The Annals of Applied Probability

Gaussianization and eigenvalue statistics for random quantum channels (III)

Benoît Collins and Ion Nechita

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In this paper, we present applications of the calculus developed in Collins and Nechita [Comm. Math. Phys. 297 (2010) 345–370] and obtain an exact formula for the moments of random quantum channels whose input is a pure state thanks to Gaussianization methods. Our main application is an in-depth study of the random matrix model introduced by Hayden and Winter [Comm. Math. Phys. 284 (2008) 263–280] and used recently by Brandao and Horodecki [Open Syst. Inf. Dyn. 17 (2010) 31–52] and Fukuda and King [J. Math. Phys. 51 (2010) 042201] to refine the Hastings counterexample to the additivity conjecture in quantum information theory. This model is exotic from the point of view of random matrix theory as its eigenvalues obey two different scalings simultaneously. We study its asymptotic behavior and obtain an asymptotic expansion for its von Neumann entropy.

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Ann. Appl. Probab., Volume 21, Number 3 (2011), 1136-1179.

First available in Project Euclid: 2 June 2011

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 15A52
Secondary: 94A17: Measures of information, entropy 94A40: Channel models (including quantum)

Random matrices Weingarten calculus quantum information theory random quantum channel


Collins, Benoît; Nechita, Ion. Gaussianization and eigenvalue statistics for random quantum channels (III). Ann. Appl. Probab. 21 (2011), no. 3, 1136--1179. doi:10.1214/10-AAP722.

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  • [1] Bhatia, R. (1997). Matrix Analysis. Graduate Texts in Mathematics 169. Springer, New York.
  • [2] Bengtsson, I. and Życzkowski, K. (2006). Geometry of Quantum States. Cambridge Univ. Press, Cambridge.
  • [3] Bożejko, M., Krystek, A. D. and Wojakowski, Ł. J. (2006). Remarks on the r and Δ convolutions. Math. Z. 253 177–196.
  • [4] Brandão, F. G. S. L. and Horodecki, M. (2010). On Hastings’ counterexamples to the minimum output entropy additivity conjecture. Open Syst. Inf. Dyn. 17 31–52.
  • [5] Braunstein, S. L. (1996). Geometry of quantum inference. Phys. Lett. A 219 169–174.
  • [6] Bryc, W. (2008). Asymptotic normality for traces of polynomials in independent complex Wishart matrices. Probab. Theory Related Fields 140 383–405.
  • [7] Coecke, B. (2006). Kindergarten quantum mechanics—lecture notes. In Quantum Theory: Reconsideration of Foundations—3. AIP Conf. Proc. 810 81–98. Amer. Inst. Phys., Melville, NY.
  • [8] Collins, B. (2003). Moments and cumulants of polynomial random variables on unitary groups, the Itzykson–Zuber integral, and free probability. Int. Math. Res. Not. 17 953–982.
  • [9] Collins, B. and Nechita, I. (2010). Random quantum channels I: Graphical calculus and the Bell state phenomenon. Comm. Math. Phys. 297 345–370.
  • [10] Collins, B. and Nechita, I. (2011). Random quantum channels II: Entanglement of random subspaces, Rényi entropy estimates and additivity problems. Adv. Math. 226 1181–1201.
  • [11] Collins, B. and Nechita, I. (2010). Eigenvalue and entropy statistics for products of conjugate random quantum channels. Entropy 12 1612–1631.
  • [12] Collins, B., Nechita, I. and Życzkowski, K. (2010). Random graph states, maximal flow and Fuss–Catalan distributions. J. Phys. A Math. Theor. 43 275303.
  • [13] Collins, B. and Śniady, P. (2006). Integration with respect to the Haar measure on unitary, orthogonal and symplectic group. Comm. Math. Phys. 264 773–795.
  • [14] Fukuda, M. and King, C. (2010). Entanglement of random subspaces via the Hastings bound. J. Math. Phys. 51 042201.
  • [15] Fukuda, M., King, C. and Moser, D. K. (2010). Comments on Hastings’ additivity counterexamples. Comm. Math. Phys. 296 111–143.
  • [16] Graczyk, P., Letac, G. and Massam, H. (2003). The complex Wishart distribution and the symmetric group. Ann. Statist. 31 287–309.
  • [17] Guionnet, A. (2009). Large Random Matrices: Lectures on Macroscopic Asymptotics. Lecture Notes in Math. 1957. Springer, Berlin.
  • [18] Hanlon, P. J., Stanley, R. P. and Stembridge, J. R. (1992). Some combinatorial aspects of the spectra of normally distributed random matrices. In Hypergeometric Functions on Domains of Positivity, Jack Polynomials, and Applications (Tampa, FL, 1991). Contemp. Math. 138 151–174. Amer. Math. Soc., Providence, RI.
  • [19] Hastings, M. B. (2009). Superadditivity of communication capacity using entangled inputs. Nature Physics 5 255.
  • [20] Hayden, P. and Winter, A. (2008). Counterexamples to the maximal p-norm multiplicity conjecture for all p>1. Comm. Math. Phys. 284 263–280.
  • [21] Jones, V. F. R. (1999). Planar Algebras. Available at arXiv:math/9909027v1.
  • [22] Mingo, J. A. and Nica, A. (2004). Annular noncrossing permutations and partitions, and second-order asymptotics for random matrices. Int. Math. Res. Not. 28 1413–1460.
  • [23] Nechita, I. (2007). Asymptotics of random density matrices. Ann. Henri Poincaré 8 1521–1538.
  • [24] Nica, A. and Speicher, R. (2006). Lectures on the Combinatorics of Free Probability. London Mathematical Society Lecture Note Series 335. Cambridge Univ. Press, Cambridge.
  • [25] Page, D. N. (1993). Average entropy of a subsystem. Phys. Rev. Lett. 71 1291–1294.
  • [26] Sommers, H.-J. and Życzkowski, K. (2004). Statistical properties of random density matrices. J. Phys. A 37 8457–8466.
  • [27] Stanley, R. P. (1997). Enumerative Combinatorics. Vol. 1. Cambridge Studies in Advanced Mathematics 49. Cambridge Univ. Press, Cambridge.
  • [28] Życzkowski, K. and Sommers, H.-J. (2001). Induced measures in the space of mixed quantum states. J. Phys. A 34 7111–7125. Quantum information and computation.
  • [29] Zvonkin, A. (1997). Matrix integrals and map enumeration: An accessible introduction. Math. Comput. Modelling 26 281–304.