Abstract
We prove additivity violation of minimum output entropy of quantum channels by straightforward application of \({\epsilon}\)-net argument and Lévy’s lemma. The additivity conjecture was disproved initially by Hastings. Later, a proof via asymptotic geometric analysis was presented by Aubrun, Szarek and Werner, which uses Dudley’s bound on Gaussian process (or Dvoretzky’s theorem with Schechtman’s improvement). In this paper, we develop another proof along Dvoretzky’s theorem in Milman’s view, showing additivity violation in broader regimes than the existing proofs. Importantly,Dvoretzky’s theorem works well with norms to give strong statements, but these techniques can be extended to functions which have norm-like structures-positive homogeneity and triangle inequality. Then, a connection between Hastings’ method and ours is also discussed. In addition, we make some comments on relations between regularized minimum output entropy and classical capacity of quantum channels.
Similar content being viewed by others
References
Aubrun G., Szarek S., Werner E.: Nonadditivity of Rényi entropy and Dvoretzky’s theorem. J. Math. Phys. 51(2), 022102–022107 (2010)
Aubrun G., Szarek S., Werner E.: Hastings’s additivity counterexample via Dvoretzky’s theorem. Commun. Math. Phys. 305(1), 85–97 (2011)
Belinschi Serban, T., Collins, B., Nechita, I.: Almost one bit violation for the additivity of the minimum output entropy. arXiv:1305.1567 [math-ph], 2013
Brandão Fernando G.S.L., Horodecki M.: On hastings’ counterexamples to the minimum output entropy additivity conjecture. Open Syst. Inf. Dyn. 17(1), 31–52 (2010)
Collins B., Fukuda M., Nechita I.: Towards a state minimizing the output entropy of a tensor product of random quantum channels. J. Math. Phys. 53(3), 032203–032220 (2012)
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. Commun. Math. Phys. 284(1), 281–290 (2008)
Collins B., Nechita I.: Random quantum channels I: graphical calculus and the Bell state phenomenon. Commun. Math. Phys. 297(2), 345–370 (2010)
Collins B., Nechita I.: Random quantum channels II: entanglement of random subspaces, Rényi entropy estimates and additivity problems. Adv. Math. 226(2), 1181–1201 (2011)
Dudley R.M.: The sizes of compact subsets of Hilbert space and continuity of Gaussian processes. J. Funct. Anal. 1, 290–330 (1967)
Dvoretzky, A.: Some results on convex bodies and Banach spaces. In: Proceeding of the International Symposium Linear Spaces (Jerusalem, 1960), pp. 123–160. Jerusalem Academic Press, Jerusalem (1961)
Fukuda M., King C.: Entanglement of random subspaces via the Hastings bound. J. Math. Phys. 51(4), 042201–042219 (2010)
Fukuda M., King C., Moser David K.: Comments on Hastings’ additivity counterexamples. Commun. Math. Phys. 296(1), 111–143 (2010)
Figiel T., Lindenstrauss J., Milman V.D.: The dimension of almost spherical sections of convex bodies. Acta Math. 139(1–2), 53–94 (1977)
Fukuda M., Nechita I.: Asymptotically well-behaved input states do not violate additivity for conjugate pairs of random quantum channels. Commun. Math. Phys. 328(3), 995–1021 (2014)
Fukuda M., Wolf Michael M.: Simplifying additivity problems using direct sum constructions. J. Math. Phys. 48(7), 072101–072107 (2007)
Grudka A., Horodecki M., Pankowski Ł.: Constructive counterexamples to the additivity of the minimum output Rényi entropy of quantum channels for all p > 2. J. Phys. A 43(42), 425304–425307 (2010)
Hastings M.B.: Superadditivity of communication capacity using entangled inputs. Nat. Phys. 5, 255 (2009)
Hayden P., Leung Debbie W., Winter A.: Aspects of generic entanglement. Commun. Math. Phys. 265(1), 95–117 (2006)
Holevo A.S.: The capacity of the quantum channel with general signal states. IEEE Trans. Inform. Theory 44(1), 269–273 (1998)
Holevo A.S.: On complementary channels and the additivity problem. Probab. Theory Appl. 51, 133–143 (2005)
Holevo, A.S.: The additivity problem in quantum information theory. In: International Congress of Mathematicians. vol. III, pp. 999–1018. European Mathematical Society, Zürich (2006)
Hayden P., Winter A.: Counterexamples to the maximal p-norm multiplicity conjecture for all p > 1. Commun. Math. Phys. 284(1), 263–280 (2008)
Jain, N.C., Marcus, M.B.: Continuity of sub-Gaussian processes. In: Probability on Banach spaces, vol. 4 of Advanced Probability Related Topics, pp. 81–196. Dekker, New York (1978)
King C., Matsumoto K., Nathanson M., Ruskai M.B.: Properties of conjugate channels with applications to additivity and multiplicativity. Markov Process. Relat. Fields 13(2), 391–423 (2007)
King C., Ruskai Mary B.: Minimal entropy of states emerging from noisy quantum channels. IEEE Trans. Inform. Theory 47(1), 192–209 (2001)
Lévy, P.: Problèmes concrets d’analyse fonctionnelle. Avec un complément sur les fonctionnelles analytiques par F. Pellegrino, 2d edn. Gauthier-Villars, Paris (1951)
Milman V.D.: A new proof of A. Dvoretzky’s theorem on cross-sections of convex bodies. Funkcional. Anal. I Priložen. 5(4), 28–37 (1971)
Montanaro A.: Weak Multiplicativity for random quantum channels. Commun. Math. Phys. 319(2), 535–555 (2013)
Milman, V.D., Schechtman, G.: Asymptotic theory of finite-dimensional normed spaces, volume 1200 of Lecture Notes in Mathematics. Springer, Berlin (1986). With an appendix by M. Gromov
Pisier, G.: The volume of convex bodies and Banach space geometry, vol. 94 of Cambridge Tracts in Mathematics. Cambridge University Press, Cambridge (1989)
Schechtman, G.: A remark concerning the dependence on \({\epsilon}\) in Dvoretzky’s theorem. In: Geometric aspects of functional analysis (1987–88), vol. 1376 of Lecture Notes in Mathematics, pp. 274–277. Springer, Berlin (1989)
Shor P.W.: Equivalence of additivity questions in quantum information theory. Commun. Math. Phys. 246(3), 453–472 (2004)
Forrest Stinespring W.: Positive functions on C*-algebras. Proc. Am. Math. Soc. 6, 211–216 (1955)
Schumacher B., Westmoreland M.D.: Sending classical information via noisy quantum channels. Phys. Rev. A 56(1), 131–138 (1997)
Werner, R.F., Holevo, A.S.: Counterexample to an additivity conjecture for output purity of quantum channels. J. Math. Phys. 43(9), 4353–4357 (2002). Quantum information theory
Author information
Authors and Affiliations
Corresponding author
Additional information
Communicated by A. Winter
Rights and permissions
About this article
Cite this article
Fukuda, M. Revisiting Additivity Violation of Quantum Channels. Commun. Math. Phys. 332, 713–728 (2014). https://doi.org/10.1007/s00220-014-2101-2
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00220-014-2101-2