Any physical theory of nature must be boundlessly multipartite nonlocal

We introduce the class of genuinely local operations and shared randomness multipartite nonlocal correlations, that is, correlations between $N$ parties that cannot be obtained from unlimited shared randomness supplemented by any composition of $(N-1)$-shared causal generalized probabilistic theory (GPT) resources. We then show that noisy $N$-partite Greenberger-Horne-Zeilinger (GHZ) quantum states as well as the three-partite $W$ quantum state can produce such correlations. This proves, if the operational predictions of quantum theory are correct, that nature's nonlocality must be boundlessly multipartite in any causal GPT. We develop a computational method which certifies that a noisy $N=3$ GHZ quantum state with fidelity $85\%$ satisfies this property, making an experimental demonstration of our results within reach. We provide our definition and contrast it with preexisting notions of genuine multipartite nonlocality. This work extends a more compact parallel Letter [X. Coiteux-Roy et al., No Bipartite-Nonlocal Causal Theory Can Explain Nature's Correlations, Phys. Rev. Lett. 127, 200401 (2021)] on the same subject and provides all the required technical proofs.

Figure 1

All non-fan-out inflations of order $K=2$ for the tetrahedron network ${\mathcal{N}}_4$ (i.e., the network with four three-way sources); ${\mathcal{N}}_4$ is composed of parties $A$ (red), $B$ (blue), $C$ (yellow), $D$ (green), and sources of colors (red, blue, green, and yellow) such that each party is connected to each source except for the one of its own color. We represent the six nonisomorphic inflations ${\mathcal{I}}_1,...,{\mathcal{I}}_6$ (which are of respective multiplicities 1,3,3,1,12,12). Let $P$ be a nonsignaling correlation over $A,B,C,D$. For $P$ to be a three-LO theory-agnostic correlation, Definition 1 requires the existence of a correlation $Q^{\left(1\right)},...,Q^{\left(6\right)}$ for each inflated ${\mathcal{I}}_1,...,{\mathcal{I}}_6$ such that (C1)–(C3) are satisfied. For example, (C1) implies that $Q_{|A^1{B}^1{C}^1{D}^1}^{\left(1\right)}=P$ and $Q_{|A^1{D}^2}^{\left(2\right)}=P_{|AD}$, but not that $Q_{|A^1{B}^1{C}^1{D}^1}^{\left(2\right)}=P$; (C2) implies that every inflation is invariant under the exchange of all copy indices, e.g., $Q_{|A^1,B^1,C^1,D^1,A^2,B^2,C^2,D^2}=Q_{|A^2,B^2,C^2,D^2,A^1,B^1,C^1,D^1}$; and (C3) implies that $Q_{|A^1{B}^1{A}^2{B}^2}^{\left(3\right)}=Q_{|A^1{B}^1}^{\left(3\right)}·Q_{|A^2{B}^2}^{\left(3\right)}$. Note that one can in principle also consider $\mathcal{J}={\mathcal{I}}_1,...,{\mathcal{I}}_6$, which is a valid inflation of ${\mathcal{N}}_4$ (but of order $K=12$) and which implies the existence of a correlation $Q$ of the parties over $\mathcal{J}$ that factorizes as the product $Q^{\left(1\right)}\times \cdots \times Q^{\left(6\right)}$ and satisfies the compatibility conditions imposed by (C2); for instance, it implies $Q_{|A^1{B}^1{C}^1{A}^2{B}^2{C}^2}^{\left(3\right)}=Q_{|A^1{B}^1{C}^1{A}^2{B}^2{C}^2}^{\left(6\right)}$.

Figure 2

In this inflation $\mathcal{I}$, each party $A_i$ is connected to the original sources ${\omega }_j$ for $j>i$ and to the cloned sources ${\omega }_j^{\prime }$ for $j<i$. Assume by contradiction that there exists an arbitrary setup, with some causal GPT, that allows us to simulate a shared random bit in ${\mathcal{N}}_N$. In $\mathcal{I}$, (C1) imposes that two consecutive parties $A_j$ and $A_{j+1}$ share an identical random bit. This implies that any chain of consecutive parties should all together share the same random bit. In particular, $A_1$ and $A_N$ share the same random bit, which is in contradiction with (C2) as they do not have any sources in common.

Figure 3

A tripartite distribution is genuinely tripartite nonlocal according to Definition 3 if it is not a 2-LOSR theory-agnostic correlation, that is, if it cannot be realized by the above scenario, where the output of each player is determined by local operations (such as joint measurements) on (1) its input, (2) the three-way randomness, and (3) two-way GPT resources.

Figure 4

The inflation technique consists of duplicating and rearranging players, sources, and input distributions. Here we inflate the (not genuinely tripartite nonlocal) triangle scenario of Fig. 3 as to have the players play two parallel games ($\mathsf{Bell}$ and $\mathsf{Same}$). It leads to a contradiction with the statistics of measurements on $|\mathrm{GHZ}\rangle$ and therefore to the conclusion that the $|\mathrm{GHZ}\rangle$ quantum state is a genuinely tripartite nonlocal resource. The duplicated players are indistinguishable copies of the same abstract process; hence Alice, on input $X=\mathbf{0}$, could be playing either game ($A_1$ and $A_2$ must have the same behavior). The only condition on the random inputs is that they be independent of all the sources. The figure represents a cut of a larger inflation of order 3, consisting of a triangle and a hexagon (three parties of the hexagon are here fully ignored and only the input values relevant for the contradiction are featured).

Figure 5

This four-party non-fan-out inflation cut exposes that the quantum state $|{\mathrm{GHZ}}_4\rangle ≔\left(\right|0000\rangle +|1111\rangle )/\sqrt{2}$ is a genuinely four-partite nonlocal resource.

Figure 6

Sufficient for Lemma 1, this line inflation is in fact a cut of the third-order ring inflation. Given $\lambda$, we condition on the output $B_{\lambda }^1=0=B_{\lambda }^2$ (whenever possible) and it imposes $A_{\lambda }^1{C}_{\lambda }^1\in \{01,10\}$.

Figure 7

Representing the second step of the proof of Proposition 4, this line inflation is in fact a cut of the third-order ring inflation. Given $\lambda$, we condition on the output $C_{\lambda }^2=1=A_{\lambda }^2$, which, to simulate without error a distribution in $\{001,010,100\}$, forces $B_{\lambda }^1=0=B_{\lambda }^2$.