Abstract

We analyze faces generated by points in an arbitrary convex set and their relative algebraic interiors, which are nonempty as we shall prove. We show that by intersecting a convex set with a sublevel or level set of a generalized affine functional, the dimension of the face generated by a point may decrease by at most one. We apply the results to the set of quantum states on a separable Hilbert space. Among others, we show that every state having finite expected values of any two (not necessarily bounded) positive operators admits a decomposition into pure states with the same expected values. We discuss applications in quantum information theory.

Author information

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

Maksim Shirokov

Steklov Mathematical Institute, Moscow, Russia

msh@mi.ras.ru

Keywords

  • Face generated by a point
  • extreme set
  • relative algebraic interior
  • generalized affine constraint
  • extreme point
  • generalized compactness
  • quantum state
  • pure-state decomposition
  • minimal output entropy
  • operator E-norms

Mathematics Subject Classification

52Axx, 47Axx, 81Qxx

References

  1. E. M. Alfsen, F. W. Shultz: State Spaces of Operator Algebras: Basic Theory, Orientations, and C*-Products, Birkhäuser, Boston (2001).https://doi.org/10.1007/978-1-4612-0147-2Google Scholar
  2. A. Barvinok: A Course in Convexity, American Mathematical Society, Providence (2002).https://doi.org/10.1090/gsm/054Google Scholar
  3. S. Becker, N. Datta: Convergence rates for quantum evolution and entropic continuity bounds in infinite dimensions, Commun. Math. Phys. 374 (2020) 823–871.https://doi.org/10.1007/s00220-019-03594-2Google Scholar
  4. R. Garcia-Patron, C. Navarrete-Benlloch, S. Lloyd, J. H. Shapiro, N. J. Cerf: Majorization theory approach to the Gaussian channel minimum entropy conjecture, Phys. Rev. Lett. 108/11 (2012) 110505.https://doi.org/10.1103/PhysRevLett.108.110505Google Scholar
  5. V. Giovannetti, R. Garcia-Patron, N. J. Cerf, A. S. Holevo: Ultimate classical communication rates of quantum optical channels, Nature Photonics 8/10 (2014) 796–800.https://doi.org/10.1038/nphoton.2014.216Google Scholar
  6. J. Hadamard: Sur les caractères de convergence des séries a termes positifs et sur les fonctions indéfiniment croissantes, Acta Math. 18 (1894) 319–336.https://doi.org/10.1007/BF02418282Google Scholar
  7. A. S. Holevo: Entanglement-assisted capacities of constrained quantum channels, Theory Probability Appl. 48/2 (2004) 243–255.https://doi.org/10.1137/S0040585X97980415Google Scholar
  8. A. S. Holevo: Quantum Systems, Channels, Information: A Mathematical Introduction, Studies in Mathematical Physics 16, De Gruyter, Berlin (2012).https://doi.org/10.1515/9783110273403Google Scholar
  9. A. S. Holevo, M. E. Shirokov: Continuous ensembles and the capacity of infinite-dimensional quantum channels, Theory Probability Appl. 50/1 (2006) 86–98.https://doi.org/10.1137/S0040585X97981470Google Scholar
  10. V. Klee, M. Martin: Semicontinuity of the face-function of a convex set, Comm. Math. Helvetici 46/1 (1971) 1–12.https://doi.org/10.1007/BF02566824Google Scholar
  11. K. Knopp: Theory and Application of Infinite Series, Dover Publications, New York (1990).Google Scholar
  12. G. Lindblad: Expectations and entropy inequalities for finite quantum systems, Comm. Math. Phys. 39/2 (1974) 111–119.https://doi.org/10.1007/BF01608390Google Scholar
  13. A. Mari, V. Giovannetti, A. S. Holevo: Quantum state majorization at the output of bosonic Gaussian channels, Nat. Comm. 5/1 (2014) 3826.https://doi.org/10.1038/ncomms4826Google Scholar
  14. L. Memarzadeh, S. Mancini: Minimum output entropy of a non-Gaussian quantum channel, Phys. Rev. A 94/2 (2016) 022341.https://doi.org/10.1103/PhysRevA.94.022341Google Scholar
  15. R. C. O’Brien: On the openness of the barycentre map, Math. Annalen 223/3 (1976) 207–212.https://doi.org/10.1007/BF01360953Google Scholar
  16. S. Papadopoulou: On the geometry of stable compact convex sets, Math. Annalen 229/3 (1977) 193–200.https://doi.org/10.1007/BF01391464Google Scholar
  17. V. Yu. Protasov, M. E. Shirokov: Generalized compactness in linear spaces and its applications, Sbornik Math. 200/5 (2009) 697–722.https://doi.org/10.1070/SM2009v200n05ABEH004016Google Scholar
  18. R. T. Rockafellar: Convex Analysis, Princeton University Press, Princeton (1970).https://doi.org/10.1515/9781400873173Google Scholar
  19. N. M. Roy: Extreme points of convex sets in infinite dimensional spaces, Amer. Math. Monthly 94/5 (1987) 409–422.https://doi.org/10.1080/00029890.1987.12000657Google Scholar
  20. K. Schmüdgen: Unbounded Self-adjoint Operators on Hilbert Space, Springer, Dordrecht (2012).https://doi.org/10.1007/978-94-007-4753-1Google Scholar
  21. R. Schneider: Convex Bodies: The Brunn-Minkowski Theory, 2nd expanded ed., Cambridge University Press, New York (2014).Google Scholar
  22. M. E. Shirokov: On properties of the space of quantum states and their application to the construction of entanglement monotones, Izvestiya Math. 74/4 (2010) 849–882.https://doi.org/10.1070/IM2010v074n04ABEH002510Google Scholar
  23. M. E. Shirokov: Operator E-norms and their use, Sb. Math. 211/9 (2020) 1323–1353.https://doi.org/10.1070/SM9336Google Scholar
  24. M. E. Shirokov, A. S. Holevo: On approximation of infinite-dimensional quantum channels, Problems Inf. Transmission 44/2 (2008) 73–90.https://doi.org/10.1134/S0032946008020014Google Scholar
  25. P. W. Shor: Equivalence of additivity questions in quantum information theory, Comm. Math. Phys. 246/3 (2004) 453–472.https://doi.org/10.1007/s00220-003-0981-7Google Scholar
  26. J. Watrous: The Theory of Quantum Information, Cambridge University Press, Cambridge (2018).https://doi.org/10.1017/9781316848142Google Scholar
  27. A. Wehrl: General properties of entropy, Rev. Mod. Phys. 50/2 (1978) 221–260.https://doi.org/10.1103/RevModPhys.50.221Google Scholar
  28. M. M. Wilde: Quantum Information Theory, 2nd ed., Cambridge University Press, Cambridge (2017).Google Scholar
  29. M. M. Wilde, H. Qi: Energy-constrained private and quantum capacities of quantum channels, IEEE Trans. Inf. Theory 64/12 (2018) 7802–7827.https://doi.org/10.1109/TIT.2018.2854766Google Scholar
  30. A. Winter: Energy-constrained diamond norm with applications to the uniform continuity of continuous variable channel capacities, arXiv: 1712.10267 (2017).Google Scholar

Suggested citation

S. Weis, M. Shirokov. “The Face Generated by a Point, Generalized Affine Constraints, and Quantum Theory.” Journal of Convex Analysis 28 (2021), No. 3, 847–870. https://doi.org/10.68381/jca28049

Download BibTeXPublished by Heldermann Verlag.

Published by Heldermann Verlag, 2021. Rights now held by Banach Press.