On monogamy of non-locality and macroscopic averages: examples and preliminary results. (English) Zbl 1467.81019
Coecke, Bob (ed.) et al., Proceedings of the 11th workshop on quantum physics and logic, QPL’14, Kyoto, Japan, June 4–6, 2014. Waterloo: Open Publishing Association (OPA). Electron. Proc. Theor. Comput. Sci. (EPTCS) 172, 36-55 (2014).
Summary: We explore a connection between monogamy of non-locality and a weak macroscopic locality condition: the locality of the average behaviour. These are revealed by our analysis as being two sides of the same coin.
Moreover, we exhibit a structural reason for both in the case of Bell-type multipartite scenarios, shedding light on but also generalising the results in the literature [R. Ramanathan et al., Phys. Rev. Lett. 107, No. 6, Article ID 060405, 5 p. (2011; doi:10.1103/PhysRevLett.107.060405); M. Pawłowski et al., Phys. Rev. Lett. 102, No. 3, Article ID 030403, 4 p. (2009; doi:10.1103/PhysRevLett.102.030403)]. More specifically, we show that, provided the number of particles in each site is large enough compared to the number of allowed measurement settings, and whatever the microscopic state of the system, the macroscopic average behaviour is local realistic, or equivalently, general multipartite monogamy relations hold.
This result relies on a classical mathematical theorem by N. N. Vorob’ev [Teor. Veroyatn. Primen. 7, 153–169 (1962; Zbl 0201.49102)] about extending compatible families of probability distributions defined on the faces of a simplicial complex – in the language of the sheaf-theoretic framework of S. Abramsky and A. Brandenburger [New J. Phys. 13, No. 11, Article ID 113036, 39 p. (2011; Zbl 1448.81028)], such families correspond to no-signalling empirical models, and the existence of an extension corresponds to locality or non-contextuality. Since Vorob’ev’s theorem depends solely on the structure of the simplicial complex, which encodes the compatibility of the measurements, and not on the specific probability distributions (i.e. the empirical models), our result about monogamy relations and locality of macroscopic averages holds not just for quantum theory, but for any empirical model satisfying the no-signalling condition.
In this extended abstract, we illustrate our approach by working out a couple of examples, which convey the intuition behind our analysis while keeping the discussion at an elementary level.
For the entire collection see [Zbl 1434.03010].
Moreover, we exhibit a structural reason for both in the case of Bell-type multipartite scenarios, shedding light on but also generalising the results in the literature [R. Ramanathan et al., Phys. Rev. Lett. 107, No. 6, Article ID 060405, 5 p. (2011; doi:10.1103/PhysRevLett.107.060405); M. Pawłowski et al., Phys. Rev. Lett. 102, No. 3, Article ID 030403, 4 p. (2009; doi:10.1103/PhysRevLett.102.030403)]. More specifically, we show that, provided the number of particles in each site is large enough compared to the number of allowed measurement settings, and whatever the microscopic state of the system, the macroscopic average behaviour is local realistic, or equivalently, general multipartite monogamy relations hold.
This result relies on a classical mathematical theorem by N. N. Vorob’ev [Teor. Veroyatn. Primen. 7, 153–169 (1962; Zbl 0201.49102)] about extending compatible families of probability distributions defined on the faces of a simplicial complex – in the language of the sheaf-theoretic framework of S. Abramsky and A. Brandenburger [New J. Phys. 13, No. 11, Article ID 113036, 39 p. (2011; Zbl 1448.81028)], such families correspond to no-signalling empirical models, and the existence of an extension corresponds to locality or non-contextuality. Since Vorob’ev’s theorem depends solely on the structure of the simplicial complex, which encodes the compatibility of the measurements, and not on the specific probability distributions (i.e. the empirical models), our result about monogamy relations and locality of macroscopic averages holds not just for quantum theory, but for any empirical model satisfying the no-signalling condition.
In this extended abstract, we illustrate our approach by working out a couple of examples, which convey the intuition behind our analysis while keeping the discussion at an elementary level.
For the entire collection see [Zbl 1434.03010].
MSC:
| 81P40 | Quantum coherence, entanglement, quantum correlations |
| 81P13 | Contextuality in quantum theory |
| 13F55 | Commutative rings defined by monomial ideals; Stanley-Reisner face rings; simplicial complexes |
| 03C90 | Nonclassical models (Boolean-valued, sheaf, etc.) |
