Abstract

The state space of an operator system of n-by-n matrices has, in a sense, many normal cones. This wealth of normal cones is a smoothness quality and it implies that each exposed face is an intersection of maximal exposed faces. A clustering property of exposed faces follows. We work on minimizing the assumptions which allow an arbitrary convex set of finite dimension to have analogous properties. One motivation for this work is an isomorphism from the exposed faces of the state space to the ground state spaces of the operator system.

Author information

Contact details are reproduced from the original publication and may be historical.

Stephan Weis

Departamento de Matemática, Inst. Matemática e Estatística, Univ. Estadual de Campinas, Rua S. Buarque de Holanda 651, Campinas-SP / CEP 13083-859, Brazil
and: Centre for Quantum Information, École Polytechnique, Université Libre, 50 av. F.D. Roosevelt - CP165/59, 1050 Bruxelles, Belgium

maths@stephan-weis.info

Keywords

  • Normal cone
  • exposed ray
  • exposed face
  • coatom
  • operator system
  • state space
  • convex support
  • joint numerical range
  • ground state
  • coorbitope

Mathematics Subject Classification

52A20, 52B05, 47L07, 51D25, 47A12; 81P16, 14P10

References

  1. E. M. Alfsen, F. W. Shultz: State Spaces of Operator Algebras: Basic Theory, Orientations, and C*-Products, Birkhäuser, Boston (2010).Google Scholar
  2. L. Arrachea, N. Canosa, A. Plastino, M. Portesi, R. Rossignoli: Maximum-entropy approach to critical phenomena in ground states of finite systems, Phys. Rev. A 45 (1992) 7104–7110.https://doi.org/10.1103/PhysRevA.45.7104Google Scholar
  3. Y. H. Au-Yeung, Y. T. Poon: A remark on the convexity and positive definiteness concerning Hermitian matrices, Southeast Asian Bull. Math. 3 (1979) 85–92.Google Scholar
  4. G. P. Barker, D. Carlson: Cones of diagonally dominant matrices, Pac. J. Math. 57 (1975) 15–32.https://doi.org/10.2140/pjm.1975.57.15Google Scholar
  5. G. P. Barker: Faces and duality in convex cones, Linear Multilinear A 6 (1978) 161–169.https://doi.org/10.1080/03081087808817234Google Scholar
  6. O. Barndorff-Nielsen: Information and Exponential Families in Statistical Theory, John Wiley, Chichester (1978).Google Scholar
  7. I. Bengtsson and K. Życzkowski: Geometry of Quantum States, Cambridge University Press, Cambridge (2006).https://doi.org/10.1017/CBO9780511535048Google Scholar
  8. A. Berman, A. Ben-Israel: Linear equations over cones with interior: A solvability theorem with applications to matrix theory, Linear Algebra Appl. 7 (1973) 139–149.https://doi.org/10.1016/0024-3795(73)90048-7Google Scholar
  9. G. Birkhoff: Lattice Theory, 3rd ed., American Mathematical Society, Providence (1967).Google Scholar
  10. J. Chen, Z. Ji, D. Kribs, Z. Wei, B. Zeng: Ground-state spaces of frustrationfree Hamiltonians, J. Math. Phys. 53 (2012) 102201.https://doi.org/10.1063/1.4748527Google Scholar
  11. J. Chen, Z. Ji, C.-K. Li, Y.-T. Poon, Y. Shen, N. Yu, B. Zeng, D. Zhou: Discontinuity of maximum entropy inference and quantum phase transitions, New J. Phys. 17 (2015) 083019.https://doi.org/10.1088/1367-2630/17/8/083019Google Scholar
