Network Quantum Steering

The development of large-scale quantum networks promises to bring a multitude of technological applications as well as shed light on foundational topics, such as quantum nonlocality. It is particularly interesting to consider scenarios where sources within the network are statistically independent, which leads to so-called network nonlocality, even when parties perform fixed measurements. Here we promote certain parties to be trusted and introduce the notion of network steering and network local hidden state (NLHS) models within this paradigm of independent sources. In one direction, we show how the results from Bell nonlocality and quantum steering can be used to demonstrate network steering. We further show that it is a genuinely novel effect by exhibiting unsteerable states that nevertheless demonstrate network steering based upon entanglement swapping yielding a form of activation. On the other hand, we provide no-go results for network steering in a large class of scenarios by explicitly constructing NLHS models.

Figure 1

Network steering scenarios. Green circles represent trusted parties, and red squares represent untrusted parties. (a) Standard steering scenario. (b) Steering scenario without inputs. (c) Triangle scenario with a trusted party. (d) Triangle scenario interpreted as a line. (e) Entanglement swapping scenario with trusted end points. (f) Generalized line scenario with trusted end points.

Figure 2

Classifying the structure of some NLHS models. Green circles represent trusted parties, and red squares represent untrusted parties who perform a fixed measurement. (a) In the scenario of Fig. 1, when one source is separable (SEP), this acts as an input to the adjacent measurements, and by taking the second source as unsteerable (${\mathrm{UNS}}_{\to }$) then this always leads to a NLHS model. (b) Similar results hold in the “unwrapped” triangle scenario [Fig. 1], where now sources can also be taken as local (LOC). We expand and detail this further in Supplemental Material Sec. I [35].