Probability Theory and Related Fields

, Volume 152, Issue 1–2, pp 299–320 | Cite as

Random repeated quantum interactions and random invariant states

Article

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.

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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Copyright information

© Springer-Verlag 2010

Authors and Affiliations

  1. 1.Institut Camille JordanUniversité de LyonVilleurbanne-CedexFrance
  2. 2.School of Physics and National Institue for Theoretical PhysicsUniversity of KwaZulu NatalDurbanSouth Africa

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