Nonasymptotic bounds for quantum purity amplification

Abstract: 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.

• The paper derives a nonasymptotic bound on the number of copies needed to achieve a target fidelity for pure quantum state extraction.
• It employs advanced combinatorial techniques, using the RSK algorithm and Schur-Weyl duality to translate representation-theoretic problems into probabilistic guarantees.
• The results are dimension-independent and practical for NISQ devices, offering explicit resource guidelines for high-fidelity quantum state distillation.

Nonasymptotic Quantum Purity Amplification: Finite-Sample Guarantees

Introduction

The problem of quantum purity amplification addresses a fundamental task in quantum information: extracting high-fidelity pure states from multiple copies of a noisy mixed state. Formally, given $n$ independently prepared copies of a $d$-dimensional mixed quantum state $\rho$ with unique principal eigenstate $\ket{v_d}$, the goal is to produce $k$ copies of $\ket{v_d}$ with high fidelity. While the asymptotic information-theoretic optimal sample complexities have been previously established, practical quantum systems operate in explicitly finite regimes, demanding nonasymptotic bounds that hold for finite $n$ and non-infinitesimal error $\delta$. This paper delivers such nonasymptotic guarantees, providing a sharp dimension-independent finite-sample complexity for arbitrary state spectra and general $k$.

Problem Formulation and Historical Context

Given $n$ copies of a mixed state $d$0 with eigenvalues $d$1 and associated eigenstates $d$2, one desires to distill $d$3 high-fidelity copies of $d$4. Previous results for the special cases of qubits and depolarizing noise analyzed protocols for $d$5 and provided asymptotic scaling, e.g., required $d$6 for target infidelity $d$7 as $d$8 [CEM99, LFI+25]. However, these bounds lacked guarantees for fixed, finite $d$9 and general $\rho$0, leaving a gap for any realistic use in laboratory settings.

The primary challenge in extending these results to arbitrary $\rho$1 has historically been the difficulty of generalizing the representation-theoretic techniques used in the asymptotic analyses, especially beyond the highly symmetric depolarizing scenario.

Main Results: Sharp Nonasymptotic Sample Complexity Bounds

The principal claim established in this work is the following nonasymptotic sample complexity upper bound, holding for all spectra and any target error $\rho$2: $\rho$3 suffices to output a quantum state with fidelity at least $\rho$4 with respect to the ideal pure state $\rho$5. Notably, this bound:

• Does not depend on the dimension $\rho$6: it applies equally for all $\rho$7 so long as $\rho$8.
• Holds for arbitrary state spectra: no symmetry or degeneracy assumptions are imposed, aside from uniqueness of the principal eigenvalue.
• Matches optimal asymptotic scaling for the physically-motivated case of depolarizing noise, as in [LFI+25], for all finite $\rho$9.

A rigorous combinatorial and probabilistic analysis supports this result. The proof leverages the combinatorics of Young diagrams and longest increasing subsequences (RSK correspondence) in random words for translating representation-theoretic statements into concrete probabilistic bounds. The arguments yield explicit constants and directly track error probabilities to ensure that the required $\ket{v_d}$0 delivers the specified target fidelity.

Technical Framework and Algorithmic Construction

The proposed protocol is a natural generalization of the Schur-Weyl-based purity amplification strategy. The procedure:

1. Performs weak Schur sampling on $\ket{v_d}$1, measuring the irreducible representation label $\ket{v_d}$2.
2. Projects to the $\ket{v_d}$3 irrep, discarding permutation information.
3. Applies a $\ket{v_d}$4-fold dual Clebsch–Gordan transform to extract the multiplicity space supporting the desired pure state, and executes a covariant CPTP map that outputs $\ket{v_d}$5 qudits.

Crucially, the protocol is unitarily covariant and block-diagonal in the Schur basis, simplifying both the theoretical analysis and potential implementation.

The main combinatorial tool is a refined analysis of the tableau row-lengths produced by the RSK algorithm on i.i.d. samples from the state spectrum $\ket{v_d}$6. Lemmas bounding the first two rows of random Young diagrams quantifiably control the probability that the sample does not concentrate on the principal eigenvalue, directly relating this probability to the spectral gap $\ket{v_d}$7 and failure probability $\ket{v_d}$8. The explicit form of the CPTP map and the corresponding analysis ensure the protocol's action always yields a valid quantum channel.

Comparison to Prior and Concurrent Work

The new nonasymptotic bound strictly strengthens previous results:

• Existing asymptotic results (e.g., [CEM99], [LFI+25]) did not provide concrete error bounds for finite $\ket{v_d}$9.
• The approach handles arbitrary spectra, not requiring depolarizing symmetry.
• The sample complexity is dimension-free, improving significantly over any result with explicit $k$0 dependence.
• The bound matches the lower limits concomitantly derived by Li et al. (Li et al., 20 May 2026, Li et al., 20 May 2026) and is tight up to constants for the i.i.d. setup.

The proofs also highlight that, contrary to previous conjectures, no additional logarithmic or dimension-dependent factors are necessary to achieve nonasymptotic optimality for the purity amplification task in the standard model.

Implications and Future Directions

Practically, these results give design guidelines for quantum state distillation protocols in NISQ-era and near-term quantum devices, setting explicit resource requirements to guarantee pure-state outputs under noise. They provide rigorous engineering bounds: so long as the principal eigenvalue $k$1 is gapped from $k$2, highly pure $k$3-copy outputs are always achievable with $k$4 input copies.

Theoretically, the work exemplifies the interplay of combinatorics, representation theory, and quantum information—specifically, the use of Young diagram statistics to derive sharp nonasymptotic quantum information results. The techniques are likely to impact quantum tomography, quantum error correction analysis, and hypothesis testing, wherever Schur-Weyl duality governs the structure of joint quantum state distributions.

Future research may extend these results to adaptive or streaming settings, more general noise models, or interactive protocols, and may further refine the constants in the nonasymptotic regime. Computational tractability—given the challenging implementation of full Schur sampling for large $k$5—remains an engineering concern, suggesting a fruitful direction for quantum algorithm design.

Conclusion

This work establishes precise, dimension-independent nonasymptotic sample complexity bounds for quantum purity amplification for arbitrary spectra and arbitrary number of output copies. Through an overview of combinatorial and quantum information-theoretic methods, the authors demonstrate that explicit fidelity guarantees can be met in the nonasymptotic regime, completely characterizing the resources required for quantum state purification. These results provide both a practical benchmark for experimental protocols and foundational insight into entanglement distillation and state extraction in quantum information science.

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References:

• "Nonasymptotic bounds for quantum purity amplification" (2605.27262)
• Li, Zhaoyi et al., "Quantum purity amplification for arbitrary eigenstates and multiple outputs" (Li et al., 20 May 2026)
• Cirac, J. I., Ekert, A. K., Macchiavello, C. (1999), Physical Review Letters, 82(21): 4344 ("Optimal purification of single qubits")
• Li, Zhaoyi et al., "Optimal quantum purity amplification" (Li et al., 2024)