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Multipartite Entanglement Detection Via Projective Tensor Norms

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Abstract

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.

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Notes

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

References

  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)

    Article  MathSciNet  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)

    Article  MathSciNet  Google Scholar 

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

    Article  ADS  MathSciNet  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)

    Article  MathSciNet  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)

    Article  ADS  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)

    Article  ADS  MathSciNet  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)

    Article  MathSciNet  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)

    Article  ADS  MathSciNet  Google Scholar 

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

    Article  MathSciNet  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)

    Article  ADS  MathSciNet  Google Scholar 

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

    MATH  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

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

    Article  ADS  MathSciNet  Google Scholar 

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

    Article  ADS  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)

    Article  MathSciNet  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)

    Article  ADS  MathSciNet  Google Scholar 

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

    Article  ADS  MathSciNet  Google Scholar 

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

    Article  ADS  MathSciNet  Google Scholar 

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

    Article  MathSciNet  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)

    Article  MathSciNet  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)

    Article  ADS  Google Scholar 

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

    Article  ADS  MathSciNet  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)

    Article  ADS  MathSciNet  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)

    Article  ADS  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)

    Article  ADS  Google Scholar 

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

    Article  ADS  Google Scholar 

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Acknowledgements

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.

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Authors and Affiliations

  1. Department of Mathematics, Politehnica University of Timişoara, Victoriei Square 2, 300006, Timişoara, Romania

    Maria Anastasia Jivulescu

  2. Institut Fourier & CNRS, Université Grenoble Alpes, 118 rue des maths, 38610, Gières, France

    Cécilia Lancien

  3. Laboratoire de Physique Théorique & CNRS, Université Paul Sabatier, 118 route de Narbonne, F-31062, Toulouse Cedex 9, France

    Ion Nechita

Authors
  1. Maria Anastasia Jivulescu
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  2. Cécilia Lancien
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  3. Ion Nechita
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Correspondence to Cécilia Lancien.

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Communicated by Matthias Christandl.

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Jivulescu, M.A., Lancien, C. & Nechita, I. Multipartite Entanglement Detection Via Projective Tensor Norms. Ann. Henri Poincaré 23, 3791–3838 (2022). https://doi.org/10.1007/s00023-022-01187-9

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  • DOI: https://doi.org/10.1007/s00023-022-01187-9

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