Annales Henri Poincaré

, Volume 19, Issue 8, pp 2491–2511 | Cite as

Existence of Locally Maximally Entangled Quantum States via Geometric Invariant Theory

  • Jim Bryan
  • Zinovy Reichstein
  • Mark Van Raamsdonk


We study a question which has natural interpretations both in quantum mechanics and in geometry. Let \(V_{1},\cdots , V_{n}\) be complex vector spaces of dimension \(d_{1},\ldots ,d_{n}\) and let \(G= {\text {SL}}_{d_{1}} \times \cdots \times {\text {SL}}_{d_{n}}\). Geometrically, we ask: Given \((d_{1},\ldots ,d_{n})\), when is the geometric invariant theory quotient \(\mathbb {P}(V_{1}\otimes \cdots \otimes V_{n})/\!/G\) non-empty? This is equivalent to the quantum mechanical question of whether the multipart quantum system with Hilbert space \(V_{1}\otimes \cdots \otimes V_{n}\) has a locally maximally entangled state, i.e., a state such that the density matrix for each elementary subsystem is a multiple of the identity. We show that the answer to this question is yes if and only if \(R(d_{1},\cdots ,d_{n})\geqslant 0\) where
$$\begin{aligned} R(d_{1},\cdots ,d_{n}) = \prod _{i}d_{i} +\sum _{k=1}^{n} (-1)^{k}\sum _{1\leqslant i_{1}<\cdots <i_{k}\leqslant n} \left( \gcd (d_{i_{1}},\cdots ,d_{i_{k}}) \right) ^{2}. \end{aligned}$$
We also provide a simple recursive algorithm which determines the answer to the question, and we compute the dimension of the resulting quotient in the non-empty cases.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Andreev, E.M., Vinberg, E.B., Elashvili, A.G.: Orbits of highest dimension of semisimple linear Lie groups. Funkcional. Anal. i Priložen. 1(4), 3–7 (1967). English translation in Functional Anal. Appl. 1, 257–261 (1967)Google Scholar
  2. 2.
    Arnaud, L., Cerf, N.J.: Exploring pure quantum states with maximally mixed reductions. Phys. Rev. A 87, 012319 (2013)ADSCrossRefGoogle Scholar
  3. 3.
    Bryan, J., Leutheusser, S., Reichstein, Z., Van Raamsdonk, M.: Locally maximally entangled states of multipart quantum systems. arXiv:1801.03508
  4. 4.
    Elashvili, A.G.: Stationary subalgebras of points of general position for irreducible linear Lie groups. Funkcional. Anal. i Priložen. 6(2), 65–78 (1972). English translation in Functional Anal. Appl. 6, 139–148 (1972)Google Scholar
  5. 5.
    Facchi, P., Florio, G., Parisi, G., Pascazio, S.: Maximally multipartite entangled states. Phys. Rev. A 77(6), 060304 (2008)ADSMathSciNetCrossRefGoogle Scholar
  6. 6.
    Gelfand, I.M., Kapranov, M.M., Zelevinsky, A.V.: Discriminants, Resultants, and Multidimensional Determinants. Mathematics: Theory and Applications. Birkhäuser Boston, Inc., Boston (1994)CrossRefzbMATHGoogle Scholar
  7. 7.
    Gour, G., Wallach, N.R.: Classification of multipartite entanglement of all finite dimensionality. Phys. Rev. Lett. 111, 060502 (2013)ADSCrossRefGoogle Scholar
  8. 8.
    Goyeneche, D., Bielawski, J., Zyczkowski, K.: Multipartite entanglement in heterogeneous systems. Phys. Rev. A 94, 012346 (2016)ADSCrossRefGoogle Scholar
  9. 9.
    Hoskins, V.: Geometric Invariant Theory and Symplectic Quotients. Lecture Notes (2012). Accessed May 2017
  10. 10.
    Klyachko, A.A.: Coherent states, entanglement, and geometric invariant theory (2002). arXiv:quant-ph/0206012
  11. 11.
    Klyachko, A.A.: Quantum marginal problem and representations of the symmetric group (2004). arXiv:quant-ph/0409113
  12. 12.
