Unitary reformulation of the thermofield double state and limits of cyclic multi-mode squeezing

We investigate the structure and uniqueness of squeezed vacuum states defined by annihilation conditions of the form $(a - \alpha a^\dagger)|\psi\rangle = 0$ and their multimode generalizations, with applications to the Thermofield Double (TFD) state in quantum field theory. For $N=1$ and $N=2$, we demonstrate that these conditions uniquely define the single- and two-mode squeezed vacua, generated by unitary squeezing operators. A key result is the unitary reformulation of the TFD state, expressed as a product of two-mode squeezing operators, ensuring invertibility and resolving the non-unitary paradox in the Minkowski--Rindler vacuum correspondence. Extending to cyclic annihilation conditions $(a_i - \alpha_i a_{i+1}^\dagger)|\psi\rangle = 0$ with $a_{N+1} \equiv a_1$, we find that non-trivial squeezed states exist only for $N=2$. For $N > 2$, we establish a no-go theorem, proving no normalizable, non-trivial solutions exist, revealing a fundamental limit on cyclic multi-mode entanglement. These results highlight the bipartite nature of TFD-like entanglement and constrain multipartite generalizations in multi-region quantum field theories.