  12. J. Chen, Z. Ji, M. B. Ruskai, B. Zeng, D.-L. Zhou: Comment on some results of Erdahl and the convex structure of reduced density matrices, J. Math. Phys. 53 (2012) 072203.https://doi.org/10.1063/1.4736842Google Scholar
  13. J. Chen, Z. Ji, B. Zeng, D. L. Zhou: From ground states to local Hamiltonians, Phys. Rev. A 86 (2012) 022339.https://doi.org/10.1103/PhysRevA.86.022339Google Scholar
  14. W.-S. Cheung, X. Liu, T.-Y. Tam: Multiplicities, boundary points, and joint numerical ranges, Oper. Matrices 1 (2011) 41–52.https://doi.org/10.7153/oam-05-02Google Scholar
  15. M.-T. Chien, H. Nakazato: Joint numerical range and its generating hypersurface, Linear Algebra Appl. 432 (2010) 173–179.https://doi.org/10.1016/j.laa.2009.07.034Google Scholar
  16. R. M. Erdahl: The convex structure of the set of N-representable reduced 2-matrices, J. Math. Phys. 13 (1972) 1608–1621.https://doi.org/10.1063/1.1665885Google Scholar
  17. B. Grünbaum: Convex Polytopes, 2nd ed., Springer-Verlag, New York (2003).https://doi.org/10.1007/978-1-4613-0019-9Google Scholar
  18. E. Gutkin, E. A. Jonckheere, M. Karow: Convexity of the joint numerical range: topological and differential geometric viewpoints, Linear Algebra Appl. 376 (2004) 143–171.https://doi.org/10.1016/j.laa.2003.06.011Google Scholar
  19. E. Gutkin, K. Życzkowski: Joint numerical ranges, quantum maps, and joint numerical shadows, Linear Algebra Appl. 438 (2013) 2394–2404.https://doi.org/10.1016/j.laa.2012.10.043Google Scholar
  20. T. Heinosaari, L. Mazzarella, M. M. Wolf: Quantum tomography under prior information, Commun. Math. Phys. 318 (2013) 355–374.https://doi.org/10.1007/s00220-013-1671-8Google Scholar
  21. A. Holevo: Probabilistic and Statistical Aspects of Quantum Theory, 2nd ed., Edizioni della Normale, Pisa (2011).https://doi.org/10.1007/978-88-7642-378-9Google Scholar
  22. E. Jaynes: Information theory and statistical mechanics. II, Phys. Rev. 108 (1957) 171–190.https://doi.org/10.1103/PhysRev.108.171Google Scholar
  23. D. S. Keeler, L. Rodman, I. M. Spitkovsky: The numerical range of 3 × 3 matrices, Linear Algebra Appl. 252 (1997) 115–139.https://doi.org/10.1016/0024-3795(95)00674-5Google Scholar
  24. R. Kippenhahn: Über den Wertevorrat einer Matrix, Math. Nachr. 6 (1951) 193–228.https://doi.org/10.1002/mana.19510060306Google Scholar
  25. N. Krupnik, I. M. Spitkovsky: Sets of matrices with given joint numerical range, Linear Algebra Appl. 419 (2006) 569–585.https://doi.org/10.1016/j.laa.2006.05.019Google Scholar
  26. C.-K. Li, Y.-T. Poon: Convexity of the joint numerical range, SIAM J. Matrix Anal. A 21 (2000) 668–678.https://doi.org/10.1137/S0895479898343516Google Scholar
  27. R. Loewy, B.-S. Tam: Complementation in the face lattice of a proper cone, Linear Algebra Appl. 79 (1986) 195–207.https://doi.org/10.1016/0024-3795(86)90299-5Google Scholar
  28. V. Magron, D. Henrion, J.-B. Lasserre: Semidefinite approximations of projections and polynomial images of semialgebraic sets, SIAM J. Optimization 25 (2015) 2143–2164.https://doi.org/10.1137/140992047Google Scholar
  29. V. Müller: The joint essential numerical range, compact perturbations, and the Olsen problem, Stud. Math. 197 (2010) 275–290.https://doi.org/10.4064/sm197-3-5Google Scholar
