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Matrix problems in quantum information science

Quantum information science is an active research area,
and there are interesting problems suitable for undergraduate
research. Here are a few of them.
1)  Decomposition of quantum gates.
     We try to decompose a unitary matrix as the product
      of simple unitary matrices so that they can be easily
      implemented. One may see the pdf file of Li's presentation
      and the arXiv papers 
        arXiv:1311.3599 and  arXiv:1210.7366
     for more information.
2)  Interpolation problems in quantum information science.
     Quantum states are represented as density matrices,
      i.e., positive semi-definite matrices with trace 1.
     A general quantum operation has the form
         T(X) = F_1XF_1* + .... + F_rXF_r*
     for some suitable choices of m x n matrices F_1, ..., F_r
     such that F_1F_1* + .... + F_rF_r* = I. 
     Suppose A_1, ..., A_k are nxn density matrices
      and B_1, ..., B_k are mxm density matrices.
    We would like to know whether there exists a quantum
     operation of the above form such that 
        T(A_i) = B_i   for i = 1, ..., k.
     One may see the arXiv papers 
         arXiv:1203.5547 and  arXiv:1012.1675
     for more background.
(Advisers: Chi-Kwong Li, Diane Pelejo)