# Communication Strength of Correlations Violating Monogamy Relations

## 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.

## Keywords

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

### 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.

## References

- 1.Masanes, L., Acín, A., Gisin, N.: General properties of nonsignaling theories. Phys. Rev. A
**73**, 012112 (2006)ADSCrossRefGoogle Scholar - 2.Barrett, J.: Information processing in generalized probabilistic theories. Phys. Rev. A
**75**, 032304 (2007)ADSCrossRefGoogle Scholar - 3.Brunner, N., Cavalcanti, D., Pironio, S., Scarani, V., Wehner, S.: Bell nonlocality. Rev. Mod. Phys.
**86**, 419 (2014)ADSCrossRefGoogle Scholar - 4.Popescu, S., Rohrlich, D.: Quantum nonlocality as an axiom. Found. Phys.
**24**, 379 (1994)ADSMathSciNetCrossRefGoogle Scholar - 5.Pawłowski, M., Paterek, T., Kaszlikowski, D., Scarani, V., Winter, A., Żukowski, M.: Information causality as a physical principle. Nature
**461**, 1101 (2009)ADSCrossRefGoogle Scholar - 6.Navascués, M., Wunderlich, H.: A glance beyond the quantum model. Proc. Roy. Soc. A
**466**, 881 (2010)ADSMathSciNetCrossRefzbMATHGoogle Scholar - 7.Fritz, T., Sainz, A.B., Augusiak, R., Brask, J.B., Chaves, R., Leverrier, A., Acín, A.: Local orthogonality as a multipartite principle for quantum correlations. Nat. Commun.
**4**, 2263 (2013)ADSCrossRefGoogle Scholar - 8.Navascués, M., Guryanova, Y., Hoban, M.J., Acín, A.: Almost quantum correlations. Nat. Commun.
**6**, 6288 (2015)ADSCrossRefGoogle Scholar - 9.Toner, B.: Monogamy of non-local quantum correlations. Proc. R. Soc. A
**465**, 59 (2009)ADSMathSciNetCrossRefzbMATHGoogle Scholar - 10.Pawłowski, M., Brukner, C.: Monogamy of Bells inequality violations in nonsignaling theories. Phys. Rev. Lett.
**102**, 030403 (2009)ADSMathSciNetCrossRefGoogle Scholar - 11.Augusiak, R., Demianowicz, M., Pawłowski, M., Tura, J., Acín, A.: Elemental and tight monogamy relations in nonsignaling theories. Phys. Rev. A
**90**, 052323 (2014)ADSCrossRefGoogle Scholar - 12.Ramanathan, R., Horodecki, P.: Strong monogamies of no-signaling violations for bipartite correlation Bell inequalities. Phys. Rev. Lett.
**113**, 210403 (2014)ADSCrossRefGoogle Scholar - 13.Clauser, J.F., Horne, M.A., Shimony, A., Holt, R.A.: Proposed experiment to test local hidden-variable theories. Phys. Rev. Lett.
**23**, 880 (1969)ADSCrossRefGoogle Scholar - 14.Braunstein, S.L., Caves, C.M.: Wringing out better Bell inequalities. Ann. Phys.
**202**, 22 (1990)ADSMathSciNetCrossRefzbMATHGoogle Scholar - 15.Barrett, J., Hardy, L., Kent, A.: No signaling and quantum key distribution. Phys. Rev. Lett
**95**, 010503 (2005)ADSCrossRefGoogle Scholar - 16.Colbeck, R., Renner, R.: Free randomness can be amplified. Nat. Phys.
**8**, 450 (2012)CrossRefGoogle Scholar - 17.Gallego, R., Masanes, L., de la Torre, G., Dhara, C., Aolita, L., Acín, A.: Full randomness from arbitrarily deterministic events. Nat. Commun.
**4**, 2654 (2013)ADSCrossRefGoogle Scholar - 18.Brandão, F.G.S.L., Ramanathan, R., Grudka, A., Horodecki, K., Horodecki, M., Horodecki, P.: Robust device-independent randomness amplification with few devices. arXiv:1310.4544
- 19.Almheiri, A., Marolf, D., Polchinski, J., Sully, J.: Black holes: complementarity or firewalls? JHEP
**02**, 062 (2013)ADSMathSciNetCrossRefGoogle Scholar - 20.Oppenheim, J., Unruh, B.: Firewalls and flat mirrors: an alternative to the AMPS experiment which evades the Harlow-Hayden obstacle. JHEP
**03**, 120 (2014)ADSCrossRefGoogle Scholar - 21.Preskill, J., Lloyd, S.: Unitarity of black hole evaporation in final-state projection models, JHEP
**08**, 126 (2014)Google Scholar - 22.Grudka, A., Hall, M.J.W., Horodecki, M., Horodecki, R., Oppenheim, J., Smolin, J.: arXiv:1506.07133
- 23.Horowitz, G.T., Maldacena, J.: The black hole final state. JHEP
**02**, 008 (2004)ADSMathSciNetCrossRefGoogle Scholar - 24.Moser, S.M.: Error probability analysis of binary asymmetric channels, final report of NSC project ‘Finite Blocklength Capacity’, http://moser-isi.ethz.ch/docs/papers/smos-2012-4.pdf
- 25.Shannon, C.E.: Collected Papers. Wiley-IEEE Press, New York (1993)zbMATHGoogle Scholar
- 26.Shannon, C.E., Weaver, W.: The Mathematical Theory of Communication. University of Illinois Press, Urbana (1949)zbMATHGoogle Scholar