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I'm hoping to use this stackexchange to solve a gate decomposition problem, but I am stumped, so hoping for help.
These definitions come from the Xanadu Strawberry Fields whitepaper on Table VI and VII.

I have a two-mode squeezed gate defined by $ S_{2}(z) = \exp \left(z^{*} \hat{a}_{1} \hat{a}_{2}-z \hat{a}_{1}^{\dagger} \hat{a}_{2}^{\dagger}\right) $ and the supposed decomposition is $$ S_{2}(z) =B^{\dagger}(\pi / 4,0)[S(z) \otimes S(-z)] B(\pi / 4,0) $$ where $$B^{\dagger}(\pi / 4,0) = \exp \left(\pi/4\left(\hat{a}_{1}^{\dagger} \hat{a}_{2}- \hat{a}_{1} \hat{a}_{2}^{\dagger}\right)\right) $$ and $$ S(z) = \exp \left(\left(z^{*} \hat{a}^{2}-z \hat{a}^{+2}\right) / 2\right) $$ Just dealing with the tensor product, I think I can write it ( with the identity implicit ) $$ [S(z) \otimes S(-z)] = \exp\left(\left(z^{*} \hat{a}_1^{2}-z \hat{a}_1^{+2}\right) / 2 + \left(z \hat{a}_2^{+2}-z^{*} \hat{a}_2^{2}\right) / 2\right)$$ but now I am still lost, because multiplying the beam splitter exponentials by this gives me a sum of the operators on either side and things do not seem to cancel at all. What am I doing wrong? Thanks in advance.

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1 Answer 1

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You need two things:

  • If $U$ is a unitary then it is true that $U^\dagger \exp(A) U = \exp( U^\dagger A U) $

  • $B^\dagger(\pi/4,0) a B(\pi/4,0) = \frac{1}{\sqrt{2}} (a+b)$ and $B^\dagger(\pi/4,0) b B(\pi/4,0) = \frac{1}{\sqrt{2}} (a-b)$

Now note that $ \left(\frac{1}{\sqrt{2}} (a+b)\right)^2 - \left(\frac{1}{\sqrt{2}} (a-b) \right)^2 = 2 a b$ and I think the identity follows.

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