We use projection methods to construct (global) quantum states with prescribed reduced (marginal) states, and possibly with some special properties such as having specific eigenvalues, having specific rank and extreme von Neumann or Renyi entropy. Using convex analysis, optimization techniques on matrix manifolds, we obtain algorithms to solve the problem. Matlab programs are written based on these algorithms and numerical examples are illustrated. The numerical results reveal new patterns leading to new insights and research problems on the topic.
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