The Elegant Joint Measurement is Non-Classical in the Triangle Network

Abstract

When quantum systems are shared by multiple parties in a network, the measurement outcomes of the parties can exhibit non-classical correlations, i.e., correlations that cannot be obtained if the parties shared classical systems instead. This phenomenon is known as quantum nonlocality and is typically demonstrated in the Bell scenario. However, the Bell scenario is fundamentally simpler to investigate than general networks, since the latter come with non-convex optimization problems that are often intractable. The triangle network is one of the simplest networks exhibiting this non-convexity due to the presence of three independent sources. Although some special cases of quantum nonlocality are known in the triangle network, general methods to certify classical incompatibility are still lacking, which suggests that our understanding of networks is still rather limited. For instance, the Elegant Joint Measurement (EJM) distribution is a simple and highly-symmetric outcome distribution that can be obtained with quantum systems and measurements in the triangle network. This distribution was conjectured to be non-classical eight years ago. In this article, we provide the first proof of non-classicality of the EJM distribution. To do so, we show how to combine inflation, a causal inference technique, with powerful symmetry reductions and Frank-Wolfe algorithms for large-scale optimization. We then use these methods to obtain computer-assisted proofs of non-classicality in exact arithmetic.

Figures (4)

• Figure 1: The triangle network features three observers (yellow squares), connected by three independent bipartite sources (green circles): Alice ($\mathrm{A}$) has access to the $\beta,\gamma$ sources, while Bob ($\mathrm{B}$) has access to $\gamma,\alpha$ and Charlie ($\mathrm{C}$) to $\alpha,\beta$.
• Figure 2: A fanout inflation of the triangle network of \ref{['fig:triangle']}, consisting of two copies of each source and four copies of each party. This inflation characterizes classical causal models in the triangle network: the fanning-out of each source indicate that the corresponding classical value sent by the source is copied and broadcast to the parties the source is connected to. Note that all the edges of the graph start from a source and end at a party (we draw some edges behind the parties for convenience).
• Figure 3: The subgraph of the parties ${\boldsymbol{A}}_0 = \{\mathrm{A}_{00},\mathrm{B}_{00},\mathrm{C}_{00}\}$ and ${\boldsymbol{A}}_1 = \{\mathrm{A}_{11},\mathrm{B}_{11},\mathrm{C}_{11}\}$ together with their parent sources in the inflation graph of \ref{['fig:inf_graph_222']}, We observe the $d$-separation Pearl_2009 of ${\boldsymbol{A}}_0$ and ${\boldsymbol{A}}_1$, i.e., there is no source connecting both a party in ${\boldsymbol{A}}_0$ and a party in ${\boldsymbol{A}}_1$). We also observe that ${\boldsymbol{A}}_0$ and ${\boldsymbol{A}}_1$ are so-called injectable sets wolfe_inflation_2019, i.e., they are connected in a manner equivalent to the triangle network of \ref{['fig:triangle']}).
• Figure 4: A visualization of the Frank-Wolfe algorithm described in \ref{['algo:fw maintext']}. We here pretend that $\ref{['target:quovecspace']}_{{\boldsymbol{A}}_0{\boldsymbol{A}}_1}^{\space c,{p}}$ has dimension $2$. The red cross represents $0 \in \ref{['target:quovecspace']}_{{\boldsymbol{A}}_0{\boldsymbol{A}}_1}^{\space c,{p}}$ while the green circles represent the vertices $\ref{['target:totconstraintmap']}^{c,{p}}(\ref{['target:detdistr']}^{\space\omega_i})$. We here assume that $\ref{['target:redinfevents']}^{{p}}_{\ref{['target:infparties']}} = \{\omega_i\}_{i\in\ref{['target:ints']}5\ref{['target:ints']}}$. \ref{['fig:fw2', 'fig:fw3', 'fig:fw4']} refer to the state of the active set $\mathcal{S}$, candidate incompatibility certificate $f_\textup{p} \in \ref{['target:quovecspace']}_{{\boldsymbol{A}}_0{\boldsymbol{A}}_1}^{\space c,{p}}$ and gap $s_\textup{global} \in \mathbb{Z}$ after completing \ref{['algoline:global gap']} of \ref{['algo:fw maintext']}.

Theorems & Definitions (8)

• Definition 3.1: Explicit inflation problem
• Proposition 3.2
• proof
• Definition 4.1
• Definition 4.2
• Definition 4.3
• Proposition 5.1
• proof