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Can every set of incompatible measurements lead to genuine multipartite steering?

Lucas E. A. Porto, Lucas Tendick, Daniel Cavalcanti, Roope Uola, Marco Túlio Quintino

Abstract

Measurement incompatibility and bipartite quantum steering are known to display a strong connection: a set of measurements is incompatible if and only if it can lead to bipartite steering. Despite such a close link between these concepts in bipartite scenarios, little is known in the multipartite setting, where notions of genuine multipartite correlations play major roles. In this work we prove that, as in the bipartite case, incompatibility is also necessary and sufficient for genuine multipartite steering in any multipartite scenario with a single uncharacterised party. Interestingly, genuine multipartite steering can be extracted from any set of incompatible measurements using states which are not SLOCC equivalent, such as GHZ and W states. In contrast, we prove that this result does not hold in scenarios with more than one uncharacterised party, by presenting a set of incompatible measurements that can never lead to genuine multipartite steering in these cases. In order to obtain our main results, we introduce methods tailored for multipartite correlations, paving the way to understanding the role of measurement incompatibility beyond bipartite scenarios.

Can every set of incompatible measurements lead to genuine multipartite steering?

Abstract

Measurement incompatibility and bipartite quantum steering are known to display a strong connection: a set of measurements is incompatible if and only if it can lead to bipartite steering. Despite such a close link between these concepts in bipartite scenarios, little is known in the multipartite setting, where notions of genuine multipartite correlations play major roles. In this work we prove that, as in the bipartite case, incompatibility is also necessary and sufficient for genuine multipartite steering in any multipartite scenario with a single uncharacterised party. Interestingly, genuine multipartite steering can be extracted from any set of incompatible measurements using states which are not SLOCC equivalent, such as GHZ and W states. In contrast, we prove that this result does not hold in scenarios with more than one uncharacterised party, by presenting a set of incompatible measurements that can never lead to genuine multipartite steering in these cases. In order to obtain our main results, we introduce methods tailored for multipartite correlations, paving the way to understanding the role of measurement incompatibility beyond bipartite scenarios.

Paper Structure

This paper contains 6 sections, 7 theorems, 16 equations, 1 figure.

Table of Contents

  1. End Matter
  2. Definitions for n-partite scenarios
  3. Useful lemmas
  4. Proof of \ref{['thm:one-sided_incomp_implies_gms']}
  5. Proof of \ref{['thm:two_sided_incomp_but_no_gms']}
  6. Proof of \ref{['thm:bi_partite_jm_iff_no_two-sided_gms']}

Key Result

Theorem 1

Let $\ket{\psi^{(1 \ldots n)}} \in \mathbb{C}^{d_1} \otimes \mathbb{C}^{d^{\otimes (n-1)}}$ be an $n$-partite state whose Schmidt decomposition in the bipartition $1|2\ldots n$ has the form $\ket{\psi} = \sum_{i = 0}^{d_1 - 1}\sqrt{p_i} \ket{\phi_i^{(1)}} \otimes \ket{\zeta_i^{(2\ldots n)}}$, where

Figures (1)

  • Figure 1: Scheme of tripartite steering scenarios where black boxes represent uncharacterised measurements. In scenarios with a single uncharacterised party (left), we show that any set of incompatible measurements enables genuine multipartite steering. In contrast, in scenarios where more than one party performs uncharacterised measurements (right), we provide sets of incompatible measurements which never lead to genuine multipartite steering.

Theorems & Definitions (13)

  • Theorem 1
  • proof
  • Theorem 2
  • proof
  • Theorem 3
  • Lemma 1
  • proof
  • Lemma 2
  • proof
  • Lemma 3
  • ...and 3 more