Random Real Valued and Complex Valued States Cannot be Efficiently Distinguished
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arXiv
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| Format: | Preprint |
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2024
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| _version_ | 1866916448953171968 |
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| author | Schatzki, Louis |
| author_facet | Schatzki, Louis |
| contents | In this short note we show that the ensemble $\{O \vert 0\rangle \langle 0 \vert O^\top \ \vert \ O \in \mathbb{O(d)}\}$, where $O$ is drawn from the Haar measure on $\mathbb{O}(d)$ cannot be distinguished from $t$ copies of a Haar random state unless $t = Ω(\sqrt{d})$. Our proof has the benefit of exactly computing the trace distance, which scales as $Θ(t^2/d)$ for $t = O(\sqrt{d})$, between the moments as well as being surprisingly short. Lastly, we show that twirling certain states with orthogonal matrices yields exact $t=3$ designs, yet the same cannot be true for $t>3$. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2410_17213 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Random Real Valued and Complex Valued States Cannot be Efficiently Distinguished Schatzki, Louis Quantum Physics In this short note we show that the ensemble $\{O \vert 0\rangle \langle 0 \vert O^\top \ \vert \ O \in \mathbb{O(d)}\}$, where $O$ is drawn from the Haar measure on $\mathbb{O}(d)$ cannot be distinguished from $t$ copies of a Haar random state unless $t = Ω(\sqrt{d})$. Our proof has the benefit of exactly computing the trace distance, which scales as $Θ(t^2/d)$ for $t = O(\sqrt{d})$, between the moments as well as being surprisingly short. Lastly, we show that twirling certain states with orthogonal matrices yields exact $t=3$ designs, yet the same cannot be true for $t>3$. |
| title | Random Real Valued and Complex Valued States Cannot be Efficiently Distinguished |
| topic | Quantum Physics |
| url | https://arxiv.org/abs/2410.17213 |