Abstract
We investigate the structure and uniqueness of squeezed vacuum states defined by annihilation conditions of the form (a - α a^†)|ψ = 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 - α_i a_i+1^†)|ψ = 0 with a_N+1 ≡ a₁, 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.