Additivity rates and PPT property for random quantum channels
Annales Mathématiques Blaise Pascal, Tome 22 (2015) no. 1, pp. 1-72.

Inspired by Montanaro’s work, we introduce the concept of additivity rates of a quantum channel $L$, which give the first order (linear) term of the minimum output $p$-Rényi entropies of ${L}^{\otimes r}$ as functions of $r$. We lower bound the additivity rates of arbitrary quantum channels using the operator norms of several interesting matrices including partially transposed Choi matrices. As a direct consequence, we obtain upper bounds for the classical capacity of the channels. We study these matrices for random quantum channels defined by random subspaces of a bipartite tensor product space. A detailed spectral analysis of the relevant random matrix models is performed, and strong convergence towards free probabilistic limits is shown. As a corollary, we compute the threshold for random quantum channels to have the positive partial transpose (PPT) property. We then show that a class of random PPT channels violate generically additivity of the $p$-Rényi entropy for all $p\ge 30.95$.

DOI : https://doi.org/10.5802/ambp.345
Classification : 46L54,  60B20,  81P45
Mots clés : Random matrix, Free Probability, Quantum Channel, Entropy, Additivity
@article{AMBP_2015__22_1_1_0,
author = {Motohisa Fukuda and Ion Nechita},
title = {Additivity rates and PPT property for random quantum channels},
journal = {Annales Math\'ematiques Blaise Pascal},
pages = {1--72},
publisher = {Annales math\'ematiques Blaise Pascal},
volume = {22},
number = {1},
year = {2015},
doi = {10.5802/ambp.345},
zbl = {1338.46072},
mrnumber = {3361563},
language = {en},
url = {https://ambp.centre-mersenne.org/articles/10.5802/ambp.345/}
}
Motohisa Fukuda; Ion Nechita. Additivity rates and PPT property for random quantum channels. Annales Mathématiques Blaise Pascal, Tome 22 (2015) no. 1, pp. 1-72. doi : 10.5802/ambp.345. https://ambp.centre-mersenne.org/articles/10.5802/ambp.345/

[1] G.G. Amosov; A.S. Holevo; R.F. Werner On some additivity problems in quantum information theory, Problems of Information Transmission, Volume 36 (2000) no. 4, pp. 305-313 | MR 1813649 | Zbl 0983.81004

[2] O. Arizmendi; I. Nechita; C. Vargas-Obieta Block modified random matrices (In preparation.)

[3] G. Aubrun Partial transposition of random states and non-centered semicircular distributions, Random Matrices: Theory and Applications, Volume 01 (2012) no. 02, 1250001 pages | Article | MR 2934718 | Zbl 1239.60002

[4] G. Aubrun; I. Nechita Realigning random states, J. Math. Phys., Volume 53 (2012) no. 10, 102210, 16 pages | Article | MR 3050579 | Zbl 1278.81022

[5] G. Aubrun; S. Szarek; E. Werner Hastings’s additivity counterexample via Dvoretzky’s theorem, Comm. Math. Phys., Volume 305 (2011) no. 1, pp. 85-97 | Article | MR 2802300 | Zbl 1222.81131

[6] T. Banica; I. Nechita Asymptotic eigenvalue distributions of block-transposed Wishart matrices, J. Theoret. Probab., Volume 26 (2013) no. 3, pp. 855-869 | Article | MR 3090554 | Zbl 1291.60014

[7] C. Beck; F. Schlögl Thermodynamics of chaotic systems, Cambridge Nonlinear Science Series, Volume 4, Cambridge University Press, Cambridge, 1993, xx+286 pages | Article | MR 1237638 | Zbl 0847.58051

[8] S.T. Belinschi; B. Collins; I. Nechita Laws of large numbers for eigenvectors and eigenvalues associated to random subspaces in a tensor product, Inventiones Mathematicae, Volume 190 (2012) no. 3, pp. 647-697 | Article | MR 2995183 | Zbl 1276.15007

[9] S.T. Belinschi; B. Collins; I. Nechita Almost one bit violation for the additivity of the minimum output entropy (2013) (http://arxiv.org/abs/1305.1567)

[10] F. G. S. L. Brandão; J. Eisert; M. Horodecki; D. Yang Entangled Inputs Cannot Make Imperfect Quantum Channels Perfect, Phys. Rev. Lett., Volume 106 (2011), 230502 pages http://link.aps.org/doi/10.1103/PhysRevLett.106.230502 | Article

[11] Man Duen Choi Completely positive linear maps on complex matrices, Linear Algebra and Appl., Volume 10 (1975), pp. 285-290 | Article | MR 376726 | Zbl 0327.15018

[12] B. Collins Moments and cumulants of polynomial random variables on unitary groups, the Itzykson-Zuber integral, and free probability, Int. Math. Res. Not. (2003) no. 17, pp. 953-982 | Article | MR 1959915 | Zbl 1049.60091

[13] B. Collins; M. Fukuda; I. Nechita Towards a state minimizing the output entropy of a tensor product of random quantum channels, J. Math. Phys., Volume 53 (2012) no. 3, 032203, 20 pages | Article | MR 2798211 | Zbl 1274.81044

[14] B. Collins; C. Gonzalez-Guillen; D. Perez-Garcia Matrix product states, random matrix theory and the principle of maximum entropy, Comm. Math. Phys., Volume 320 (2013) no. 3, pp. 663-677 | Article | MR 3057186 | Zbl 1270.15024

[15] B. Collins; C. Male The strong asymptotic freeness of Haar and deterministic matrices, Ann. Sci. Éc. Norm. Supér. (4), Volume 47 (2014) no. 1, pp. 147-163 | MR 3205602 | Zbl 1303.15043

[16] B. Collins; I. Nechita Eigenvalue and Entropy Statistics for Products of Conjugate Random Quantum Channels, Entropy, Volume 12 (2010) no. 6, pp. 1612-1631 http://www.mdpi.com/1099-4300/12/6/1612 | Article | MR 2659411 | Zbl 1229.81044

[17] B. Collins; I. Nechita Random quantum channels II: entanglement of random subspaces, Rényi entropy estimates and additivity problems, Adv. Math., Volume 226 (2011) no. 2, pp. 1181-1201 | Article | MR 2737781 | Zbl 1203.81022

[18] B. Collins; I. Nechita; K. Życzkowski Random graph states, maximal flow and Fuss-Catalan distributions, J. Phys. A, Volume 43 (2010) no. 27, 275303, 39 pages | Article | MR 2658283 | Zbl 1193.81031

[19] B. Collins; P. Śniady Integration with respect to the Haar measure on unitary, orthogonal and symplectic group, Comm. Math. Phys., Volume 264 (2006) no. 3, pp. 773-795 | Article | MR 2217291 | Zbl 1108.60004

[20] Benoît Collins; Motohisa Fukuda; Ion Nechita Low entropy output states for products of random unitary channels, Random Matrices Theory Appl., Volume 2 (2013) no. 1, 1250018, 36 pages | Article | MR 3039822 | Zbl 1270.15023

[21] Benoît Collins; Motohisa Fukuda; Ion Nechita On the convergence of output sets of quantum channels, Journal of Operator Theory, Volume 73 (2015) no. 2, pp. 333-360 | Article

[22] Benoît Collins; Ion Nechita Random quantum channels I: graphical calculus and the Bell state phenomenon, Comm. Math. Phys., Volume 297 (2010) no. 2, pp. 345-370 | Article | MR 2651902 | Zbl 1191.81050

[23] T. Cubitt; A. W. Harrow; D. Leung; A. Montanaro; A. Winter Counterexamples to additivity of minimum output $p$-Rényi entropy for $p$ close to 0, Comm. Math. Phys., Volume 284 (2008) no. 1, pp. 281-290 | Article | MR 2443306 | Zbl 1154.94007

