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 (ρ I )+E F (ρ 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.
Similar content being viewed by others
References
Hayden, P.M., Horodecki, M., Terhal, B.M.: J. Phys. A 34(35), 6891–6898 (2001)
Wootters, W. : Phys. Rev. Lett. 80, 2245 (1998)
Benatti, F., Narnhofer, H.: Phys. Rev. A 63, 042306 (2001)
Vollbrecht, KG.H., Werner, R.F.: Phys. Rev. A 64, 062307 (2001)
Vidal, G., Dür, W., Cirac, J.I.: Phys. Rev. Lett. 89, 027901 (2002)
Horodecki, M., Sen De, A. Sen, U.: quant-ph/0207031, 2002
Matsumoto, K., Shimono, T., Winter, A.: quant-ph/0206148, 2002
Heng Fan: quant-ph/0210169, 2002
Rockafellar, R.T.: Convex Analysis. Princeton, NJ: Princeton University Press, 1970
Boyd, S., Vandenberghe, L.: Convex Optimization. Available online at http://www.stanford.edu/∼boyd/cvxbook.html, 2002
Bennett, C.H., DiVincenzo, D.P., Smolin, J., Wootters, W.K.: Phys. Rev. A 54, 3824 (1996)
Lockhart, R.B.: J. Math. Phys. 41(10), 6766–6771 (2000)
Audenaert, K.M.R., Verstraete, F., DeMoor, B.: Phys. Rev. A 64, 052304 (2001)
Uhlmann, A.: quant-ph/9704017, 1997
Hiai, F., Petz, D.: Lin. Alg. Appl. 181, 153–185 (1993)
Ando, T., Hiai, F.: Lin. Alg. Appl. 197, 198, 113–131 (1994)
Apostol, T.M.: Mathematical Analysis. Reading MA: Addison-Wesley, 1974
Amosov, G.G., Holevo, A.S., Werner, R.F.: Problems in Information Transmission 36, 25–34 (2000) and math-ph/0003002 (2000)
Werner, R.F., Holevo, A.S.: J. Math. Phys. 43(9), 4353–4357 (2002)
King, C.: quant-ph/0212057, 2002
Horn, R.A., Johnson, C.R.: Topics in Matrix Analysis. Cambridge: Cambridge University Press, 1991
King, C.: J. Math. Phys. 43(9), 4334–4340 (2002)
King, C.: quant-ph/0204172 (2002)
Shor, P.W.: quant-ph/0305035 (2003)
Author information
Authors and Affiliations
Corresponding author
Additional information
Communicated by M.B. Ruskai
Rights and permissions
About this article
Cite this article
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
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00220-003-0987-1