- Physical Review Journals
All JournalsPhysics Magazine

Physical Review A

Reuse & Permissions

It is not necessary to obtain permission to reuse this article or its components as it is available under the terms of the Creative Commons Attribution 4.0 International license. This license permits unrestricted use, distribution, and reproduction in any medium, provided attribution to the author(s) and the published article's title, journal citation, and DOI are maintained. Please note that some figures may have been included with permission from other third parties. It is your responsibility to obtain the proper permission from the rights holder directly for these figures.

Export citation

Export citation

Choose format for download:

Download Citation
  • Open Access

Nonexistence of maximally entangled mixed states for a fixed spectrum

Gonzalo Camacho1,* and Julio I. de Vicente2,3,†

  • *Contact author: gonzalo.camacho@dlr.de
  • †Contact author: jdvicent@math.uc3m.es
PDF

Phys. Rev. A 113, 022416 – Published 10 February, 2026

DOI: https://doi.org/10.1103/2m8l-kfnm

Abstract

The existence of a maximally entangled pure state is a cornerstone result of entanglement theory that has paramount consequences in quantum information theory. A natural generalization of this property is to consider whether a notion of maximal entanglement is possible among all states with the same spectrum (where the aforementioned case of pure states corresponds to the particular choice in which the spectrum is a delta distribution, i.e., rank-1 states). Despite positive evidence in the past that such a notion might exist at least in the case of two-qubit states, it was recently shown by de Vicente [Phys. Rev. Lett. 133, 050202 (2024)] that the answer to the above question is negative. This reference proved this for particular choices of the spectrum in the case of rank-2 two-qubit density matrices. While this settles the problem in general, it still leaves open whether there are other choices of the spectrum outside the case of pure states where a maximally entangled state for a fixed spectrum might exist. In this work, we extend this impossibility result to all rank-2 and rank-3 two-qubit states as well as for a large class of eigenvalue distributions in the case where the rank equals four.

View figure in article

Physics Subject Headings (PhySH)

Article Text

References (44)

  1. M. B. Plenio and S. Virmani, Quantum Inf. Comput. 7, 1 (2007).
  2. R. Horodecki, P. Horodecki, M. Horodecki, and K. Horodecki, Rev. Mod. Phys. 81, 865 (2009).
  3. C. Eltschka and J. Siewert, J. Phys. A: Math. Theor. 47, 424005 (2014).
  4. E. Chitambar and G. Gour, Rev. Mod. Phys. 91, 025001 (2019).
  5. M. A. Nielsen, Phys. Rev. Lett. 83, 436 (1999).
  6. W. Dür, G. Vidal, J. I. Cirac, N. Linden, and S. Popescu, Phys. Rev. Lett. 87, 137901 (2001).
  7. B. Kraus and J. I. Cirac, Phys. Rev. A 63, 062309 (2001).
  8. F. Verstraete, K. Audenaert, and B. De Moor, Phys. Rev. A 64, 012316 (2001).
  9. C. H. Bennett, D. P. DiVincenzo, J. A. Smolin, and W. K. Wootters, Phys. Rev. A 54, 3824 (1996).
  10. V. Vedral, M. B. Plenio, M. A. Rippin, and P. L. Knight, Phys. Rev. Lett. 78, 2275 (1997).
  11. G. Vidal and R. F. Werner, Phys. Rev. A 65, 032314 (2002).
  12. J. I. de Vicente, Phys. Rev. Lett. 133, 050202 (2024).
  13. G. Vidal, Phys. Rev. Lett. 83, 1046 (1999).
  14. G. Vidal, D. Jonathan, and M. A. Nielsen, Phys. Rev. A 62, 012304 (2000).
  15. D. Jonathan and M. B. Plenio, Phys. Rev. Lett. 83, 1455 (1999).
  16. G. Vidal, Phys. Rev. A 62, 062315 (2000).
  17. A. W. Harrow and M. A. Nielsen, Phys. Rev. A 68, 012308 (2003).
  18. F. G. S. L. Brandão and M. B. Plenio, Nat. Phys. 4, 873 (2008).
  19. F. G. S. L. Brandão and M. B. Plenio, Commun. Math. Phys. 295, 829 (2010).
  20. F. G. S. L. Brandão and N. Datta, IEEE Trans. Inf. Theory 57, 1754 (2011).
  21. P. Contreras-Tejada, C. Palazuelos, and J. I. de Vicente, Phys. Rev. Lett. 122, 120503 (2019).
  22. E. Chitambar, J. I. de Vicente, M. W. Girard, and G. Gour, J. Math. Phys. 61, 042201 (2020).
  23. L. Lami and B. Regula, Nat. Phys. 19, 184 (2023).
  24. M. Berta, F. G. S. L. Brandão, G. Gour, L. Lami, M. B. Plenio, B. Regula, and M. Tomamichel, Quantum, 7, 1103 (2023).
  25. L. Lami and B. Regula, Nat. Commun. 15, 10120 (2024).
  26. E. M. Rains, arXiv:quant-ph/9707002.
  27. E. M. Rains, Phys. Rev. A 60, 173 (1999).
  28. V. Gheorghiu and R. B. Griffiths, Phys. Rev. A 78, 020304(R) (2008).
  29. E. Chitambar and R. Duan, Phys. Rev. Lett. 103, 110502 (2009).
  30. G. Gour and N. R. Wallach, New J. Phys. 13, 073013 (2011).
  31. M. Hebenstreit, C. Spee, and B. Kraus, Phys. Rev. A 93, 012339 (2016).
  32. M. Hebenstreit, M. Englbrecht, C. Spee, J. I. de Vicente, and B. Kraus, New J. Phys. 23, 033046 (2021).
  33. S. Boyd and L. Vandenberghe, Convex Optimization (Cambridge University Press, Cambridge, 2004).
  34. P. Horodecki, Phys. Lett. A 232, 333 (1997).
  35. A. Peres, Phys. Rev. Lett. 77, 1413 (1996).
  36. M. Horodecki, P. Horodecki, and R. Horodecki, Phys. Lett. A 223, 1 (1996).
  37. J. Watrous, The Theory of Quantum Information (Cambridge University Press, Cambridge, 2018).
  38. E. Chitambar, D. Leung, L. Mancinska, M. Ozols, and A. Winter, Commun. Math. Phys. 328, 303 (2014).
  39. R. Horodecki and M. Horodecki, Phys. Rev. A 54, 1838 (1996).
  40. M. Horodecki and P. Horodecki, Phys. Rev. A 59, 4206 (1999).
  41. A. Sanpera, R. Tarrach, and G. Vidal, Phys. Rev. A 58, 826 (1998).
  42. W. Dür, G. Vidal, and J. I. Cirac, Phys. Rev. A 62, 062314 (2000).
  43. https://qci.dlr.de/iqda/.
  44. G. Camacho and J. I. de Vicente, Figure data for Nonexistence of maximally entangled mixed states for a fixed spectrum [Data set], Zenodo (2026), doi: https://doi.org/10.5281/zenodo.18373612.

Outline

Information

Phys. Rev. A 113, 022416– Published 10 February, 2026
  • Received 21 November 2025
  • Accepted 22 January 2026
Crossmark - Check for updates button
Creative Commons Logo

Published by the American Physical Society under the terms of the Creative Commons Attribution 4.0 International license. Further distribution of this work must maintain attribution to the author(s) and the published article's title, journal citation, and DOI.

Published by the American Physical Society

Sign In to Your Journals Account

Filter

Filter

Article Lookup

Enter a citation