[24] P. Di Francesco; O. Golinelli; E. Guitter Meander, folding, and arch statistics, Math. Comput. Modelling, Volume 26 (1997) no. 8-10, pp. 97-147 (Combinatorics and physics (Marseilles, 1995)) | Article | MR 1492504 | Zbl 1185.82026

[25] R. O. W. Franz; B. A. Earnshaw A constructive enumeration of meanders, Ann. Comb., Volume 6 (2002) no. 1, pp. 7-17 | Article | MR 1923083 | Zbl 1009.05005

[26] S. Friedland Additive invariants on quantum channels and regularized minimum entropy, Topics in operator theory. Volume 2. Systems and mathematical physics (Oper. Theory Adv. Appl.) Volume 203, Birkhäuser Verlag, Basel, 2010, pp. 237-245 | Article | MR 2683243 | Zbl 1189.81273

[27] M. Fukuda Extending additivity from symmetric to asymmetric channels, J. Phys. A, Volume 38 (2005) no. 45, p. L753-L758 | Article | MR 2198760 | Zbl 1079.81011

[28] M. Fukuda Revisiting Additivity Violation of Quantum Channels, Comm. Math. Phys., Volume 332 (2014) no. 2, pp. 713-728 | Article | MR 3257660 | Zbl 1300.81016

[29] M. Fukuda; P. Śniady Partial transpose of random quantum states: Exact formulas and meanders, Journal of Mathematical Physics, Volume 54 (2013) no. 4, 042202 pages http://link.aip.org/link/?JMP/54/042202/1 | Article | MR 3088232 | Zbl 1285.81005

[30] Motohisa Fukuda; Ion Nechita Asymptotically well-behaved input states do not violate additivity for conjugate pairs of random quantum channels, Comm. Math. Phys., Volume 328 (2014) no. 3, pp. 995-1021 | Article | MR 3201218 | Zbl 1295.81029

[31] A. Grudka; M. Horodecki; L. Pankowski Constructive counterexamples to the additivity of the minimum output Rényi entropy of quantum channels for all $p>2$, J. Phys. A, Volume 43 (2010) no. 42, 425304, 7 pages | Article | MR 2726721 | Zbl 1201.81026

[32] Michael J. W. Hall Random quantum correlations and density operator distributions, Phys. Lett. A, Volume 242 (1998) no. 3, pp. 123-129 | Article | MR 1626863 | Zbl 0940.82031

[33] M.B. Hastings Superadditivity of communication capacity using entangled inputs, Nature Physics, Volume 5 (2009), 255 pages | Article

[34] P. Hayden; A. Winter Counterexamples to the maximal $p$-norm multiplicity conjecture for all $p>1$, Comm. Math. Phys., Volume 284 (2008) no. 1, pp. 263-280 | Article | MR 2443305 | Zbl 1201.94066

[35] R. Hildebrand Positive partial transpose from spectra, Phys. Rev. A, Volume 76 (2007), 052325 pages http://link.aps.org/doi/10.1103/PhysRevA.76.052325 | Article

[36] A. S. Holevo The capacity of the quantum channel with general signal states, IEEE Trans. Inform. Theory, Volume 44 (1998) no. 1, pp. 269-273 | Article | MR 1486663 | Zbl 0897.94008

[37] A. S. Holevo Additivity conjecture and covariant channels, International Journal of Quantum Information, Volume 03 (2005) no. 01, pp. 41-47 http://www.worldscientific.com/doi/abs/10.1142/S0219749905000530 | Article | Zbl 1133.81320

[38] A. S. Holevo On complementary channels and the additivity problem, Prob. Th. and Appl., Volume 51 (2005), pp. 133-143 | MR 2324171 | Zbl 1113.81022

[39] A. S. Holevo The additivity problem in quantum information theory, International Congress of Mathematicians. Vol. III, Eur. Math. Soc., Zürich, 2006, pp. 999-1018 | MR 2275716 | Zbl 1100.94007

