Dimension Independent Disentanglers from Unentanglement and Applications
Fernando G. Jeronimo, Pei Wu
TL;DR
This work introduces a dimension-independent disentangler that converts bipartite unentangled inputs into highly multipartite, approximately product outputs, with an error that scales as $\tilde{O}((k^3/\ell)^{1/4})$ and exact recovery on product inputs. The core mechanism, PAPO, underpins a versatile framework that leverages disentangling to enable advanced property testing, including a Super Swap/Product Test pair, and yields strong gap amplification results for unentangled quantum-proof systems. The authors connect these tools to quantum complexity, showing that NEXP can be captured by almost-$\textup{QMA}^{\mathbb{R}}(k)$ with mild bias and that a near-optimal gap amplification exists for $\textup{QMA}^{+}(k)$ up to a critical threshold. Collectively, the results provide dimension-free methods for controlling entanglement, with wide-ranging implications for testing multipartite separability and for the boundaries between major complexity classes in quantum computation.
Abstract
Quantum entanglement is a key enabling ingredient in diverse applications. However, the presence of unwanted adversarial entanglement also poses challenges in many applications. In this paper, we explore methods to "break" quantum entanglement. Specifically, we construct a dimension-independent k-partite disentangler (like) channel from bipartite unentangled input. We show: For every $d,\ell\ge k$, there is an efficient channel $Λ: \mathbb{C}^{d\ell} \otimes \mathbb{C}^{d\ell} \to \mathbb{C}^{dk}$ such that for every bipartite separable state $ρ_1\otimes ρ_2$, the output $Λ(ρ_1\otimesρ_2)$ is close to a k-partite separable state. Concretely, for some distribution $μ$ on states from $\mathbb{C}^d$, $$ \left\|Λ(ρ_1 \otimes ρ_2) - \int | ψ\rangle \langle ψ|^{\otimes k} dμ(ψ)\right\|_1 \le \tilde O \left(\left(\frac{k^{3}}{\ell}\right)^{1/4}\right). $$ Moreover, $Λ(| ψ\rangle \langle ψ|^{\otimes \ell}\otimes | ψ\rangle \langle ψ|^{\otimes \ell}) = | ψ\rangle \langle ψ|^{\otimes k}$. Without the bipartite unentanglement assumption, the above bound is conjectured to be impossible. Leveraging our disentanglers, we show that unentangled quantum proofs of almost general real amplitudes capture NEXP, greatly relaxing the nonnegative amplitudes assumption in the recent work of QMA^+(2)=NEXP. Specifically, our findings show that to capture NEXP, it suffices to have unentangled proofs of the form $| ψ\rangle = \sqrt{a} | ψ_+ \rangle + \sqrt{1-a} | ψ_- \rangle$ where $| ψ_+ \rangle$ has non-negative amplitudes, $| ψ_- \rangle$ only has negative amplitudes and $| a-(1-a) | \ge 1/poly(n)$ with $a \in [0,1]$. Additionally, we present a protocol achieving an almost largest possible gap before obtaining QMA^R(k)=NEXP$, namely, a 1/poly(n) additive improvement to the gap results in this equality.