I Correlation Matrix of Quadratic Hamiltonian

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The discussion focuses on the difficulty of rederiving equations (61) and (62) from a specific paper, particularly regarding the evaluation of terms like αεα^T using equation (58). It is noted that the authors do not explicitly solve for α, which complicates understanding. The matrix of phases (U) is suggested to be reabsorbed into the definition of α, making α's dependency on U crucial, especially since U is set to 1 in this context. The conversation highlights the need for clearer explanations before equation (60) to aid comprehension. Overall, the paper is appreciated for its content, despite the complexity of its presentation.
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I am struggling to rederive equations (61) and (62) from the following paper, namely I just want to understand how they evaluated terms like ##\alpha\epsilon\alpha^{T}## using (58). It seems like they don't explicitly solve for ##\alpha## right?
 
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First off, very neat paper. As for (58), if they are reabsorbing the matrix of phases (U) into the definition of α, then anything having to do with α is dependent on what U is set to (in this case = 1). I think this is informed by (51) and the instructions for (60). You are correct in saying they dont explicitly solve for α, but they could have shown their work a bit more before (60), those instructions are unnecessarily packed.
 

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