Dimension Independent Disentanglers from Unentanglement and Applications
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2024
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| _version_ | 1866909118283907072 |
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| author | Jeronimo, Fernando G. Wu, Pei |
| author_facet | Jeronimo, Fernando G. Wu, Pei |
| contents | 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. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2402_15282 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Dimension Independent Disentanglers from Unentanglement and Applications Jeronimo, Fernando G. Wu, Pei Quantum Physics Computational Complexity 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. |
| title | Dimension Independent Disentanglers from Unentanglement and Applications |
| topic | Quantum Physics Computational Complexity |
| url | https://arxiv.org/abs/2402.15282 |