreasing.