Dimension Independent Disentanglers from Unentanglement and Applications

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 $\Lambda: \mathbb{C}^{d\ell} \otimes \mathbb{C}^{d\ell} \to \mathbb{C}^{dk}$ such that for every bipartite separable state $\rho_1\otimes \rho_2$, the output $\Lambda(\rho_1\otimes\rho_2)$ is close to a k-partite separable state. Concretely, for some distribution $\mu$ on states from $\mathbb{C}^d$, $$ \left\|\Lambda(\rho_1 \otimes \rho_2) - \int | \psi \rangle \langle \psi |^{\otimes k} d\mu(\psi)\right\|_1 \le \tilde O \left(\left(\frac{k^{3}}{\ell}\right)^{1/4}\right). $$ Moreover, $\Lambda(| \psi \rangle \langle \psi |^{\otimes \ell}\otimes | \psi \rangle \langle \psi |^{\otimes \ell}) = | \psi \rangle \langle \psi |^{\otimes k}$. 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 $| \psi \rangle = \sqrt{a} | \psi_+ \rangle + \sqrt{1-a} | \psi_- \rangle$ where $| \psi_+ \rangle$ has non-negative amplitudes, $| \psi_- \rangle$ 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.