## Abstract

In any theory satisfying the no-signaling principle correlations generated among spatially separated parties in a Bell-type experiment are subject to certain constraints known as monogamy relations. Recently, in the context of the black hole information loss problem it was suggested that these monogamy relations might be violated. This in turn implies that correlations arising in such a scenario must violate the no-signaling principle and hence can be used to send classical information between parties. Here, we study the amount of information that can be sent using such correlations. To this aim, we first provide a framework associating them with classical channels whose capacities are then used to quantify the usefulness of these correlations in sending information. Finally, we determine the minimal amount of information that can be sent using signaling correlations violating the monogamy relation associated to the chained Bell inequalities.

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## Notes

- 1.
For simplicity we consider only the situation in which the observables have two outcomes, but this is not a serious restriction.

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## Acknowledgments

We thank M. Horodecki, R. Horodecki, P. Kurzyński, M. Lewenstein, J. Łodyga and A. Wójcik for helpful discussions. W. K. and A. G. were supported by the Polish Ministry of Science and Higher Education Grant no. IdP2011 000361. M. O. was supported by the ERC Advanced Grant QOLAPS, START scholarship granted by Foundation for Polish Science, the Polish National Science Centre grant under Contract No. DEC-2011/01/M/ST2/00379, the John Templeton Foundation, the Spanish MINECO grant FOQUS, the “Severo Ochoa” Programme (SEV-2015-0522), and the Generalitat de Catalunya grant SGR875. R. A. was supported by the ERC Advanced Grant OSYRIS, the EU IP SIQS, the John Templeton Foundation, the Spanish Ministry project FOQUS (FIS2013-46768) and the Generalitat de Catalunya project 2014 SGR 874. W. K. thanks the Foundation of Adam Mickiewicz University in Poznań for the support from its scholarship programme. A.G. thanks ICFO–Institut de Ciències Fotòniques for hospitality.

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## Appendices

### Appendix 1: Conditions for Correlators from the Violation of Monogamy Relation (4)

We will now prove that for a particular violation \(\Delta \), the inequalities (26)–(29) along with the trivial conditions

satisfied by any pair *X*, *Y* constitute the only restrictions on the two-body correlators \(\mathbf {c}=(x_A^1,y_A^1,x_B^0,y_B^0,x_B^1,y_B^1)\) in the sense that for any such \(\mathbf {c}\) satisfying inequalities (26)-(29), there is a box \(\{p(A_iB_jE)\}\) realizing these correlators and violating the monogamy (4) by \(\Delta \).

Before passing to the proof, let us first introduce some additional notions. Let again *B* be the convex set of all tripartite boxes \(\{p(A_iB_jE)\}\) whose all one and three-partite expectation values vanish. Notice that such boxes are fully characterized by twelve two-body correlators \(\langle A_iB_j\rangle _E\), \(\langle A_iE\rangle _{B_j}\), and \(\langle B_jE\rangle _{A_i}\) with \(i,j=0,1\), that is,

for every *a*, *b*, *e* and *i*, *j*. For further benefits we also arrange the above expectation values in a vector \(\mathbf {p}\).

Let then \(\mathcal {P}\) be a subset of *B* consisting of boxes for which the value of the right-hand side of (4) is \(M(\mathbf {p})\in [4,6]\), i.e., elements of \(\mathcal {P}\) either saturate the monogamy relation (4) or violate it. Moreover, by \(\mathcal {P}_{\Delta }\) we denote those elements of \(\mathcal {P}\) for which the value \(M(\mathbf {p})\) is precisely \(4+\Delta \), i.e.,

Clearly, \(\mathcal {P}\) and \(\mathcal {P}_{\Delta }\) are polytopes whose vertices can easily be found, and, in particular, the vertices of \(\mathcal {P}\) belong to either \(\mathcal {P}_0\) or \(\mathcal {P}_2\).

Let finally \(\phi :B\rightarrow \mathbb {R}^6\) be a vector-valued function associating a vector of six correlators \(\mathbf {c}=(x_A^1,y_A^1,x_B^0,y_B^0,x_B^1,y_B^1)\) to any element of *B*. With the aid of this mapping we can associate to \(P_{\Delta }\) the following polytope

On the other hand, let us introduce the polytope \(\widetilde{\mathcal {Q}}_{\Delta }\) of vectors of the form \((\mathbf {c},\Delta )\) with \(\mathbf {c}\) satisfying the inequalities (26)-(29) for some fixed \(\Delta \) along with the trivial conditions (51). By definition, \(\mathcal {Q}_{\Delta }\subseteq \widetilde{\mathcal {Q}}_{\Delta }\) for any \(\Delta \) and our aim now is to prove that \(\mathcal {Q}_{\Delta }=\widetilde{\mathcal {Q}}_{\Delta }\). In particular, we want to show that any \(\mathbf {c}\in \mathcal {Q}_{\Delta }\) with some fixed \(\Delta \ge 0\) can always be completed to a full probability distribution \(\mathbf {p}\in \mathcal {P}_{\Delta }\) violating (4) by \(\Delta \).

