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Random repeated quantum interactions and random invariant states
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  • Published: 05 October 2010

Random repeated quantum interactions and random invariant states

  • Ion Nechita1 &
  • Clément Pellegrini2 

Probability Theory and Related Fields volume 152, pages 299–320 (2012)Cite this article

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  • 15 Citations

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Abstract

We consider a generalized model of repeated quantum interactions, where a system \({\mathcal{H}}\) is interacting in a random way with a sequence of independent quantum systems \({\mathcal{K}_n, n \geq 1}\). Two types of randomness are studied in detail. One is provided by considering Haar-distributed unitaries to describe each interaction between \({\mathcal{H}}\) and \({\mathcal{K}_n}\). The other involves random quantum states describing each copy \({\mathcal{K}_n}\). In the limit of a large number of interactions, we present convergence results for the asymptotic state of \({\mathcal{H}}\). This is achieved by studying spectral properties of (random) quantum channels which guarantee the existence of unique invariant states. Finally this allows to introduce a new physically motivated ensemble of random density matrices called the asymptotic induced ensemble.

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References

  1. Arveson W.: The probability of entanglement. Commun. Math. Phys. 286(1), 283–312 (2009)

    Article  MATH  MathSciNet  Google Scholar 

  2. Attal S., Pautrat Y.: From repeated to continuous quantum interactions. Ann. Henri Poincaré 7(1), 59–104 (2006)

    Article  MATH  MathSciNet  Google Scholar 

  3. Bengtsson, I., Życzkowski, K.: Geometry of Quantum States. An Introduction to Quantum Entanglement. Cambridge University Press, Cambridge, xii+466 p (2006)

  4. Bhatia R.: Matrix Analysis. Graduate Texts in Mathematics, 169. Springer, New York (1997)

    Google Scholar 

  5. Bruneau L., Joye A., Merkli M.: Asymptotics of repeated interaction quantum systems. J. Funct. Anal. 239(1), 310–344 (2006)

    Article  MATH  MathSciNet  Google Scholar 

  6. Bruneau L., Joye A., Merkli M.: Infinite products of random matrices and repeated interaction dynamics. Ann. Inst. H. Poincaré Probab. Statist. 46(2), 442–464 (2010)

    Article  MATH  MathSciNet  Google Scholar 

  7. Bruneau L., Joye A., Merkli M.: Random repeated interaction quantum systems. Commun. Math. Phys. 284(2), 553–581 (2008)

    Article  MATH  MathSciNet  Google Scholar 

  8. Braunstein S.L.: Geometry of quantum inference. Phys. Lett. A 219(3–4), 169–174 (1996)

    Article  MATH  MathSciNet  Google Scholar 

  9. Cubitt T., Montanaro A., Winter A.: On the dimension of subspaces with bounded Schmidt rank. J. Math. Phys. 49, 022107 (2008)

    Article  MathSciNet  Google Scholar 

  10. Dunford N., Schwartz J.T.: Linear Operators. Part I. General Theory, pp. xiv+858. Wiley Inc., New York (1988)

    MATH  Google Scholar 

  11. Evans D., Hoegh-Krohn R.: Spectral properties of positive maps on C *-algebras. J. London Math. Soc. (2) 17(2), 345–355 (1978)

    Article  MATH  MathSciNet  Google Scholar 

  12. Faraut J.: Analysis on Lie Groups. An Introduction. Cambridge Studies in Advanced Mathematics, vol. 110. Cambridge University Press, Cambridge (2008)

    Book  Google Scholar 

  13. Farenick D.R.: Irreducible positive linear maps on operator algebras. Proc. Am. Math. Soc. 124(11), 3381–3390 (1996)

    Article  MATH  MathSciNet  Google Scholar 

  14. George A., Ikramov K.D.: Common invariant subspaces of two matrices. Linear Algebra Appl. 287(1–3), 171–179 (1999)

    Article  MATH  MathSciNet  Google Scholar 

  15. Groh U.: The peripheral point spectrum of Schwarz operators on C *-algebras. Math. Z. 176(3), 311–318 (1981)

    Article  MATH  MathSciNet  Google Scholar 

  16. Hayden P., Leung D.W., Winter A.: Aspects of generic entanglement. Commun. Math. Phys. 265(1), 95–117 (2006)

    Article  MATH  MathSciNet  Google Scholar 

  17. Hayden P., Winter A.: Counterexamples to the maximal p-norm multiplicativity conjecture for all p > 1. Commun. Math. Phys. 284(1), 263–280 (2008)

