Abstract
For all p > 1, we demonstrate the existence of quantum channels with non-multiplicative maximal output p-norms. Equivalently, for all p > 1, the minimum output Rényi entropy of order p of a quantum channel is not additive. The violations found are large; in all cases, the minimum output Rényi entropy of order p for a product channel need not be significantly greater than the minimum output entropy of its individual factors. Since p = 1 corresponds to the von Neumann entropy, these counterexamples demonstrate that if the additivity conjecture of quantum information theory is true, it cannot be proved as a consequence of any channel-independent guarantee of maximal p-norm multiplicativity. We also show that a class of channels previously studied in the context of approximate encryption lead to counterexamples for all p > 2.
Similar content being viewed by others
References
Schumacher B.: Quantum coding. Phys. Rev. A 51, 2738–2747 (1995)
Jozsa R., Schumacher B.: A new proof of the quantum noiseless coding theorem. J. Mod. Opt. 41, 2343–2349 (1994)
Pierce J.: The early days of information theory. IEEE Transactions on Information Theory 19(1), 3–8 (1973)
Gordon, J.P.: Noise at optical frequencies; information theory. In: Miles P.A. ed., Quantum electronics and coherent light; Proceedings of the international school of physics Enrico Fermi, Course XXXI, New York: Academic Press, 1964 pp. 156–181
Holevo, A.S.: Information theoretical aspects of quantum measurements. Probl. Info. Transm. (USSR), 9(2), 31–42 (1973). Translation: Probl. Info. Transm. 9, 177–183 (1973)
Hausladen P., Jozsa R., Schumacher B., Westmoreland M., Wootters W.K.: Classical information capacity of a quantum channel. Phys. Rev. A 54, 1869–1876 (1996)
Holevo A.S.: The capacity of the quantum channel with general signal states. IEEE Trans. Inf. Theory 44, 269–273 (1998)
Schumacher B., Westmoreland M.D.: Sending classical information via noisy quantum channels. Phys. Rev. A 56, 131–138 (1997)
Shor P.W.: Equivalence of additivity questions in quantum information theory. Commun. Math. Phys. 246, 453–472 (2004)
Pomeransky A.A.: Strong superadditivity of the entanglement of formation follows from its additivity. Physical Review A 68(3), 032317 (2003)
Audenaert K.M.R., Braunstein S.L.: On strong superadditivity of the entanglement of formation. Commun. Math. Phys. 246, 443–452 (2004)
Matsumoto K., Shimono T., Winter A.: Remarks on additivity of the Holevo channel capacity and of the entanglement of formation. Commun. Math. Phys. 246, 427–442 (2004)
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)
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)
Vidal G., Dür W., Cirac J.I.: Entanglement cost of bipartite mixed states. Phys. Rev. Lett. 89(2), 027901 (2002)
Matsumoto K., Yura F.: Entanglement cost of antisymmetric states and additivity of capacity of some quantum channels. J. Phys. A: Math. Gen. 37, L167–L171 (2004)
Vollbrecht K.G.H., Werner R.F.: Entanglement measures under symmetry. Phys. Rev. A 64(6), 062307 (2001)
King C., Ruskai M.B.: Minimal entropy of states emerging from noisy quantum channels. IEEE Trans. Inf. Th. 47(1), 192–209 (2001)
Osawa S., Nagaoka H.: Numerical experiments on the capacity of quantum channel with entangled input states. IEICE Trans. Fund. Elect., Commun. and Comp. Sci. E84(A10), 2583–2590 (2001)
Amosov G.G., Holevo A.S., Werner R.F.: On some additivity problems of quantum information theory. Probl. Inform. Transm. 36(4), 25 (2000)
Amosov, G.G., Holevo, A.S.: On the multiplicativity conjecture for quantum channels. http://arxiv.org/list/:math-ph/0103015, 2001
King C.: Additivity for unital qubit channels. J. Math. Phys. 43(10), 4641–4643 (2002)
Fujiwara A., Hashizumé T.: Additivity of the capacity of depolarizing channels. Phys. Lett. A 299, 469–475 (2002)
King C.: The capacity of the quantum depolarizing channel. IEEE Trans. Inf. Th. 49(1), 221–229 (2003)
Holevo A.S.: Quantum coding theorems. Russ. Math. Surv. 53, 1295–1331 (1998)
King, C.: Maximization of capacity and p-norms for some product channels. http://arxiv.org/list/:quant-ph/0103086, 2001
Shor P.W.: Additivity of the classical capacity of entanglement-breaking quantum channels. J. Math. Phys. 43, 4334–4340 (2002)
Devetak I., Shor P.W.: The capacity of a quantum channel for simultaneous transmission of classical and quantum information. Commun. Math. Phys. 256, 287–303 (2005)
King, C., Matsumoto, K., Nathanson, M., Ruskai, M.B.: Properties of conjugate channels with applications to additivity and multiplicativity. http://arxiv.org/list/:quant-ph/0509126, 2005., to appear in special issue of Markov processes and Related Fields in memory of J.F. leuis
Cortese J.: Holevo-Schumacher-Westmoreland channel capacity for a class of qudit unital channels. Phys. Rev. A 69(2), 022302 (2004)
Datta, N., Holevo, A.S., Suhov, Y.M.: A quantum channel with additive minimum output entropy. http://arxiv.org/list/:quant-ph/0403072, 2004
Fukuda M.: Extending additivity from symmetric to asymmetric channels. J. Phys. A: Math. Gen. 38, L753–L758 (2005)
