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Information causality beyond the random access code model

Baichu Yu, Valerio Scarani

TL;DR

This work closes some gaps in information causality using a new quantifier for IC, based on the notion of “redundant information”, which is still obeyed by quantum correlations.

Abstract

Information causality (IC) was one of the first principles that have been invoked to bound the set of quantum correlations. For some families of correlations, this principle recovers exactly the boundary of the quantum set; for others, there is still a gap. We close some of these gaps using a new quantifier for IC, based on the notion of ``redundant information''. This progress was made possible by the recognition that the principle of IC can be captured without referring to the success criterion of random access codes. We give strong numerical evidence that the new definition is still obeyed by quantum correlations in the same scenario.

Information causality beyond the random access code model

TL;DR

This work closes some gaps in information causality using a new quantifier for IC, based on the notion of “redundant information”, which is still obeyed by quantum correlations.

Abstract

Information causality (IC) was one of the first principles that have been invoked to bound the set of quantum correlations. For some families of correlations, this principle recovers exactly the boundary of the quantum set; for others, there is still a gap. We close some of these gaps using a new quantifier for IC, based on the notion of ``redundant information''. This progress was made possible by the recognition that the principle of IC can be captured without referring to the success criterion of random access codes. We give strong numerical evidence that the new definition is still obeyed by quantum correlations in the same scenario.
Paper Structure (3 sections, 20 equations, 3 figures, 1 table)

This paper contains 3 sections, 20 equations, 3 figures, 1 table.

Table of Contents

  1. Sampling and Heristic Optimisation of IC Values
  2. Sampling
  3. Optimisation

Figures (3)

  • Figure 1: Slices \ref{['box']} of the non-signalling polytope studied in this work: from top to bottom, $R_{NS}=P_{NL}^{010}$, $R_{NS}=P_{NL}^{110}$, and $R_{NS}=P_{L}^{0000}$. In all figures, the top left corner is the PR box $P_{NL}^{000}$, the bottom line is the facet $CHSH=2$. ICO represents IC for the original definition \ref{['ICoriginal']}; these are the curves found in allcock2009recovering. ICR represents IC for our definition based on redundant information \ref{['modifiedIC']}. The quantum boundary is the Tsirelson-Landau-Masanes bound (see scaranibook) for the first two figures, and a straight line for the third goh2018geometry.
  • Figure 2: Distributions of (a) $IC_{\mathrm{red}}$ and (b) $IC_{\mathrm{RAC}}$ values over the sampled extremal correlations. The $IC_{\mathrm{red}}$ values are bounded by the channel capacity $k$, with a lower cutoff at $1.4 \times 10^{-8}$. In contrast, $IC_{\mathrm{RAC}}$ values show a higher concentration near $k$ but extend down to zero.
  • Figure 3: Frequencies of $IC_{red}$ values 300000 random mixture for $M=2,10,20,30,40,50$ random extremal points. We see that when the number of states to be mixed increases, the IC values tend to become smaller as a collective behavior. And when $M\geq 40$, the distribution becomes stable.