Skip to main content

Multipartite Entanglement Detection Via Projective Tensor Norms


We introduce and study a class of entanglement criteria based on the idea of applying local contractions to an input multipartite state, and then computing the projective tensor norm of the output. More precisely, we apply to a mixed quantum state a tensor product of contractions from the Schatten class \(S_1\) to the Euclidean space \(\ell _2\), which we call entanglement testers. We analyze the performance of this type of criteria on bipartite and multipartite systems, for general pure and mixed quantum states, as well as on some important classes of symmetric quantum states. We also show that previously studied entanglement criteria, such as the realignment and the SIC POVM criteria, can be viewed inside this framework. This allows us to answer in the positive two conjectures of Shang, Asadian, Zhu, and Gühne by deriving systematic relations between the performance of these two criteria.

This is a preview of subscription content, access via your institution.

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5
Fig. 6
Fig. 7


  1. Note that this norm relation holds in general for any tester \({\mathcal {E}}\): \(\Vert {\mathcal {E}}^{\otimes 2}(\rho )\Vert _{\ell _2^{d^2} \otimes _\pi \ell _2^{d^2}} = \Vert {\hat{E}} X {\hat{E}}^*\Vert _1\).


  1. Appleby, D.M., Fuchs, C., Zhu, H.: Group theoretic, Lie algebraic and Jordan algebraic formulations of the SIC existence problem. Quantum Inform. Comput 15, 12 (2013)

    MathSciNet  Google Scholar 

  2. Aubrun, Gu., Szarek, S.: Alice and bob meet banach: the interface of asymptotic geometric analysis and quantum information theory, volume 223. Am. Math. Soc. (2017)

  3. Aubrun, G.: Personal communication, (2020)

  4. Chen, K., Ling-An, W.: A matrix realignment method for recognizing entanglement. Quantum Inform. Comput. 3, 193–202 (2003)

    MathSciNet  Article  Google Scholar 

  5. Derksen, H., Friedland, S., Lim, L.-H., Wang, L.: Theoretical and computational aspects of entanglement. arXiv preprint: arXiv:1705.07160v1, (2017)

  6. Friedland, S., Lim, L.-H.: Nuclear norm of higher-order tensors. Math. Comput. 87(311), 1255–1281 (2018)

    MathSciNet  Article  Google Scholar 

  7. Graydon, M.A., Appleby, D.M.: Quantum conical designs. J. Phys. A: Math. Theor. 49(8), 085301 (2016)

    ADS  MathSciNet  Article  Google Scholar 

  8. Gharibian, S.: Strong NP-hardness of the quantum separability problem. Quantum Inf. Comput. 10(3), 343–360 (2010)

    MathSciNet  MATH  Google Scholar 

  9. Gour, G., Kalev, A.: Construction of all general symmetric informationally complete measurements. J. Phys. A: Math. Theor. 47(33), 335302 (2014)

    MathSciNet  Article  Google Scholar 

  10. Horodecki, M., Horodecki, P.: Reduction criterion of separability and limits for a class of distillation protocols. Phys. Rev. A 59, 4206 (1999)

    ADS  Article  Google Scholar 

  11. Horodecki, M., Horodecki, P., Horodecki, R.: Separability of mixed states: necessary and sufficient conditions. Phys. Lett. A 223(1), 1–8 (1996)

    ADS  MathSciNet  Article  Google Scholar 

  12. Horodecki, M., Horodecki, P., Horodecki, R.: Separability of mixed quantum states: linear contractions and permutation criteria. Open Syst. Inform. Dynam. 13(1), 103–111 (2006)

    MathSciNet  Article  Google Scholar 

  13. Jivulescu, M.A., Nechita, I., Găvruţa, P.: On symmetric decompositions of positive operators. J. Phys. A: Math. Theor. 50(16), 165303 (2017)

    ADS  MathSciNet  Article  Google Scholar 

  14. Johnston, N.: Characterizing operations preserving separability measures via linear preserver problems. Linear and Multilinear Algebra 59(10), 1171–1187 (2011)

    MathSciNet  Article  Google Scholar 

  15. Lai, L.-M., Li, T., Fei, S.-M., Wang, Z.-X.: Entanglement criterion via general symmetric informationally complete measurements. Quantum Inf. Process. 17(11), 314 (2018)

