Abstract
We reduce the number of open additivity problems in quantum information theory by showing that four of them are equivalent. Namely, we show that the conjectures of additivity of the minimum output entropy of a quantum channel, additivity of the Holevo expression for the classical capacity of a quantum channel, additivity of the entanglement of formation, and strong superadditivity of the entanglement of formation, are either all true or all false.
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Koenraad, M.R., Audenaert, Braunstein, S.L.: On strong superadditivity of the entanglement of formation. quant-ph/0303045
Bell, J.: On the Einstein Podolsky Rosen paradox. Physics 1, 195–200 (1964)
Benatti, F., Narnhofer, H.: Additivity of the entanglement of formation. Phys. Rev. A 63, art. 042306 (2001)
Bennett, C.H., Bernstein, H.J., Popescu, S., Schumacher, B.: Concentrating partial entanglement by local operations. Phys. Rev. A 53, 2046–2052 (1996)
Bennett, C.H., DiVincenzo, D.P., Smolin, J.A., Wootters, W.K.: Mixed-state entanglement and quantum error correction. Phys. Rev. A 54, 3824–3851 (1996), quant-ph/9604024
Gordon, J.P.: Noise at optical frequencies: Information theory. In: Proceedings of the International School of Physics Enrico Fermi. Course XXXI: Quantum Electronics and Coherent Light, P.A. Mills, (ed.), New York: Academic Press, 1964), pp. 156–181
Hayden, P.M., Horodecki, M., Terhal, B.M.: The asymptotic entanglement cost of preparing a quantum state. J. Phys. A: Math. Gen. 34, 6891–6898 (2001)
Holevo, A.S.: Bounds for the quantity of information transmitted by a quantum communication channel. Probl. Peredachi Inf. 9(3), 3–11 (1973) [in Russian; English translation in Probl. Inf. Transm. (USSR) 9, 177–183 (1973)]
Holevo, A.S.: The capacity of the quantum channel with general signal states. IEEE Trans. Info. Theory 44, 269–273 (1998)
King, C., Ruskai, M.B.: Minimal entropy of states emerging from noisy quantum channels. IEEE Trans. Info. Theory 47, 192–209 (2001)
Levitin, L.B.: On the quantum measure of the amount of information. In: Proceedings of the Fourth All-Union Conference on Information Theory, Tashkent (1969), pp. 111–115, (in Russian)
Matsumoto, K., Shimono, T., Winter, A.: Remarks on additivity of the Holevo channel capacity and of the entanglement of formation. quant-ph/0206148
Pomeransky, A.: Strong superadditivity of the entanglement of formation follows from its additivity. quant-ph/0305056
Ruskai, M.B.: Some bipartite states do not arise from channels. quant-ph/0303141
Schumacher, B., Westmoreland.: Sending classical information via a noisy quantum channel. Phys. Rev. A 56, 131–138 (1997)
Shor, P.W.: Capacities of quantum channels and how to find them. quant-ph/0304102
Vidal, G., Dür, W., Cirac, J.I.: Entanglement cost of mixed states. Phys. Rev. Lett. 89, art. 027901 (2002)
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Communicated By M.B. Ruskai
An erratum to this article can be found athttp://dx.doi.org/10.1007/s00220-004-1071-1
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Shor, P. Equivalence of Additivity Questions in Quantum Information Theory. Commun. Math. Phys. 246, 453–472 (2004). https://doi.org/10.1007/s00220-003-0981-7
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DOI: https://doi.org/10.1007/s00220-003-0981-7