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In Michael A. Nielsen & Isaac L. Chuang, Quantum Computation and Quantum Information, 10th Anniversary Edition, the proof of Theorem 9.2 (Trace-preserving quantum operations are contractive) on page 406-407 introduces positive matrices with orthogonal support $Q$ and $S$.

  1. How are these matrices constructed?
  2. Why do Equation (9.38-9.43) hold?
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