Holevo, A.S.: Additivity of classical capacity and related problems. Available online at: http://www.imaph.tu-bs.de/qi/problems/10.pdf, 2004
Holevo, A.S.: The additivity problem in quantum information theory. In: Proceedings of the International Congress of Mathematicians, (Madrid, Spain, 2006), Zurich:Publ. EMS, 2007, pp. 999–1018
King C., Ruskai M.B.: Comments on multiplicativity of maximal p-norms when p = 2. Quantum Inf. and Comput. 4, 500–512 (2004)
King C., Nathanson M., Ruskai M.B.: Multiplicativity properties of entrywise positive maps. Linear alge. Applications. 404, 367–379 (2005)
Serafini A., Eisert J., Wolf M.M.: Multiplicativity of maximal output purities of Gaussian channels under Gaussian inputs. Phys. Rev. A 71(1), 012320 (2005)
Giovannetti V., Lloyd S.: Additivity properties of a Gaussian channel. Phys. Rev. A 69, 062307 (2004)
Devetak I., Junge M., King C., Ruskai M.B.: Multiplicativity of completely bounded p-norms implies a new additivity result. Commun. Math. Phys. 266, 37–63 (2006)
Michalakis, S.: Multiplicativity of the maximal output 2-norm for depolarized Werner-Holevo channels. http://arxiv.org/list/:0707.1722, 2007
Werner R.F., Holevo A.S.: Counterexample to an additivity conjecture for output purity of quantum channels. J. Math. Phys. 43, 4353–4357 (2002)
Alicki R., Fannes M.: Note on multiple additivity of minimal Renyi entropy output of the Werner-Holevo channels. Open Syst. Inf. Dyn. 11(4), 339–342 (2005)
Datta, N.: Multiplicativity of maximal p-norms in Werner-Holevo channels for 1 < p < 2. http://arxiv.org/list/:quant-ph/0410063, 2004
Giovannetti V., Lloyd S., Ruskai M.B.: Conditions for multiplicativity of maximal p -norms of channels for fixed integer p. J. Math. Phys. 46, 042105 (2005)
Winter, A.: The maximum output p-norm of quantum channels is not multiplicative for any p > 2. http://arxiv.org/abs/:0707.0402, 2007
Hayden, P.: The maximal p-norm multiplicativity conjecture is false. arXiv.org:0707.3291, 2007
Hayden P., Leung D., Shor P.W., Winter A.: Randomizing Quantum States: Constructions and Applications. Commun. Math. Phys. 250, 371–391 (2004)
Aubrun, G.: On almost randomizing channels with a short Kraus decomposition. http://arxiv.org/abs/:0805.2900v2, 2008
Paulsen, V.I.: Completely bounded maps and dilations. New York: Longman Scientific and Technical, 1986
Hayden P., Leung D.W., Winter A.: Aspects of generic entanglement. Commun. Math. Phys. 265, 95–117 (2006)
Bennett C.H., Hayden P., Leung D.W., Shor P.W., Winter A.: Remote preparation of quantum states. IEEE Trans. Inf. Th. 51(1), 56–74 (2005)
Geman S.: A Limit Theorem for the Norm of Random Matrices. Ann. Prob. 8(2), 252–261 (1980)
Johnstone I.M.: On the distribution of the largest eigenvalue in principal components analysis. Ann. Stat. 29(2), 295–327 (2001)
Davidson, K.R., Szarek, S.J.: Local Operator Theory, Random Matrices and Banach Spaces. In: Johnson W.B., Lindenstrauss J. eds., Handbook of the Geometry of Banach Spaces, Vol. I, Chap. 8, London:Elsevier, 2001, pp. 317–366
Ledoux, M.: The concentration of measure phenomenon, Vol. 89 of Mathematical Surveys and Monographs. Providence, RI: American Mathematical Society, 2001
King C.: Maximal p-norms of entanglement breaking channels. Quantum Inf. and Comp. 3(2), 186–190 (2003)
Wolf M.M., Eisert J.: Classical information capacity of a class of quantum channels. New J. Phys. 7, 93 (2005)
Cubitt, T., Harrow, A.W., Leung, D., Montanaro, A., Winter, A.: Counterexamples to additivity of minimum output p-Rényi entropy for p close to 0. http://arxiv.org/abs/:0712.3628v2, 2007, Commun. Math. Phys. doi:10.1007/s00220-008-0625-z
Ambainis, A., Smith, A.: Small pseudo-random families of matrices: Derandomizing approximate quantum encryption. In: Proc. RANDOM, LNCS 3122, Berlin-Heidelberg-NewYork: Springer, 2004, pp. 249–260
Ben-Aroya, A., Ta-Shma, A.: Quantum expanders and the quantum entropy difference problem. http://arxiv.org/abs/:quant-ph/0702129, 2007
Hastings M.B.: Random unitaries give quantum expanders. Phys. Rev. A 76, 032315 (2007)
Pérez-García D., Wolf M.M., Palazuelos C., Villanueva I., Junge M.: Unbounded Violation of Tripartite Bell Inequalities. Commun. Math. Phys. 279(2), 455–486 (2008)
Aubert S., Lam C.S.: Invariant integration over the unitary group. J. Math. Phys. 44, 6112–6131 (2003)
Aubert S., Lam C.S.: Invariant and group theoretical integrations over the U(n) group. J. Math. Phys. 45, 3019–3039 (2004)
Collins B., Śniady P.: Integration with respect to the Haar measure on unitary, orthogonal and symplectic group. Commun. Math. Phys. 264, 773–795 (2006)
Author information
Authors and Affiliations
Corresponding author
Additional information
Communicated by M.B. Ruskai
Rights and permissions
About this article
Cite this article
Hayden, P., Winter, A. Counterexamples to the Maximal p-Norm Multiplicativity Conjecture for all p > 1. Commun. Math. Phys. 284, 263–280 (2008). https://doi.org/10.1007/s00220-008-0624-0
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00220-008-0624-0