Extent of multiparticle quantum nonlocality

It is well known that entangled quantum states are nonlocal: the corrrelations between local measurements carried out on these states cannot be reproduced by local hidden variable models. Svetlichny, followed by others, showed that multipartite quantum states are more nonlocal than bipartite ones in the sense that even some nonlocal classical models with (super-luminal) communication between some of the parties cannot reproduce the quantum correlations. Here we study in detail the kinds of nonlocality present in quantum states. More precisely, we enquire what kinds of classical communication patterns cannot reproduce quantum correlations. By studying the extremal points of the space of all multiparty probability distributions, in which all parties can make one of a pair of measurements each with two possible outcomes, we find a necessary condition for classical nonlocal models to reproduce the statistics of all quantum states. This condition extends and generalizes work of Svetlichny and others in which it was showed that a particular class of classical nonlocal models, the “separable” models, cannot reproduce the statistics of all multiparticle quantum states. Our condition shows that the nonlocality present in some entangled multiparticle quantum states is much stronger than previously thought. We also study the sufficiency of our condition.

Figure 1

(i) This graph represents the set of extreme points identifiable with lhv models—i.e., $a_i\left(x_i\right)$. Bell considered convex combinations of these points. (ii) (a) and (b) These graphs represent functions $\vec{a}\left(\vec{x}\right)$ with the structure $a_1(x_1,\wedge ,x_3,\wedge ),a_3(x_1,\wedge ,x_3,\wedge ),$ $a_2(\wedge ,x_2,\wedge ,x_4),$ $a_4(\wedge ,x_2,\wedge ,x_4)$ and $a_1(x_1,x_2,x_3,\wedge ),$ $a_3(x_1,x_2,x_3,\wedge ),$ $a_2(x_1,x_2,x_3,\wedge ),$ $a_4(\wedge ,\wedge ,\wedge ,x_4)$, respectively. Reference 8 considers correlations which are convex combinations of extremal points represented by separable graphs of this form and show that such models have $⟨S_4⟩⩽2$ (see Sec. 4). (iii) This graph represents functions $\vec{a}\left(\vec{x}\right)$ which have $a_1(x_1,x_2,x_3,\wedge ),$ $a_3(x_1,x_2,x_3,\wedge ),$ $a_2(x_1,x_2,\wedge ,x_4),$ $a_4(x_1,x_2,\wedge ,x_4)$. Parties 3 and 4 are separated in the sense that no party receives arrows from both. We will show that convex combinations of these extreme points yield a value of $⟨S_4⟩⩽2$. We call graphs of type (i)–(iii) “partially paired.” (iv) Examples of graphs in which there is always a party which knows the settings of any pair — we call these “totally paired.” Certain convex combinations of the extremal points represented by these graphs can achieve the algebraic maximum $⟨S_4⟩=4$, as described in the text. (a) $a_1(x_1,x_2,x_3,x_4),$ $a_3(\wedge ,\wedge ,x_3,\wedge ),$ $a_2(\wedge ,x_2,\wedge ,\wedge ),$  $a_4(\wedge ,\wedge ,\wedge ,x_4)$ and (b) $a_1(x_1,\wedge ,x_3,\wedge ),$ $a_3(\wedge ,x_2,x_3,x_4),$ $a_2(x_1,x_2,\wedge ,\wedge ),$ $a_4(x_1,x_2,\wedge ,x_4)$. (v) This graph represents the set of all extreme points and all possible four-party graphs are its subgraphs.

Figure 2

Graphical representation of the most general Partially Paired (PP) $m$-party graph. One circle contains parties $\{2,\dots ,k\}$ and the other circle contains parties $\{k+1,\dots ,m-1\}$. All parties within a circle can send and receive arrows from all other parties within the same circle. Each party in a circle can send and receive arrows to all parties in the other circle. Party 1 can send and receive arrows from parties $\{2,\dots ,k\}$. Party 1 can receive arrows from (but does not send arrows to) parties $\{k+1,\dots ,m-1\}$ Party $m$ can receive and send arrows to parties $\{k+1,\dots ,m-1\}$. Party $m$ can receive arrows from (but does not send arrows to) parties $\{2,\dots ,k\}$. There are no arrows between parties 1 and $m$. With these restrictions there is no party that receives arrows both from party 1 and party $m$. The subgraph composed only of the arrows emanating from parties 1 and $m$ is therefore separable. The graph thus belongs to the PP class.