Skip to main content

Entangleability of cones


We solve a long-standing conjecture by Barker, proving that the minimal and maximal tensor products of two finite-dimensional proper cones coincide if and only if one of the two cones is generated by a linearly independent set. Here, given two proper cones \({\mathcal {C}}_1\), \({\mathcal {C}}_2\), their minimal tensor product is the cone generated by products of the form \(x_1\otimes x_2\), where \(x_1\in {\mathcal {C}}_1\) and \(x_2\in {\mathcal {C}}_2\), while their maximal tensor product is the set of tensors that are positive under all product functionals \(\varphi _1\otimes \varphi _2\), where \(\varphi _1|_{{\mathcal {C}}_1}\geqslant 0\) and \(\varphi _2|_{{\mathcal {C}}_2}\geqslant 0\). Our proof techniques involve a mix of convex geometry, elementary algebraic topology, and computations inspired by quantum information theory. Our motivation comes from the foundations of physics: as an application, we show that any two non-classical systems modelled by general probabilistic theories can be entangled.

This is a preview of subscription content, access via your institution.

Fig. 1
Fig. 2
Fig. 3


  1. Alternatively, the reader will find at a SageMath script which checks the correctness of (7).


  1. G. Aubrun, L. Lami, and C. Palazuelos. Universal entangleability of non-classical theories (2019). Preprint arXiv:1910.04745

  2. G. Aubrun, L. Lami, C Palazuelos, and M. Plávala. Entanglement and superposition are equivalent concepts. In preparation

  3. G. Aubrun, L. Lami, C. Palazuelos, S. J. Szarek, and A. Winter. Universal gaps for XOR games from estimates on tensor norm ratios. Comm. Math. Phys., (1)375 (2020), 679–724

  4. G. Aubrun and S. J. Szarek. Alice and Bob meet Banach, volume 223 of Mathematical Surveys and Monographs. American Mathematical Society, Providence, RI, (2017). The interface of asymptotic geometric analysis and quantum information theory

  5. G. P. Barker. Monotone norms and tensor products. Linear Multilinear Algebra, (3)4 (1976), 191–199

  6. G. P. Barker. Theory of cones. Linear Algebra Appl., 39 (1981), 263–291

    Article  MathSciNet  Google Scholar 

  7. G. P. Barker and R. Loewy. The structure of cones of matrices. Linear Algebra Appl., (1)12 (1975) 87–94

    Article  MathSciNet  Google Scholar 

  8. H. Barnum, J. Barrett, M. Leifer, and A. Wilce. Generalized No-broadcasting theorem. Phys. Rev. Lett., (24)99 (2007), 240501

    Article  Google Scholar 

  9. D. A. Birnbaum. Cones in the tensor product of locally convex lattices. Am. J. Math., (4)98 (1976),1049–1058

    Article  MathSciNet  Google Scholar 

  10. A. Bluhm, A. Jenčová, and I. Nechita. Incompatibility in General Probabilistic Theories, Generalized Spectrahedra, and Tensor Norms. (2020) Preprint arXiv:2011.06497

  11. D. Cariello. Does symmetry imply PPT property? Quantum Inf. Comput., (9-10)15 (2015), 812–824

    MathSciNet  Google Scholar 

  12. J. F. Clauser, M. A. Horne, A. Shimony, and R. A. Holt. Proposed experiment to test local hidden-variable theories. Phys. Rev. Lett., 23(1969), 880–884

    Article  Google Scholar 

  13. E. G. Effros. Structure in simplexes. Acta Math., 117 (1967), 103–121

    Article  MathSciNet  Google Scholar 

  14. E. G. Effros. Injectives and tensor products for convex sets and\(C^*\)-algebras. NATO Advanced Study Institute, University College of Swansea (1972)

  15. T. Fritz, T. Netzer, and A. Thom. Spectrahedral containment and operator systems with finite-dimensional realization. SIAM J. Appl. Algebra Geom., (1)1 (2017), 556–574

