Abstract
Generalizing the notions of the row and the column stochastic matrices, we introduce the multidimensional Q-stochastic tensors. We prove that every Q-stochastic tensor can be decomposed as a convex combination of finitely many binary Q-stochastic tensors and that the binary Q-stochastic tensors are exactly the extreme points of the compact convex set of all Q-stochastic tensors with the same size. Applications to characterizing the Bell locality of a quantum state in a multipartite system are demonstrated. Algorithms for computing the convex decompositions of Q-stochastic tensors are provided.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Bell, J.S.: On the Einstein Podolsky Rosen paradox. Physics 1, 195–200 (1964)
Birkhoff, G.: Three observations on linear algebra. Univ. Nac. Tucumán Rev. Ser. A 5, 147–151 (1946)
Brualdi, R.A., Csima, J.: Extremal plane stochastic matrices of dimension three. Linear Algebra Appl. 11, 105–133 (1975)
Brunner, N., Cavalcanti, D., Pironio, S., Scarani, V., Wehner, S.: Bell nonlocality. Rev. Mod. Phys. 86, 419–478 (2014)
Cao, H.X., Chen, H.Y., Guo, Z.H., Lee, T.L.: The convex decomposition of row-stochastic matrices. Ann. Math. Sci. Appl. 8(2), 289–306 (2023)
Cao, H.X., Guo, Z.H.: Characterizing Bell nonlocality and EPR steering. Sci. China Phys. Mech. Astron. 62(3), 030311 (2019)
Cavalcanti, D., Skrzypczyk, P.: Quantum steering: a review. Rep. Prog. Phys. 80, 024001 (2017)
Chang, H.X., Paksoy, V.E., Zhang, F.Z.: Polytopes of stochastic tensors. Ann. Funct. Anal. 7(3), 386–393 (2016)
Chen, H.B., Qi, L., Caccetta, L., Zhou, G.L.: Birkhoff–von Neumann theorem and decomposition for doubly stochastic tensors. Linear Algebra Appl. 583, 119–133 (2019)
Conway, J.B.: A Course in Functional Analysis, 2nd edn. Springer, New York (2007)
Cui, L.B., Li, W., NG, M.K.: Birkhoff–von Neumann theorem for multistochastic tensors. SIAM J. Matrix Anal. Appl. 35(3), 956–973 (2014)
Einstein, A., Podolsky, B., Rosen, N.: Can quantum-mechanical description of physical reality be considered complete? Phys. Rev. 47, 777–780 (1935)
Fischer, P., Swart, E.R.: Three dimensional line stochastic matrices and extreme points. Linear Algebra Appl. 69, 179–203 (1985)
Jurkat, W.B., Ryser, H.J.: Extremal configurations and decomposition theorems. Int. J. Algebra 8, 194–222 (1966)
Lee, H.Y., Im, B.H., Smith, D.H.: Stochastic tensors and approximate symmetry. Discrete Math. 340, 1335–1350 (2017)
Li, C.K., Tam, B.S., Tsing, N.K.: Linear maps preserving permutation and stochastic matrices. Linear Algebra Appl. 341, 5–22 (2002)
Li, Z.S., Zhang, F.Z., Zhang, X.D.: On the number of vertices of the stochastic tensor polytope. Linear Multilinear Algebra 65(10), 2064–2075 (2017)
Schrödinger, E.: Discussion of probability relations between separated systems. Math. Proc. Camb. Philos. Soc. 31, 555–563 (1935)
Author information
Authors and Affiliations
Corresponding author
Additional information
Communicated by Jani Virtanen.
Dedicated to Professor Chi-Kwong Li on the occasion of his 65th birthday.
Rights and permissions
Springer Nature or its licensor (e.g. a society or other partner) holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law.
About this article
Cite this article
Cao, HX., Chen, HY., Guo, ZH. et al. Convex decompositions of Q-stochastic tensors and Bell locality in a multipartite system. Adv. Oper. Theory 9, 19 (2024). https://doi.org/10.1007/s43036-024-00316-x
Received:
Accepted:
Published:
DOI: https://doi.org/10.1007/s43036-024-00316-x