×

Random quantum channels. II: Entanglement of random subspaces, Rényi entropy estimates and additivity problems. (English) Zbl 1203.81022

Summary: We obtain new bounds for the minimum output entropies of random quantum channels. These bounds rely on random matrix techniques arising from free probability theory. We then revisit the counterexamples developed by P. Hayden and A. Winter [Commun. Math. Phys. 284, No. 1, 263–280 (2008; Zbl 1201.94066)] to get violations of the additivity equalities for minimum output Rényi entropies. We show that random channels obtained by randomly coupling the input to a qubit violate the additivity of the \(p\)-Rényi entropy, for all \(p>1\). For some sequences of random quantum channels, we compute almost surely the limit of their Schatten \(S_{1}\rightarrow S_p\) norms.
[For part I, cf. Commun. Math. Phys. 297, No. 2, 345–370 (2010; Zbl 1191.81050).]

MSC:

81P45 Quantum information, communication, networks (quantum-theoretic aspects)
94A17 Measures of information, entropy
15B52 Random matrices (algebraic aspects)
94A40 Channel models (including quantum) in information and communication theory
PDFBibTeX XMLCite
Full Text: DOI arXiv

References:

[1] Aubrun, G.; Nechita, I., Catalytic majorization and \(l_p\) norms, Comm. Math. Phys., 278, 1, 133-144 (2008) · Zbl 1140.81318
[2] Aubrun, G.; Nechita, I., Stochastic domination for iterated convolutions and catalytic majorization, Ann. Inst. H. Poincaré Probab. Statist., 45, 3, 611-625 (2009) · Zbl 1179.60008
[3] Bhatia, R., Matrix Analysis, Grad. Texts in Math., vol. 169 (1997), Springer-Verlag: Springer-Verlag New York
[4] Brandao, F.; Horodecki, M. S.L., On Hastings’ counterexamples to the minimum output entropy additivity conjecture, Open Syst. Inf. Dyn., 17, 31-52 (2010) · Zbl 1206.81024
[5] Collins, B., Product of random projections, Jacobi ensembles and universality problems arising from free probability, Probab. Theory Related Fields, 133, 3, 315-344 (2005) · Zbl 1100.46036
[6] Collins, B.; Nechita, I., Random quantum channels I: Graphical calculus and the Bell state phenomenon, Comm. Math. Phys., 297, 2, 345-370 (2010) · Zbl 1191.81050
[7] Collins, B.; Śniady, P., Integration with respect to the Haar measure on unitary, orthogonal and symplectic group, Comm. Math. Phys., 264, 3, 773-795 (2006) · Zbl 1108.60004
[8] Fukuda, M.; King, C., Entanglement of random subspaces via the Hastings bound, J. Math. Phys., 51, 042201 (2010) · Zbl 1310.81028
[9] Fukuda, M.; King, C.; Moser, A., Comments on Hastings’ additivity counterexamples, Comm. Math. Phys., 296, 1, 111-143 (2010) · Zbl 1194.81037
[10] Haagerup, U.; Thorbjørnsen, S., A new application of random matrices: \(Ext(C_{red}^\ast(F_2))\) is not a group, Ann. of Math. (2), 162, 2, 711-775 (2005) · Zbl 1103.46032
[11] Hastings, M. B., Superadditivity of communication capacity using entangled inputs, Nat. Phys., 5, 255-257 (2009)
[12] Hayden, P., The maximal p-norm multiplicativity conjecture is false
[13] Hayden, P.; Leung, D.; Winter, A., Aspects of generic entanglement, Comm. Math. Phys., 265, 95-117 (2006) · Zbl 1107.81011
[14] Hayden, P.; Winter, A., Counterexamples to the maximal p-norm multiplicativity conjecture for all \(p > 1\), Comm. Math. Phys., 284, 1, 263-280 (2008) · Zbl 1201.94066
[15] Ledoux, M., Differential operators and spectral distributions of invariant ensembles from the classical orthogonal polynomials part I: the continuous case, Electron. J. Probab., 9, 177-208 (2004) · Zbl 1073.60037
[16] Nechita, I., Asymptotics of random density matrices, Ann. H. Poincaré, 8, 8, 1521-1538 (2007) · Zbl 1135.82017
[17] Soshnikov, A., Determinantal random point fields, Uspekhi Mat. Nauk. Uspekhi Mat. Nauk, Russian Math. Surveys, 55, 5, 923-975 (2000), (in Russian); translation in: · Zbl 0991.60038
[18] Voiculescu, D. V., A strengthened asymptotic freeness result for random matrices with applications to free entropy, Int. Math. Res. Not. IMRN, 1, 41-63 (1998) · Zbl 0895.60004
[19] Voiculescu, D. V.; Dykema, K. J.; Nica, A., Free Random Variables (1992), Amer. Math. Soc. · Zbl 0795.46049
[20] Werner, R.; Holevo, A., Counterexample to an additivity conjecture for output purity of quantum channels, J. Math. Phys., 43, 4353-4357 (2002) · Zbl 1060.94008
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.