Communications in Mathematical Physics

, Volume 352, Issue 3, pp 881–904

An Improved Semidefinite Programming Hierarchy for Testing Entanglement



We present a stronger version of the Doherty–Parrilo–Spedalieri (DPS) hierarchy of approximations for the set of separable states. Unlike DPS, our hierarchy converges exactly at a finite number of rounds for any fixed input dimension. This yields an algorithm for separability testing that is singly exponential in dimension and polylogarithmic in accuracy. Our analysis makes use of tools from algebraic geometry, but our algorithm is elementary and differs from DPS only by one simple additional collection of constraints.


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  1. 1.
    Grötschel, M., Lovász, L. Schrijver, A.: Geometric Algorithms and Combinatorial Optimization, Algorithms and Combinatorics, vol. 2, second corrected edition edn., Springer (1993)Google Scholar
  2. 2.
    Liu, Y.K.: The complexity of the consistency and N-representability problems for quantum states. Ph.D. thesis, University of California, San Diego (2007) arXiv:0712.3041
  3. 3.
    Harrow, A.W., Montanaro, A.: Testing product states, quantum Merlin–Arthur games and tensor optimization. J. ACM 60(1), 3:1 (2013). arXiv:1001.0017
  4. 4.
    Beigi, S., Shor, P.W.: Approximating the set of separable states using the positive partial transpose test. J. Math. Phys. 51(4), 042202 (2010) arXiv:0902.1806
  5. 5.
    Gall, F.L., Nakagawa, S., Nishimura, H.: On QMA protocols with two short quantum proofs. Q. Inf. Comp. 12, 589 (2012) arXiv:1108.4306
  6. 6.
    Cubitt, T.S., Perez-Garcia, D., Wolf, M.: Undecidability of the spectral gap problem (2014). (In preparation) Google Scholar
  7. 7.
    Ito, T., Kobayashi, H., Watrous, J.: Quantum Interactive Proofs with Weak Error Bounds. In: Proceedings of the 3rd Innovations in Theoretical Computer Science Conference (2012), ITCS ’12, pp. 266–275. arXiv:1012.4427
  8. 8.
    Basu S., Pollack R., Roy M.F.: On the combinatorial and algebraic complexity of quantifier elimination. J. ACM 43, 1002 (1996)MathSciNetCrossRefMATHGoogle Scholar
  9. 9.
    Doherty, A.C., Parrilo, P.A., Spedalieri, F.M.: A complete family of separability criteria (2003) arXiv:quant-ph/0308032
  10. 10.
    Barak, B., Steurer D.: Sum-of-squares proofs and the quest toward optimal algorithms (2014) arXiv:1404.5236
  11. 11.
    Nie J.: An exact Jacobian SDP relaxation for polynomial optimization. Math. Program. 137, 225 (2013) arXiv:1006.2418 MathSciNetCrossRefMATHGoogle Scholar
  12. 12.
    Gharibian S.: Strong NP-hardness of the quantum separability problem. QIC 10, 343 (2010) arXiv:0810.4507 MathSciNetMATHGoogle Scholar
  13. 13.
    Aaronson, S., Impagliazzo, R., Moshkovitz, D.: AM with multiple merlins. In: Computational Complexity (CCC), 2014 IEEE 29th Conference on (2014), pp. 44–55. arXiv:1401.6848
  14. 14.
    Barak, B., Brandão, F.G.S.L., Harrow, A.W., Kelner, J., Steurer, D., Zhou, Y.: Hypercontractivity, sum-of-squares proofs, and their applications. In: Proceedings of the 44th symposium on Theory of Computing (2012), STOC ’12, pp. 307–326 arXiv:1205.4484
  15. 15.
    Impagliazzo, R., Paturi, R., Zane, F.: Which problems have strongly exponential complexity?. In: Foundations of Computer Science, 1998. Proceedings. 39th Annual Symposium on (IEEE, 1998), pp. 653–662Google Scholar
  16. 16.
    Gurvits, L.: Classical deterministic complexity of Edmonds’ problem and quantum entanglement (2003) arXiv:quant-ph/0303055
  17. 17.
    Blier, H., Tapp, A.: All languages in NP have very short quantum proofs. In: First International Conference on Quantum, Nano, and Micro Technologies. IEEE Computer Society, Los Alamitos, CA, USA, (2009), pp. 34–37 arXiv:0709.0738
  18. 18.
    Chiesa, A., Forbes, M.A.: Improved soundness for QMA with multiple provers. Chic. J. Theor. Comput. Sci. 2013(1) (2013) arXiv:1108.2098
  19. 19.
    Navascués M., Owari M., Plenio M.B.: Power of symmetric extensions for entanglement detection. Phys. Rev. A 80, 052306 (2009) arXiv:0906.2731 ADSCrossRefMATHGoogle Scholar
  20. 20.
    Aaronson, S., Beigi, S., Drucker, A., Fefferman, B., Shor, P.: The power of unentanglement. Annual IEEE Conference on Computational Complexity 0, 223 (2008) arXiv:0804.0802
  21. 21.
    Chen, J., Drucker, A.: Short multi-prover quantum proofs for SAT without entangled measurements (2010) arXiv:1011.0716
  22. 22.
