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On Strong Superadditivity of the Entanglement of Formation

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Abstract

We employ a basic formalism from convex analysis to show a simple relation between the entanglement of formation E F and the conjugate function E * of the entanglement function E(ρ)=S(Tr A ρ). We then consider the conjectured strong superadditivity of the entanglement of formation E F (ρ)≥E F I )+E F II ), where ρ I and ρ II are the reductions of ρ to the different Hilbert space copies, and prove that it is equivalent with subadditivity of E *. Furthermore, we show that strong superadditivity would follow from multiplicativity of the maximal channel output purity for quantum filtering operations, when purity is measured by Schatten p-norms for p tending to 1.

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References

  1. Hayden, P.M., Horodecki, M., Terhal, B.M.: J. Phys. A 34(35), 6891–6898 (2001)

    Article  MATH  Google Scholar 

  2. Wootters, W. : Phys. Rev. Lett. 80, 2245 (1998)

    Article  Google Scholar 

  3. Benatti, F., Narnhofer, H.: Phys. Rev. A 63, 042306 (2001)

    Article  Google Scholar 

  4. Vollbrecht, KG.H., Werner, R.F.: Phys. Rev. A 64, 062307 (2001)

    Article  Google Scholar 

  5. Vidal, G., Dür, W., Cirac, J.I.: Phys. Rev. Lett. 89, 027901 (2002)

    Article  Google Scholar 

  6. Horodecki, M., Sen De, A. Sen, U.: quant-ph/0207031, 2002

  7. Matsumoto, K., Shimono, T., Winter, A.: quant-ph/0206148, 2002

  8. Heng Fan: quant-ph/0210169, 2002

  9. Rockafellar, R.T.: Convex Analysis. Princeton, NJ: Princeton University Press, 1970

  10. Boyd, S., Vandenberghe, L.: Convex Optimization. Available online at http://www.stanford.edu/∼boyd/cvxbook.html, 2002

  11. Bennett, C.H., DiVincenzo, D.P., Smolin, J., Wootters, W.K.: Phys. Rev. A 54, 3824 (1996)

    Article  MathSciNet  Google Scholar 

  12. Lockhart, R.B.: J. Math. Phys. 41(10), 6766–6771 (2000)

    Article  MATH  Google Scholar 

  13. Audenaert, K.M.R., Verstraete, F., DeMoor, B.: Phys. Rev. A 64, 052304 (2001)

    Article  Google Scholar 

  14. Uhlmann, A.: quant-ph/9704017, 1997

  15. Hiai, F., Petz, D.: Lin. Alg. Appl. 181, 153–185 (1993)

    Article  MathSciNet  MATH  Google Scholar 

  16. Ando, T., Hiai, F.: Lin. Alg. Appl. 197, 198, 113–131 (1994)

    MATH  Google Scholar 

  17. Apostol, T.M.: Mathematical Analysis. Reading MA: Addison-Wesley, 1974

  18. Amosov, G.G., Holevo, A.S., Werner, R.F.: Problems in Information Transmission 36, 25–34 (2000) and math-ph/0003002 (2000)

  19. Werner, R.F., Holevo, A.S.: J. Math. Phys. 43(9), 4353–4357 (2002)

    Article  Google Scholar 

  20. King, C.: quant-ph/0212057, 2002

  21. Horn, R.A., Johnson, C.R.: Topics in Matrix Analysis. Cambridge: Cambridge University Press, 1991

  22. King, C.: J. Math. Phys. 43(9), 4334–4340 (2002)

    Article  Google Scholar 

  23. King, C.: quant-ph/0204172 (2002)

  24. Shor, P.W.: quant-ph/0305035 (2003)

Download references

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Correspondence to Koenraad M.R. Audenaert.

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Communicated by M.B. Ruskai

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Audenaert, K., Braunstein, S. On Strong Superadditivity of the Entanglement of Formation. Commun. Math. Phys. 246, 443–452 (2004). https://doi.org/10.1007/s00220-003-0987-1

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  • DOI: https://doi.org/10.1007/s00220-003-0987-1

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