Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Composable simultaneous purification: when all communication scenarios reduce to spatial correlations

Matilde Baroni, Dominik Leichtle, Ivan Šupić, Damian Markham, Marco Túlio Quintino

Abstract

Bell non-locality is a powerful framework to distinguish classical, quantum and post-quantum resources, which relies on non-communicating players. Under which restriction can we have the same separations, if we allow for communication? Non-signalling state assemblages, and the fact that they can always be simultaneously purified, turned out to be the key element to restrict the simplest bipartite communication scenario, the prepare-and-measure, to the standard bipartite Bell scenario. Yet, many distinctive features of quantum theory are genuinely multipartite and cannot be reduced to two-party behaviour. In this work we are interested in extending this simultaneous purification inspired result to all multipartite communication schemes. As a first step, we unify and extend the simultaneous purification result from states to instruments and super-instruments, which are composable structures, and open up the possibility to explore more complex communication scenarios. Our main contribution is to establish that arbitrary compositions of non-signalling assemblages cannot escape the standard spatial quantum Bell correlations set. As a consequence, any interactive quantum realization of correlations outside of this set must involve at least one signalling assemblage of quantum operations, even when the resulting correlations are non-signalling.

Composable simultaneous purification: when all communication scenarios reduce to spatial correlations

Abstract

Bell non-locality is a powerful framework to distinguish classical, quantum and post-quantum resources, which relies on non-communicating players. Under which restriction can we have the same separations, if we allow for communication? Non-signalling state assemblages, and the fact that they can always be simultaneously purified, turned out to be the key element to restrict the simplest bipartite communication scenario, the prepare-and-measure, to the standard bipartite Bell scenario. Yet, many distinctive features of quantum theory are genuinely multipartite and cannot be reduced to two-party behaviour. In this work we are interested in extending this simultaneous purification inspired result to all multipartite communication schemes. As a first step, we unify and extend the simultaneous purification result from states to instruments and super-instruments, which are composable structures, and open up the possibility to explore more complex communication scenarios. Our main contribution is to establish that arbitrary compositions of non-signalling assemblages cannot escape the standard spatial quantum Bell correlations set. As a consequence, any interactive quantum realization of correlations outside of this set must involve at least one signalling assemblage of quantum operations, even when the resulting correlations are non-signalling.
Paper Structure (2 sections, 5 theorems, 32 equations, 3 figures)

This paper contains 2 sections, 5 theorems, 32 equations, 3 figures.

Table of Contents

  1. Simultaneous purification of non-signalling assemblages of arbitrary quantum objects
  2. Further communication scenarios

Key Result

Theorem 1

Let $\mathsf{k} \in \mathbb{N}$. The arbitrary composition of $\mathsf{k}$ non-signalling quantum assemblages always produces correlations which admit a $\mathsf{k}$-partite Bell quantum model.

Figures (3)

  • Figure 1: Decompositions for (a) state, (b) instrument, and (c) super-instrument assemblages. The first equality (i) is true for any valid assemblage. Decomposition (ii) is valid if and only if the assemblage is non-signalling. The last decomposition (iii) refers to unsteerable assemblages. These separations reflect the separation between non-signalling, quantum, and classical sets of correlations. The decompositions can be generalized beyond super-instruments to arbitrary quantum objects, as shown in Appendix \ref{['app:SGHJW-proofs']}.
  • Figure 2: Different communication schemes. Each party is represented by a square, the thin vertical arrows are classical inputs and outputs, and the thick arrows are quantum channels. The grey 'H' in (c) represents a process matrix Oreshkov_2012.
  • Figure 3: A bipartite scenario in which the network is modelled by a multi-round process matrix, and the players are described by assemblages of quantum combs. Time is flowing from left to right.

Theorems & Definitions (9)

  • Theorem 1
  • Theorem 2
  • Theorem 3
  • proof
  • Theorem 4
  • proof
  • Definition 1
  • Theorem 5
  • proof