    Brandão, F.G.S.L., Christandl, M., Yard, J.: Faithful squashed entanglement. Comm. Math. Phys. 306, 805 (2011) arXiv:1010.1750
  23. 23.
    Li, K., Winter, A.: Relative entropy and squashed entanglement. Comm. Math. Phys. 326, 63 (2014) arXiv:1210.3181
  24. 24.
    Brandão, F.G.S.L., Harrow, A.W.: Quantum de Finetti theorems under local measurements with applications. In: Proceedings of the 45th annual ACM Symposium on theory of computing. (2013), STOC ’13, pp. 861–870 arXiv:1210.6367
  25. 25.
    Shi, Y., Wu, X.: Epsilon-net method for optimizations over separable states. In: ICALP12. Springer, (2012), pp. 798–809 arXiv:1112.0808
  26. 26.
    Brandão, F.G., Harrow, A.W.: Estimating injective tensor norms using nets (2014). (In preparation)Google Scholar
  27. 27.
    Li, K., Smith, G.: Quantum de Finetti theorem measured with fully one-way LOCC norm (2014) arXiv:1408.6829
  28. 28.
    Putinar M.: Positive polynomials on compact semi-algebraic sets. Indiana Univ. Math. J. 42, 969 (1993)MathSciNetCrossRefMATHGoogle Scholar
  29. 29.
    Buhrman, H., Regev, O., Scarpa, G., de Wolf, R.: Near-optimal and explicit Bell inequality violations. In: Proceedings of the 2011 IEEE 26th Annual Conference on Computational Complexity. (2011), CCC ’11, pp. 157–166. arXiv:1012.5043
  30. 30.
    Klerk, E., Laurent, M., Parrilo, P.: On the equivalence of algebraic approaches to the minimization of forms on the simplex. In: Henrion, D., Garulli, A. (eds.) Positive Polynomials in Control, Lecture Notes in Control and Information Science, vol. 312, Springer Berlin Heidelberg, (2005), pp. 121–132.Google Scholar
  31. 31.
    Löfberg, J.: YALMIP : A toolbox for modeling and optimization in MATLAB. In: In Proceedings of the CACSD Conference Taipei, Taiwan, (2004)Google Scholar
  32. 32.
    Löfberg J.: Pre- and post-processing sum-of-squares programs in practice. IEEE Trans. Autom. Control 54, 1007 (2009)MathSciNetCrossRefGoogle Scholar
  33. 33.
    MOSEK ApS: The MOSEK optimization toolbox for MATLAB manual. Version 7.1 (Revision 28). (2015)Google Scholar
  34. 34.
    Permenter, F., Parrilo, P.: Partial facial reduction: simplified, equivalent SDPs via approximations of the PSD cone (2014) arXiv:1408.4685
  35. 35.
    Laurent, M.: Sums of squares, Moment matrices and optimization over polynomials. In: Putinar, M., Sullivant, S. (eds) Emerging Applications of Algebraic Geometry, The IMA Volumes in Mathematics and its Applications, vol. 149, Springer New York, (2009), pp. 157–270Google Scholar
  36. 36.
    Nie J., Ranestad K.: Algebraic degree of polynomial optimization. SIAM J. Optim. 20, 485 (2009). doi:10.1137/080716670 MathSciNetCrossRefMATHGoogle Scholar
  37. 37.
    Trnovská M.: Strong duality conditions in semidefinite programming. J. Electr. Eng. 56, 1 (2005)MATHGoogle Scholar
  38. 38.
    Cédric Josz, C.(INRIA), Henrion, Didier: (LAAS. Strong duality in Lasserre’s hierarchy for polynomial optimization. Optim. lett. 10, 3 (2016) arXiv:1405.7334
  39. 39.
    Shapiro, A.: First and second order analysis of nonlinear semidefinite programs. Mathematical Programming pp. 301 (1997)Google Scholar
  40. 40.
    Strelchuk S., Oppenheim J.: Hybrid zero-capacity channels. Phys. Rev. A 86, 022328 (2012) arXiv:1207.1084 ADSCrossRefGoogle Scholar
  41. 41.
    Pereszlényi, A.: Multi-prover quantum Merlin–Arthur proof systems with small gap. (2012) arXiv:1205.2761
  42. 42.
    Cox, D., little, J., O’Shea, D.: Ideals, Varieties, and Algorithms: An Introduction to Computational Algebraic Geometry and Commutative Algebra, second edition edn. Undergarduate texts in mathematics. Springer, (1996)Google Scholar
  43. 43.
    Harris, J.: Algebraic geometry: a first course. Graduate texts in mathematics. Springer, (1992)Google Scholar
  44. 44.
    Buchberger B.: Ein algorithmisches Kriterum für die Lösbarkeit eines algebraisches Gleichungssystems. Aequationes Mathematicae 4, 374 (1970)MathSciNetCrossRefMATHGoogle Scholar
  45. 45.
    Mayr, E.W., Ritscher, S.: Degree bounds for GröBner bases of low-dimensional polynomial ideals. In: Proceedings of the 2010 International Symposium on Symbolic and Algebraic Computation. ACM, New York, NY, USA, (2010), ISSAC ’10, pp. 21–27.Google Scholar

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© Springer-Verlag Berlin Heidelberg 2017

Authors and Affiliations

  1. 1.MIT Center for Theoretical PhysicsCambridgeUSA
  2. 2.Computer and Information Science DepartmentUniversity of OregonEugeneUSA

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