Correlation Matrix of Quadratic Hamiltonian

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SUMMARY

The discussion focuses on the derivation of equations (61) and (62) from a specific paper concerning the correlation matrix of a quadratic Hamiltonian. The participant highlights the evaluation of terms like ##\alpha\epsilon\alpha^{T}## using equation (58) and notes that the authors do not explicitly solve for ##\alpha##. It is established that the matrix of phases (U) is reabsorbed into the definition of ##\alpha##, which is contingent on the value of U, set to 1 in this context. The participant critiques the clarity of the instructions leading to equation (60), suggesting that they are overly condensed.

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thatboi
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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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