Nonlocality free wirings and the distinguishability between Bell boxes

Bell nonlocality can be formulated in terms of a resource theory with local-hidden variable models as resourceless objects. Two such theories are known, one built upon local operations assisted by shared randomness (LOSRs) and the other one allowing, in addition, for prior-to-input classical communication. We show that prior communication, although unable to create nonlocality, leads to wirings not only beyond LOSRs but also not contained in a much broader class of (nonlocality-generating) global wirings. Technically, this is shown by proving that it can improve the statistical distinguishability between Bell correlations optimized over all fixed measurement choices. This has implications in nonlocality quantification, and leads us to a natural universal definition of Bell nonlocality measures. To end up with, we also consider the statistical strength of nonlocality proofs. We point out some issues of its standard definition in the resource-theoretic operational framework, and suggest simple fixes for them. Our findings reveal nontrivial features of the geometry of the set of wirings and may have implications in the operational distinguishability of nonlocal behaviors.

Figure 1

Circuit representation of the three main types of Bell-box wirings. Panel (a) shows a generic $\mathsf{LOSR}$ wiring. There, the inputs of the final box ${\mathit{P}}_{\text{f}}$ are processed by an input box with local behavior ${\mathit{I}}^{\left(L\right)}$, whose outputs are input to the initial box ${\mathit{P}}_0$. The outputs produced by ${\mathit{P}}_0$ are, in turn, processed by an output box with local behavior ${\mathit{O}}^{\left(L\right)}$, which has as inputs all previously generated dits, without any exchange of dits between Alice and Bob's sides. The outputs of ${\mathit{O}}^{\left(L\right)}$ are the final outputs. Panel (b) shows an example of the class $\mathsf{WPICC}$, which allows for communication between the users provided it does not carry any information about the final inputs. In the example, Bob measures his initial local box before his final input is decided. He chooses his initial input with a single-partite box ${\mathit{D}}_Y$ without inputs. The output of his initial box is sent to Alice, who uses it, together with her final input, to choose her initial input according to a single-partite behavior ${\mathit{I}}^{\left(A\right)}$. The remaining dit processing is the same as in a $\mathsf{LOSR}$ wiring. Finally, in panel (c), a generic $\mathsf{GW}$ wiring is displayed. The circuitry there is analogous to that of $\mathsf{LOSR}$ wirings, except that the input and output boxes are governed by generic nonlocal behaviors $\mathit{I}$ and $\mathit{O}$, respectively, not restricted to $\mathsf{L}$. In fact, $\mathit{I}$ and $\mathit{O}$ need in general not even be in $\mathsf{NS}$. Important subclasses of $\mathsf{GW}$ are the no-signaling and quantum wirings, obtained when $\mathit{I}$ and $\mathit{O}$ are restricted to $\mathsf{NS}$ and $\mathsf{Q}$, respectively. While $\mathsf{LOSR}$ and $\mathsf{WPICC}$ are classes of nonlocality-free wirings, $\mathsf{GW}$ can create Bell nonlocality, i.e., it can map local behaviors into nonlocal ones.

Figure 2

Inclusion relationships among the three classes of wirings. In spite of being nonlocality free, some wirings with prior-to-input classical communication are out of the set of global wirings, which are, in general, not nonlocality free.