Dimension Independent Disentanglers from Unentanglement and Applications

Quantum entanglement, a distinctive form of quantum correlation, has become a key enabling ingredient in diverse applications in quantum computation, complexity, cryptography, etc. However, the presence of unwanted adversarial entanglement also poses challenges and even prevents the correct behaviour of many protocols and applications. In this paper, we explore methods to "break" the quantum correlations. Specifically, we construct a dimension-independent k-partite disentangler (like) channel from bipartite unentangled input. In particular, we show: For every d,𝓁 ≥ k ∈ ℕ^+, there is an efficient channel Λ : ℂ^{d𝓁} ⊗ ℂ^{d𝓁} → ℂ^{dk} such that for every bipartite separable density operator ρ₁⊗ ρ₂, the output Λ(ρ₁⊗ρ₂) is close to a k-partite separable state. Concretely, for some distribution μ on states from C^d, ║ Λ(ρ₁⊗ρ₂) - ∫ |ψ⟩⟨ψ|^{⊗k} dμ(ψ) ║₁ ≤ Õ((k³/𝓁)^{1/4}). Moreover, Λ(|ψ⟩⟨ψ|^{⊗𝓁} ⊗ |ψ⟩⟨ψ|^{⊗𝓁}) = |ψ⟩⟨ψ|^{⊗k}. Without the bipartite unentanglement assumption, the above bound is conjectured to be impossible and would imply QMA(2) = QMA. Leveraging multipartite unentanglement ensured by our disentanglers, we achieve the following: (i) a new proof that QMA(2) admits arbitrary gap amplification; (ii) a variant of the swap test and product test with improved soundness, addressing a major limitation of their original versions. More importantly, we demonstrate that unentangled quantum proofs of almost general real amplitudes capture NEXP, thereby greatly relaxing the non-negative amplitudes assumption in the recent work of QMA^+(2) = NEXP [Jeronimo and Wu, STOC 2023]. Specifically, our findings show that to capture NEXP, it suffices to have unentangled proofs of the form |ψ⟩ = √a |ψ_{+}⟩ + √{1-a} |ψ_{-}⟩ where |ψ_{+}⟩ has non-negative amplitudes, |ψ_{-}⟩ only has negative amplitudes and |a-(1-a)| ≥ 1/poly(n) with a ∈ [0,1]. Additionally, we present a protocol achieving an almost largest possible completeness-soundness gap before obtaining QMA^ℝ(k) = NEXP, namely, a 1/poly(n) additive improvement to the gap results in this equality.