No-local-broadcasting theorem for non-signalling behaviours and assemblages

TL;DR

This paper proves the conjecture that one cannot locally broadcast nonlocal boxes based on fundamental properties of the relative entropy of boxes and obtains an analogous theorem for steerable assemblages.

Abstract

The no-broadcasting theorem is a fundamental result in quantum information theory. It guarantees that a class of attacks on quantum protocols, based on eavesdropping and indiscriminate copying of quantum information, are impossible. Due to its fundamental importance, it is natural to ask whether it is an intrinsic quantum property or whether it also holds for a broader class of non-classical theories. To address this question, one could use the framework of correlation scenarios. Under this standpoint, Joshi, Grudka, and Horodecki$^{\otimes 4}$ conjectured that one cannot locally broadcast nonlocal behaviours. In this paper, we prove their conjecture based on the monotonicity of the relative entropy for behaviours. Additionally, following a similar reasoning, we obtain an analogous no-go theorem for steerable assemblages.

Figures (5)

• Figure 1: Pictorial representation of a local broadcasting scenario: Alice (A) and Bob (B) share some sort of known correlation, which is represented by them being enclosed in a box. They send some information to pairs of Alices ($\mathrm{A_0,A_1}$) and Bobs ($\mathrm{B_0,B_1}$) in some localised manner. The broadcast is successful if the lower Alice-Bob pairs with the same index share the original correlation illustrated by the dotted boxes (colours online).
• Figure 2: Pictorial representation of a (uniform) correlation scenario. Measurements are represented by the coloured buttons on the top of each box. A single outcome is returned after a measurement is performed, and schematically, one light bulb goes off (colours online).
• Figure 3: Illustration of an elementary LOSR transformation of a behaviour. The input behaviour $\mathbf{P}$ is in green in the middle, with two wavy lines between the two parts representing possibly non-classical correlations; its inputs are $x$ and $y$, while the outputs are $a$ and $b$. The red boxes above represent the preprocessing behaviour $I$, while the blue ones below represent the post-processing behaviour $O$. They have single wavy lines between them to represent classical correlations. The final behaviour $\mathbf{P}'$ has inputs $\chi,\psi$ and outputs $\alpha,\beta$ (colours online).
• Figure 4: Pictorial representation of a bipartite steering scenario. One agent gets the usual box with coloured buttons on the top and outcomes represented by light bulbs. The other agent receives a quantum system on which they can make measurements. After pressing buttons and performing measurements, every agent receives an outcome (colours online).
• Figure 5: Illustration of an LOSR transformation of an assemblage. The green box and the black dot in the middle represent the original assemblage with input $x$ and output $a$ and related quantum state $\rho_{a|x}$, with the two wavy lines between the two parts representing possibly stronger than classical correlations. The red box represents the preprocessing probability distribution $I_{\lambda}$, with a wavy line ($\lambda$) representing a classical correlation with the CPTP map $\mathcal{E}_\lambda$. The blue box represents the post-processing probability distribution $O$. The final assemblage has input $\chi$, outputs $\alpha$, and the related quantum state $\rho'_{\alpha|\chi}$ (colours online).

Theorems & Definitions (59)

• Definition 1
• Definition 2
• Definition 3
• Definition 4
• Definition 5
• Definition 6: Broadcasting of a behaviour
• Definition 7
• Definition 8
• Definition 9
• Definition 10