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| Format: | Preprint |
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2026
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| Online Access: | https://arxiv.org/abs/2605.27262 |
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| _version_ | 1866913164395806720 |
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| author | Scharnhorst, Thilo Spilecki, Jack Wright, John |
| author_facet | Scharnhorst, Thilo Spilecki, Jack Wright, John |
| contents | In quantum purity amplification, one is given $n$ copies of a noisy quantum state $ρ\in \mathbb{C}^{d \times d}$ and asked to prepare $k$ copies of its principal eigenstate $|v_d\rangle$. Several prior works have derived information-theoretically optimal algorithms for this problem, but the bounds they prove are only shown in the asymptotic regime as the number of samples $n$ tends to infinity. In this paper, we establish the following nonasymptotic guarantee: if $ρ$'s eigenvalues are sorted $p_1 \leq \cdots \leq p_d$ and $p_{d-1} < p_d$, then \begin{equation*}
n = O\Big(k + \frac{k}δ \cdot \frac{1-p_d}{(p_d-p_{d-1})^2}\Big)
\end{equation*} copies suffice to output a state with fidelity at least $1-δ$ with $|v_d^{\otimes k}\rangle$. Our bound holds for arbitrary spectra, and is independent of the dimension $d$. In the case of depolarizing noise, our finite-sample guarantee matches the optimal asymptotic scaling. Our proof is based on the combinatorics of random Young diagrams. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2605_27262 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Nonasymptotic bounds for quantum purity amplification Scharnhorst, Thilo Spilecki, Jack Wright, John Quantum Physics In quantum purity amplification, one is given $n$ copies of a noisy quantum state $ρ\in \mathbb{C}^{d \times d}$ and asked to prepare $k$ copies of its principal eigenstate $|v_d\rangle$. Several prior works have derived information-theoretically optimal algorithms for this problem, but the bounds they prove are only shown in the asymptotic regime as the number of samples $n$ tends to infinity. In this paper, we establish the following nonasymptotic guarantee: if $ρ$'s eigenvalues are sorted $p_1 \leq \cdots \leq p_d$ and $p_{d-1} < p_d$, then \begin{equation*} n = O\Big(k + \frac{k}δ \cdot \frac{1-p_d}{(p_d-p_{d-1})^2}\Big) \end{equation*} copies suffice to output a state with fidelity at least $1-δ$ with $|v_d^{\otimes k}\rangle$. Our bound holds for arbitrary spectra, and is independent of the dimension $d$. In the case of depolarizing noise, our finite-sample guarantee matches the optimal asymptotic scaling. Our proof is based on the combinatorics of random Young diagrams. |
| title | Nonasymptotic bounds for quantum purity amplification |
| topic | Quantum Physics |
| url | https://arxiv.org/abs/2605.27262 |