Comment on "Strong Quantum Darwinism and Strong Independence are Equivalent to Spectrum Broadcast Structure"

TL;DR

This paper critiques a recent claim that Spectrum Broadcast Structure is equivalent to Strong Quantum Darwinism and Strong Independence, showing the original condition is necessary but not sufficient and proposing a strengthened criterion.

Contribution

The authors identify a flaw in the previous formulation of Strong Independence and propose a revised condition that guarantees the claimed equivalence.

Findings

Original condition is necessary but not sufficient.

Counter-example demonstrates the flaw.

Revised condition ensures the equivalence.

Abstract

In a recent Letter [Phys. Rev. Lett. 122, 010403 (2019)], an equivalence is proposed between the so-called Spectrum Broadcast Structure for a system-multienvironment quantum state, and the conjunction of two information-theory notions: (a) Strong Quantum Darwinism; and (b) Strong Independence. Here, we show that the mathematical formulation of condition (b) by the authors (namely, the pairwise independence of the fragments of the environment, conditioned on the system), is necessary but not sufficient to ensure the equivalence. We propose a simple counter-example, together with a strengthened formulation of condition (b), ensuring the equivalence proposed by the authors.