    ADS  MathSciNet  Article  Google Scholar 

  16. Nielsen, M.A., Chuang, I.L.: Quantum computation and quantum information. Cambridge University Press, UK (2010)

    Book  Google Scholar 

  17. Palazuelos, C.: On the largest Bell violation attainable by a quantum state. J. Funct. Anal. 267(7), 1959–1985 (2014)

    MathSciNet  Article  Google Scholar 

  18. Peres, A.: Separability criterion for density matrices. Phys. Rev. Lett. 77(8), 1413 (1996)

    ADS  MathSciNet  Article  Google Scholar 

  19. Pérez-García, D.: Deciding separability with a fixed error. Phys. Lett. A 330(3–4), 149–154 (2004)

    ADS  Article  Google Scholar 

  20. Puchała, Z., Gawron, P., Miszczak, J.A., Skowronek, Ł, Choi, M.-D., Życzkowski, K.: Product numerical range in a space with tensor product structure. Linear Algebra Appl. 434(1), 327–342 (2011)

    MathSciNet  Article  Google Scholar 

  21. Pérez-García, D., Wolf, M.M., Petz, D., Ruskai, M.B.: Contractivity of positive and trace-preserving maps under \({L}_p\) norms. J. Math. Phys. 47(8), 083506 (2006)

    ADS  MathSciNet  Article  Google Scholar 

  22. Rudolph, O.: A separability criterion for density operators. J. Phys. A: Math. Gen. 33(21), 3951 (2000)

    ADS  MathSciNet  Article  Google Scholar 

  23. Rudolph, O.: Computable cross-norm criterion for separability. Lett. Math. Phys. 70, 57–64 (2004)

    ADS  MathSciNet  Article  Google Scholar 

  24. Rudolph, O.: Further results on the cross norm criterion for separability. Quantum Inf. Process. 4(3), 219–239 (2005)

    MathSciNet  Article  Google Scholar 

  25. Ryan, R.A.: Introduction to tensor products of banach spaces. Springer, Berlin (2002)

    Book  Google Scholar 

  26. Sokoli, F., Alber, G.: Generalized schmidt decomposability and its relation to projective norms in multipartite entanglement. J. Phys. A: Math. Theor. 47(32), 325301 (2014)

    MathSciNet  Article  Google Scholar 

  27. Shang, J., Asadian, A., Zhu, H., Gühne, O.: Enhanced entanglement criterion via symmetric informationally complete measurements. Phys. Rev. A 98(2), 022309 (2018)

    ADS  Article  Google Scholar 

  28. Shimony, A.: Degree of entanglement. Ann. N. Y. Acad. Sci. 755(1), 675–679 (1995)

    ADS  MathSciNet  Article  Google Scholar 

  29. Sarbicki, G., Scala, G., Chruściński, D.: Family of multipartite separability criteria based on a correlation tensor. Phys. Rev. A 101, 012341 (2020)

    ADS  MathSciNet  Article  Google Scholar 

  30. Tomczak-Jaegermann, N.: Banach-Mazur distances and finite-dimensional operator ideals, 38. Longman Sc & Tech, (1989)

  31. Watrous, J.: The Theory of Quantum Information. Cambridge University Press, UK (2018)

    Book  Google Scholar 

  32. Werner, R.F.: Quantum states with einstein-podolsky-rosen correlations admitting a hidden-variable mode. Phys. Rev. A 40, 4277 (1989)

    ADS  Article  Google Scholar 

  33. Wei, T.-C., Goldbart, P.M.: Geometric measure of entanglement and applications to bipartite and multipartite quantum states. Phys. Rev. A 68(4), 042307 (2003)

    ADS  Article  Google Scholar 

  34. Zhu, H., Chen, L., Hayashi, M.: Additivity and non-additivity of multipartite entanglement measures. New J. Phys. 12(8), 083002 (2010)

    ADS  Article  Google Scholar 

Download references


MAJ acknowledges the support of Université Toulouse III Paul Sabatier in the form of an invited professorship, during which this work was initiated. We would like to thank Guillaume Aubrun for the extremely valuable help in understanding map factorization questions.

Author information

Authors and Affiliations


Corresponding author

Correspondence to Cécilia Lancien.

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Communicated by Matthias Christandl.

Rights and permissions

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

Cite this article

Jivulescu, M.A., Lancien, C. & Nechita, I. Multipartite Entanglement Detection Via Projective Tensor Norms. Ann. Henri Poincaré (2022).

Download citation

  • Received:

  • Accepted:

  • Published:

  • DOI: