Multimode squeezed vacuum ($|0⟩=C\exp (\frac{1}{2}{a}^{\prime }{}^{†}M{a}^{\prime }{}^{†})|0^{\prime }⟩$)

For an invertible finite-mode Bogoliubov transformation $a=A{a}^{\prime }+B{a}^{\prime }{}^{†}$, the symmetric squeezing matrix is $M=-A^{-1}B$. The canonical commutation relations make its singular values strictly less than one. The normalized state has $|C|=\det (I-M{M}^{†}{)}^{1/4}$. Infinitely many modes additionally require bosonic mode mixing implementability.