  30. T. Netzer, D. Plaumann, M. Schweighofer: Exposed faces of semidefinitely representable sets, SIAM J Optimization 20 (2010) 1944–1955.https://doi.org/10.1137/090750196Google Scholar
  31. S. A. Ocko, X. Chen, B. Zeng, B. Yoshida, Z. Ji, M. B. Ruskai, I. L. Chuang: Quantum codes give counterexamples to the unique preimage conjecture of the N-representability problem, Phys. Rev. Lett. 106 (2011) 110501.https://doi.org/10.1103/PhysRevLett.106.110501Google Scholar
  32. V. I. Paulsen: Completely Bounded Maps and Operator Algebras, Cambridge University Press, New York (2002).https://doi.org/10.1017/CBO9780511546631Google Scholar
  33. M. Ramana, A. J. Goldman: Some geometric results in semidefinite programming, J. Global Optim. 7 (1995) 33–50.https://doi.org/10.1007/BF01100204Google Scholar
  34. R. T. Rockafellar: Convex Analysis, Princeton University Press, Princeton (1970).https://doi.org/10.1515/9781400873173Google Scholar
  35. L. Rodman, I. M. Spitkovsky, A. Szkoła, S. Weis: Continuity of the maximum-entropy inference: Convex geometry and numerical ranges approach, J. Math. Phys. 57 (2016) 015204.https://doi.org/10.1063/1.4926965Google Scholar
  36. P. Rostalski, B. Sturmfels: Dualities, in: G. Blekherman, P. Parrilo, R. Thomas (eds.), Semidefinite Optimization and Convex Algebraic Geometry, MOS-SIAM Series on Optimization, Philadelphia (2013) 203–249.https://doi.org/10.1137/1.9781611972290.ch5Google Scholar
  37. R. Sanyal, F. Sottile, B. Sturmfels: Orbitopes, Mathematika 57 (2011) 275–314.https://doi.org/10.1112/S002557931100132XGoogle Scholar
  38. R. Schneider: Convex Bodies: The Brunn-Minkowski Theory, 2nd ed., Cambridge University Press, New York (2014).https://doi.org/10.1017/CBO9781139003858Google Scholar
  39. R. Sinn, B. Sturmfels: Generic spectrahedral shadows, SIAM J Optimization 25 (2015) 1209–1220.https://doi.org/10.1137/140978478Google Scholar
  40. K. Szymański, S. Weis, K. Życzkowski: Classification of joint numerical ranges of three hermitian matrices of size three, arXiv:1603.06569 [math.FA], submitted.Google Scholar
  41. B.-S. Tam: On the duality operator of a convex cone, Linear Algebra Appl. 64 (1985) 33–56.https://doi.org/10.1016/0024-3795(85)90265-4Google Scholar
  42. S. Weis: Quantum convex support, Linear Algebra Appl. 435 (2011) 3168–3188.https://doi.org/10.1016/j.laa.2011.06.004Google Scholar
  43. S. Weis: A note on touching cones and faces, J. Convex Analysis 19 (2012) 323–353.https://doi.org/10.68381/jca19019Google Scholar
  44. S. Weis: Duality of non-exposed faces, J. Convex Analysis 19 (2012) 815–835.https://doi.org/10.68381/jca19042Google Scholar
  45. S. Weis, A. Knauf: Entropy distance: New quantum phenomena, J. Math. Phys. 53 (2012) 102206.https://doi.org/10.1063/1.4757652Google Scholar
  46. S. Weis: Continuity of the maximum-entropy inference, Commun. Math. Phys. 330 (2014) 1263–1292.https://doi.org/10.1007/s00220-014-2090-1Google Scholar
  47. G. M. Ziegler: Lectures on Polytopes, Springer-Verlag, New York (1995).https://doi.org/10.1007/978-1-4613-8431-1Google Scholar

Suggested citation

S. Weis. “Operator Systems and Convex Sets with Many Normal Cones.” Journal of Convex Analysis 25 (2018), No. 1, 41–64. https://doi.org/10.68381/jca25003

Download BibTeXPublished by Heldermann Verlag.

Published by Heldermann Verlag, 2018. Rights now held by Banach Press.