With the above goal we define two additional polytopes

and

Direct numerical computation shows that, analogously to \(\mathcal {P}\), the vertices of \(\mathcal {Q}_v\) belong to either \(\mathcal {Q}_0\) or \(\mathcal {Q}_2\). In the same way one shows that the vertices of both polytopes \(\mathcal {Q}_{v}\) and \(\widetilde{\mathcal {Q}}_{v}\) overlap, which implies that \(\mathcal {Q}_v=\widetilde{\mathcal {Q}}_{v}\). Using then the definition of these sets and the fact that the mapping \(\mathbf {p}\rightarrow (\phi (\mathbf {p}),M(\mathbf {p}))\) is linear, one obtains that \(\mathcal {Q}_{\Delta }=\widetilde{\mathcal {Q}}_{\Delta }\) for any \(\Delta \).

### Appendix 2: Analytical Computation of \(C_2\)

Here we determine analytically the capacity \(C_{\Delta }\) in the case when the monogamy relation (4) is violated maximally, i.e., for \(\Delta =2\). From Inequations (26)–(29) it immediately follows that \(x_B^0=x_B^1=1\), \(y_B^0=x_A^1\), and \(y_B^1=-y_A^1\), and the problem of determining \(C_2\) considerably simplifies to

where we have substituted \(y_B^0=\alpha \) and \(y_B^1=\beta \) and have denoted \(\widetilde{C}(\alpha ,\beta )=C((1+x)/2,(1+y)/2)\) with *C* defined in Eq. (19). To compute the above, it is useful to notice that the function \(\widetilde{C}\) satisfies \(\widetilde{C}(\alpha ,\beta )=\widetilde{C}(\alpha ,\beta ) =\widetilde{C}(-\alpha ,-\beta )\), and that it is convex in both arguments (cf. Ref. [25]). The latter implies in particular that for any \(\alpha \le 0\), \(\widetilde{C}(1,\alpha )\ge \widetilde{C}(\alpha ,\beta )\) and also \(\widetilde{C}(1,\alpha )\ge \widetilde{C}(\alpha ,-\beta )\) with \(-1\le \beta \le 1\). This observation suggests dividing the square \(-1\le \alpha ,\beta \le 1\) into four ones (closed) whose facets are given by \(\alpha =0\) and \(\beta =0\), and determining \(C_2\) in each of them. In fact, whenever \(\alpha \le 0\) or \(\beta \le 0\),

and by direct checking one obtains \(C_2=0.322\). In order to find \(C_2\) in the last region given by \(\alpha \ge 0\) and \(\beta \ge 0\), one first notices \(\widetilde{C}(1,\alpha )\ge \widetilde{C}(1,\beta )\) if, and only if \(\alpha \le \beta \). This, along with the fact that \(\widetilde{C}(\alpha ,-\beta )=\widetilde{C}(-\beta ,\alpha )= \widetilde{C}(\beta ,-\alpha )\) means that we can restrict our attention to the case \(\alpha \le \beta \), for which

In the last step we notice that for any \(0\le \beta \le 1\), \(\widetilde{C}(\alpha ,-\beta )\) and \(\widetilde{C}(1,\alpha )\) are, respectively, monotonically increasing and decreasing functions of \(\alpha \). Additionally, for any \(0 \le \alpha \le 1\), \(\widetilde{C}(\alpha ,-\beta )\) is a monotonically increasing function of \(\beta \). Then, for \(\alpha =1\), \(\widetilde{C}(1,-1)=1\), while \(\widetilde{C}(1,1)=0\) (recall that we assume that \(\alpha \le \beta \)), and for \(\alpha =0\), \(\min _{\beta \ge 0}\widetilde{C}(\alpha ,-\beta )=0\) and \(\widetilde{C}(1,0)>0\). All this means that both functions \(\widetilde{C}(1,\alpha )\) and \(\widetilde{C}(\alpha ,-\beta )\) intersect, implying that \(C_2\) lies on the line given by \(\widetilde{C}(1,\alpha )=\widetilde{C}(\alpha ,-\beta )\). Finally, as already mentioned, \(\widetilde{C}(\alpha ,-\beta )\) is a monotonically decreasing function of \(\beta \) which together with \(\alpha \le \beta \) means that \(\alpha =\beta \) has to be taken. One then arrives at the condition that \(\widetilde{C}(1,\alpha )=\widetilde{C}(\alpha ,-\alpha )\), which has a solution when for \(\alpha =0.469\) giving \(C_2=0.158\). By comparing both minima, we finally obtain that \(C_2=0.158\).

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Kłobus, W., Oszmaniec, M., Augusiak, R. *et al.* Communication Strength of Correlations Violating Monogamy Relations.
*Found Phys* **46, **620–634 (2016). https://doi.org/10.1007/s10701-015-9983-5

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### Keywords

- Monogamy relations
- No-signaling principle
- Capacities of communication channels