Abstract
Entanglement and interference are among the most fundamental properties of quantum mechanics. In this work, we investigate the role and power of interference in the context of detecting entanglement. We do so from a computational complexity lens by proving that unentanglement gives no additional power to stoquastic Merlin-Arthur verification. For every polynomial number of provers k=k(n), StoqMa(k)=StoqMa . Conceptually, the proof separates the role of entanglement from the role of interference: once destructive interference is ruled out by stoquasticity, the product-state constraint can be absorbed into a polynomially larger one-witness stoquastic verification. The main analytic ingredient is a positive, value-based de Finetti theorem for separately symmetric extensions. If M is an entrywise nonnegative positive semidefinite contraction on A₁⊗…⊗ A_k, then the nonnegative product value of M is approximated to additive error ε by the largest eigenvalue of Π_R^<k (M_A_1,1… A_k-1,1A_k⊗ I) Π_R^<k, R=O(k²Σ_ilogdim A_i/ε³), where Π_R^<k is the operator on A₁^⊗ R ⊗ … ⊗ A_k-1^⊗ R ⊗ A_k projecting to the subspace Sym^R(A₁) ⊗ … ⊗ Sym^R(A_k-1) ⊗ A_k. The spectral relaxation is then realized as an actual one-witness stoquastic verifier. After replacing the uniform permutation averages in the symmetric projectors by inverse-polynomially close dyadic inverse-invariant averages. Consequently, StoqMa(k)=StoqMa⊆AM∩PP⊆PSPACE . The positive de Finetti theorem is isolated as a standalone technique and may be useful in other nonnegative tensor-optimization and stoquastic-verification settings.