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On the partial transpose of a Haar unitary matrix

Volume 266 / 2022

James A. Mingo, Mihai Popa, Kamil Szpojankowski Studia Mathematica 266 (2022), 337-359 MSC: Primary 46L54; Secondary 60B20. DOI: 10.4064/sm210517-13-12 Published online: 6 June 2022


We consider the effect of a partial transpose on the limit $*$-distribution of a Haar distributed random unitary matrix. If we fix the number of blocks, $b$, we show that the partial transpose can be decomposed along diagonals into a sum of $b$ matrices which are asymptotically free and identically distributed. We then consider the joint effect of different block decompositions and show that under some mild assumptions we also get asymptotic freeness.


  • James A. MingoDepartment of Mathematics and Statistics
    Jeffery Hall, Queen’s University
    Kingston, Ontario, ONL K7L 3Z1
  • Mihai PopaDepartment of Mathematics
    The University of Texas at San Antonio
    One UTSA Circle
    San Antonio, TX 78249, USA
    Institute of Mathematics “Simion Stoilow”
    of the Romanian Academy
    P.O. Box 1-764
    RO-70700 Bucureşti, Romania
  • Kamil SzpojankowskiWydział Matematyki i Nauk Informacyjnych
    Politechnika Warszawska
    Koszykowa 75
    00-662 Warszawa, Poland

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