[40] C. King; K. Matsumoto; M. Nathanson; M. B. Ruskai Properties of conjugate channels with applications to additivity and multiplicativity, Mark. Proc. Rela. Fiel., Volume 13 (2007) no. 2, pp. 391-423 | MR 2343855 | Zbl 1139.81020

[41] C. King; M. B. Ruskai Minimal entropy of states emerging from noisy quantum channels, IEEE Trans. Inform. Theory, Volume 47 (2001) no. 1, pp. 192-209 | Article | MR 1819966 | Zbl 1016.94012

[42] Christopher King Maximal $p$-norms of entanglement breaking channels, Quantum Inf. Comput., Volume 3 (2003) no. 2, pp. 186-190 | MR 1965589 | Zbl 1152.81757

[43] C. Male The norm of polynomials in large random and deterministic matrices, Probab. Theory Related Fields, Volume 154 (2012) no. 3-4, pp. 477-532 | Article | MR 3000553 | Zbl 1269.15039

[44] A. Montanaro Weak Multiplicativity for Random Quantum Channels, Comm. Math. Phys., Volume 319 (2013) no. 2, pp. 535-555 | Article | MR 3037588 | Zbl 1269.81027

[45] Alexandru Nica; Roland Speicher Lectures on the combinatorics of free probability, London Mathematical Society Lecture Note Series, Volume 335, Cambridge University Press, Cambridge, 2006, xvi+417 pages | Article | MR 2266879 | Zbl 1133.60003

[46] B. Schumacher; M. D. Westmoreland Sending classical information via noisy quantum channels, Phys. Rev. A, Volume 56(1) (1997), pp. 131-138 | Article

[47] P. W. Shor Additivity of the classical capacity of entanglement-breaking quantum channels, J. Math. Phys., Volume 43 (2002) no. 9, pp. 4334-4340 | Article | MR 1924442 | Zbl 1060.94004

[48] P. W. Shor Equivalence of additivity questions in quantum information theory, Comm. Math. Phys., Volume 246 (2004) no. 3, pp. 453-472 | Article | MR 2053939 | Zbl 1070.81030

[49] G. Smith; J. Yard Quantum Communication with Zero-Capacity Channels, Science, Volume 321 (2008) no. 5897, pp. 1812-1815 | Article | MR 2456108 | Zbl 1226.94011

[50] J. M. Steele Probability theory and combinatorial optimization, CBMS-NSF Regional Conference Series in Applied Mathematics, Volume 69, SIAM, Philadelphia, PA, 1997, viii+159 pages | Article | MR 1422018 | Zbl 0916.90233

[51] W. F. Stinespring Positive functions on ${C}^{*}$-algebras, Proc. Amer. Math. Soc., Volume 6 (1955), pp. 211-216 | MR 69403 | Zbl 0064.36703

[52] D. V. Voiculescu; K. J. Dykema; A. Nica Free random variables, CRM Monograph Series, Volume 1, American Mathematical Society, Providence, RI, 1992, vi+70 pages | MR 1217253 | Zbl 0795.46049

[53] D. Weingarten Asymptotic behavior of group integrals in the limit of infinite rank, J. Mathematical Phys., Volume 19 (1978) no. 5, pp. 999-1001 | Article | MR 471696 | Zbl 0388.28013

[54] R. F. Werner; A. S. Holevo Counterexample to an additivity conjecture for output purity of quantum channels, J. Math. Phys., Volume 43 (2002) no. 9, pp. 4353-4357 (Quantum information theory) | Article | MR 1924444 | Zbl 1060.94008

[55] K. Życzkowski; K.A. Penson; I. Nechita; B. Collins Generating random density matrices, J. Math. Phys., Volume 52 (2011) no. 6, 062201, 20 pages | Article | MR 2841746

[56] K. Życzkowski; H.-J. Sommers Induced measures in the space of mixed quantum states, J. Phys. A, Volume 34 (2001) no. 35, pp. 7111-7125 | Article | MR 1863143 | Zbl 1031.81011