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Who first introduced the mathematical definition of entanglement?

The name "entanglement" and the definition for pure states comes from Schrödinger 1935. The definition for mixed states comes from Werner 1989
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What is the original reference for the Hadamard test?

I believe I have found the original reference to the Hadamard test: It is "Quantum Algorithms Revisited" by R. Cleve, A. Ekert, C. Macchiavello, and M. Mosca.
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Graph Limits in Quantum Computing

Because the adjacency matrix of undirected graphs are symmetric about the diagonal, these graphs are hermitian, and you are correct to suppose that quantum computing can be a natural vehicle for ...
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