    Klyachko, A.A.: Dynamical symmetry approach to entanglement. In: Physics and Theoretical Computer Science, Volume 7 of NATO Secur. Sci. Ser. D Inf. Commun. Secur., pp. 25–54. IOS, Amsterdam (2007). arXiv:0802.4008v1
  13. 13.
    Li, J.-L., Qiao, C.-F.: Classification of the entangled states \(2\times M\times N\). Quantum Inf. Process. 12(1), 251–268 (2013)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    Littelmann, P.: Koreguläre und äquidimensionale Darstellungen. J. Algebra 123(1), 193–222 (1989)MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    Miyake, A., Verstraete, F.: Multipartite entanglement in \(2\times 2\times n\) quantum systems. Phys. Rev. A 69, 012101 (2004)ADSMathSciNetCrossRefGoogle Scholar
  16. 16.
    Mumford, D., Fogarty, J., Kirwan, F.: Geometric Invariant Theory, Third edition, Ergebnisse der Mathematik und ihrer Grenzgebiete (2), 34, Springer, Berlin (1994)Google Scholar
  17. 17.
    Popov, V.L.: Criteria for the stability of the action of a semisimple group on the factorial of a manifold. Izv. Akad. Nauk SSSR Ser. Mat. 34, 523–531 (1970)MathSciNetGoogle Scholar
  18. 18.
    Popov, V.L., Vinberg, E.B.: Invariant theory. In: Algebraic Geometry, IV, Encyclopaedia of Mathematical Sciences, 55, Springer, Berlin, pp. 123–284 (1994)Google Scholar
  19. 19.
    Reid, M.: Graded Rings and Varieties in Projective Space. Accessed May 2017
  20. 20.
    Rosenlicht, M.: Some basic theorems on algebraic groups. Am. J. Math. 78, 401–443 (1956)MathSciNetCrossRefzbMATHGoogle Scholar
  21. 21.
    Richardson Jr., R.W.: Principal orbit types for algebraic transformation spaces in characteristic zero. Invent. Math. 16, 6–14 (1972)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  22. 22.
    Sato, M., Kimura, T.: A classification of irreducible prehomogeneous vector spaces and their relative invariants. Nagoya Math. J. 65, 1–155 (1977)MathSciNetCrossRefzbMATHGoogle Scholar
  23. 23.
    Scott, A.J.: Multipartite entanglement, quantum-error-correcting codes, and entangling power of quantum evolutions. Phys. Rev. A 69, 052330 (2004)ADSCrossRefGoogle Scholar
  24. 24.
    Wallach, N.R.: Quantum computing and entanglement for mathematicians. In: Representation Theory and Complex Analysis, Lectures Given at the C.I.M.E. Summer School held in Venice, Italy, June 10–17, 2004, pp. 345–376, Lecture Notes in Mathematices, 1931. Springer, Berlin (2008)Google Scholar
  25. 25.
    Walter, M.: Multipartite Quantum States and their Marginals. Ph.D. Thesis, ETH Zurich (2014). arXiv:1410.6820
  26. 26.
    Wang, S., Lu, Y., Long, G.-L.: Entanglement classification of \(2\times 2 \times 2 \times d\) quantum systems via the ranks of the multiple coefficient matrices. Phys. Rev. A 87, 062305 (2013)ADSCrossRefGoogle Scholar
  27. 27.
    Yu, C.-S., Zhou, L., Song, H.-S.: Genuine tripartite entanglement monotone of \((2\otimes 2\otimes n)\)-dimensional systems. Phys. Rev. A 77, 022313 (2008)ADSMathSciNetCrossRefGoogle Scholar

Copyright information

© Springer International Publishing AG, part of Springer Nature 2018

Authors and Affiliations

  • Jim Bryan
    • 1
  • Zinovy Reichstein
    • 1
  • Mark Van Raamsdonk
    • 2
  1. 1.Department of MathematicsUniversity of British ColumbiaVancouverCanada
  2. 2.Department of Physics and AstronomyUniversity of British ColumbiaVancouverCanada

Personalised recommendations