Anonymous Quantum Nonlocality

We investigate the phenomenon of anonymous quantum nonlocality, which refers to the existence of multipartite quantum correlations that are not local in the sense of being Bell-inequality-violating but where the nonlocality is—due to its biseparability with respect to all bipartitions—seemingly nowhere to be found. Such correlations can be produced by the nonlocal collaboration involving definite subset(s) of parties but to an outsider, the identity of these nonlocally correlated parties is completely anonymous. For all $n\ge 3$, we present an example of an $n$-partite quantum correlation exhibiting anonymous nonlocality derived from the $n$-partite Greenberger-Horne-Zeilinger state. An explicit biseparable decomposition of these correlations is provided for any partitioning of the $n$ parties into two groups. Two applications of these anonymous Greenberger-Horne-Zeilinger correlations in the device-independent setting are discussed: multipartite secret sharing between any two groups of parties and bipartite quantum key distribution that is robust against nearly arbitrary leakage of information.

Figure 1

Schematic representation of the various sets of tripartite correlations. Correlations biseparable with respect to party $i$ in one group and parties $j$ and $k$ in the other lie in the (light blue) rectangle labeled by “$i|jk$”. The convex hull of the three biseparable sets $i|jk$, where $i,j,k\in \{1,2,3\}$ is represented by the filled convex region and gives correlations decomposable as the right-hand side of Eq. (1). The blank region between the outermost box and the filled convex region represents correlations that are genuinely tripartite nonlocal. Intersection of the three biseparable subsets $i|jk$ gives correlations satisfying Eqs. (2a)–(2c); its subset featuring ANL is the tiled region while local correlations lie in the (cyan) rectangle $L$. Hatched regions represent biseparable correlations where classical mixture of different bipartitions is necessary for their production.

Figure 2

ANL in the quadripartite scenario. Each participating party is abstractly represented by a box labeled by the party number. The correlations were produced by having parties 1 and 2, as well as 3 and 4 collaborated nonlocally (symbolized by $\sim$). To an outsider who only has access to $\overrightarrow{a}$ and $\overrightarrow{x}$, even if one is given the promise that the correlations were produced by the four parties separated into two fixed groups, it is impossible to tell which actual partitioning of the parties generated these correlations.