    Article  MathSciNet  Google Scholar 

  16. K. H. Han. Tensor products of function systems revisited. Positivity, (1)20 (2016),235–255

    Article  MathSciNet  Google Scholar 

  17. A. Hatcher. Algebraic Topology. Cambridge University Press, Cambridge, 2002.

    MATH  Google Scholar 

  18. B. Huber and T. Netzer. A note on non-commutative polytopes and polyhedra (2018). Preprint arXiv:1809.00476

  19. E. Kirchberg. On nonsemisplit extensions, tensor products and exactness of group \(C^*\)-algebras. Invent. Math., (3)112 (1993), 449–489

    Article  MathSciNet  Google Scholar 

  20. V. Klee. Some new results on smoothness and rotundity in normed linear spaces. Math. Ann., (1)139 (1959), 51–63

    Article  MathSciNet  Google Scholar 

  21. L. Lami. Non-classical correlations in quantum mechanics and beyond. PhD thesis, Universitat Autònoma de Barcelona, (2017). Preprint arXiv:1803.02902

  22. B. Mulansky. Tensor products of convex cones. In: Multivariate approximation and splines (Mannheim, 1996), volume 125 of International Series of Numerical Mathematics, pp. 167–176. Birkhäuser, Basel (1997)

  23. I. Namioka and R. R. Phelps. Tensor products of compact convex sets. Pacific J. Math., (2)31 (1969), 469–480

    Article  MathSciNet  Google Scholar 

  24. B. Passer, O. M. Shalit, and B. Solel. Minimal and maximal matrix convex sets. J. Funct. Anal., (11)274 (2018), 3197–3253

    Article  MathSciNet  Google Scholar 

  25. G. Pisier. Tensor Products of C*-Algebras and Operator Spaces: The ConnesKirchberg Problem. London Mathematical Society Student Texts. Cambridge University Press (2020)

  26. M. Plávala. All measurements in a probabilistic theory are compatible if and only if the state space is a simplex. Phys. Rev. A, (4)94 (2016), 042108

  27. S. Popescu and D. Rohrlich. Quantum nonlocality as an axiom. Found. Phys., (3)24 (1994) 379–385

    Article  MathSciNet  Google Scholar 

  28. R. T. Rockafellar. Convex Analysis. Princeton Mathematical Series, No. 28. Princeton University Press, Princeton, N.J. (1970)

  29. R. Schneider. Convex Bodies: The Brunn–Minkowski Theory. Encyclopedia of Mathematics and its Applications. Cambridge University Press, 2 edition (2013).

  30. B. S. Tam. Some results of polyhedral cones and simplicial cones. Linear Multilinear Algebra, (4)4 (1976/77), 281–284

  31. J. van Dobben de Bruyn. Tensor Products of Convex Cones, Part II: Closed Cones in Finite-Dimensional Spaces (2020). Preprint arXiv:2009.11843

  32. G. M. Ziegler. Lectures on Polytopes, volume 152 of Graduate Texts in Mathematics. Springer-Verlag, New York (1995)

Download references


We are very grateful to Kyung Hoon Han for several remarks which helped us clarifying the manuscript. We thank also Alexander Müller-Hermes for useful comments on some of the results. GA was supported in part by ANR (France) under the grant StoQ (2014-CE25-0003). LL acknowledges financial support from the European Research Council under the Starting Grant GQCOP (Grant no. 637352), from the Foundational Questions Institute under the grant FQXi-RFP-IPW-1907, and from the Alexander von Humboldt Foundation. CP is partially supported by Spanish MINECO through Grant No. MTM2017-88385-P, by the Comunidad de Madrid through grant QUITEMAD-CM P2018/TCS4342 and by SEV-2015-0554-16-3. MP acknowledges support from grant VEGA 2/0142/20, from the grant of the Slovak Research and Development Agency under contract APVV-16-0073, from the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation - 447948357) and the ERC (Consolidator Grant 683107/TempoQ).

Author information

Authors and Affiliations


Corresponding author

Correspondence to Guillaume Aubrun.

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Supplementary Information

Below is the link to the electronic supplementary material.

Supplementary file1 (SAGE 1 kb)

Rights and permissions

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

Cite this article

Aubrun, G., Lami, L., Palazuelos, C. et al. Entangleability of cones. Geom. Funct. Anal. 31, 181–205 (2021).

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI:

Keywords and phrases

  • Tensor product of cones
  • Entangleability
  • General probabilistic theories

Mathematics Subject Classification

  • Primary: 52A20
  • 47L07
  • Secondary: 81P16