    Article  MATH  MathSciNet  Google Scholar 

  18. Humphreys J.E.: Linear algebraic groups. Graduate Texts in Mathematics, No. 21. Springer, New York-Heidelberg (1975)

    Google Scholar 

  19. Nechita I.: Asymptotics of random density matrices. Ann. Henri Poincaré 8(8), 1521–1538 (2007)

    Article  MATH  MathSciNet  Google Scholar 

  20. Parthasarathy K.R.: On the maximal dimension of a completely entangled subspace for finite level quantum systems. Proc. Math. Sci. 114(4), 365–374 (2004)

    Article  MATH  Google Scholar 

  21. Pellegrini C.: Existence, uniqueness and approximation of stochastic Schrödinger equation: the diffusive case. Ann. Prob. 36(6), 2332–2353 (2008)

    Article  MATH  MathSciNet  Google Scholar 

  22. Pellegrini C.: Existence, uniqueness and approximation of the jump-type stochastic Schrödinger equation for two-level systems. Stoch. Process. Appl. 120(9), 1722–1747 (2010)

    Article  MATH  MathSciNet  Google Scholar 

  23. Pérez-García, D. et al.: Contractivity of positive and trace-preserving maps under L p norms. J. Math. Phys. 47 (8), 083506, 5 p (2006)

  24. Shemesh D.: Common eigenvectors of two matrices. Linear Algebra Appl. 62, 11–18 (1984)

    Article  MATH  MathSciNet  Google Scholar 

  25. Terhal B., DiVincezo D.: The problem of equilibration and the computation of correlation functions on a quantum computer. Phys. Rev. A 61, 022301 (2000)

    Article  Google Scholar 

  26. Walgate J., Scott A.J.: Generic local distinguishability and completely entangled subspaces. J. Phys. A 41, 375305 (2008)

    Article  MathSciNet  Google Scholar 

  27. Bruzda W., Cappellini V., Sommers H.-J., Życzkowski K.: Random Quantum Operations. Phys. Lett. A 373(3), 320–324 (2009)

    Article  MATH  MathSciNet  Google Scholar 

  28. Życzkowski K., Sommers H.-J.: Induced measures in the space of mixed quantum states. J. Phys. A 34(35), 7111–7125 (2001)

    Article  MATH  MathSciNet  Google Scholar 

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Authors and Affiliations

  1. Institut Camille Jordan, Université de Lyon, 43 blvd du 11 novembre 1918, 69622, Villeurbanne-Cedex, France

    Ion Nechita

  2. School of Physics and National Institue for Theoretical Physics, University of KwaZulu Natal, Private Bag X54001, Durban, 4000, South Africa

    Clément Pellegrini

Authors
  1. Ion Nechita
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  2. Clément Pellegrini
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Correspondence to Ion Nechita.

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Cite this article

Nechita, I., Pellegrini, C. Random repeated quantum interactions and random invariant states. Probab. Theory Relat. Fields 152, 299–320 (2012). https://doi.org/10.1007/s00440-010-0323-6

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  • Received: 09 February 2009

  • Revised: 02 September 2010

  • Published: 05 October 2010

  • Issue Date: February 2012

  • DOI: https://doi.org/10.1007/s00440-010-0323-6

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Keywords

  • Quantum repeated interactions
  • Random quantum channels
  • Random matrices
  • Peripheral spectrum
  • Random density matrices

Mathematics Subject Classification (2000)

  • Primary 15A52
  • Secondary 94A40
  • 60F15
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