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arXiv:2605.21570 · Physical Review A

Quantum Purity Amplification for Arbitrary Eigenstates and Multiple Outputs

Zhaoyi Li, Elias Theil, Aram W. Harrow, Isaac Chuang

quant-phSubmitted May 20, 2026• Announced May 21, 2026• 0 citations• Revised Sep 29, 2026 (v2)
v1 · May 20, 2026v2 · Sep 29, 2026
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Abstract

Quantum purity amplification (QPA) is the task of coherently transforming $n$ copies of a mixed state into high-fidelity copies of a chosen eigenstate. We develop a general theory of QPA for $n$ input copies, $m$ output copies, and arbitrary choices of target eigenstate, local dimension $d$, and input spectrum. We introduce the overhang-removal protocol, derive its all-site and one-site fidelities across output regimes, and show that it is optimal in several regimes, both asymptotically and nonasymptotically. For the asymptotic analysis, we use a path-graph parametrization to show that, when the target eigenvalue has a constant spectral gap $D_{k,\mathrm{min}}$, achieving worst-case all-site error $\varepsilon$ scales as $O(m/(\varepsilon D_{k,\mathrm{min}}^2))$. When $m/n$ approaches a constant, the performance exhibits phase-like regimes, which we characterize explicitly. For the nonasymptotic analysis, we develop a theory of generalized Young diagrams that yields tight sample complexity bounds and provides the first tight dimension-uniform guarantee for optimal QPA. We also provide asymptotically efficient implementations of the overhang-removal protocol. Together, these results establish QPA as a rigorous example of coherent quantum information processing with dimension-uniform sample complexity, supplying the technical foundation for the coherent-incoherent separation developed in the companion work.

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16+73 pages, 10+3 figures, 2+0 tables

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Quantum purity amplification (QPA) is the task of coherently transforming nnn copies of a mixed state into high-fidelity copies of a chosen eigenstate. We develop a general theory of QPA for nnn input copies, mmm output copies, and arbitrary choices of target eigenstate, local dimension ddd, and input spectrum. We introduce the overhang-removal protocol, derive its all-site and one-site fidelities across output regimes, and show that it is optimal in several regimes, both asymptotically and nonasymptotically. For the asymptotic analysis, we use a path-graph parametrization to show that, when the target eigenvalue has a constant spectral gap Dk,minD_{k,\mathrm{min}}Dk,min​, achieving worst-case all-site error ε\varepsilonε scales as O(m/(εDk,min2))O(m/(\varepsilon D_{k,\mathrm{min}}^2))O(m/(εDk,min2​)). When m/nm/nm/n approaches a constant, the performance exhibits phase-like regimes, which we characterize explicitly. For the nonasymptotic analysis, we develop a theory of generalized Young diagrams that yields tight sample complexity bounds and provides the first tight dimension-uniform guarantee for optimal QPA. We also provide asymptotically efficient implementations of the overhang-removal protocol. Together, these results establish QPA as a rigorous example of coherent quantum information processing with dimension-uniform sample complexity, supplying the technical foundation for the coherent-incoherent separation developed in the companion work.
In quantum purity amplification, one is given nnn copies of a noisy quantum state ρ∈Cd×dρ\in \mathbb{C}^{d \times d}ρ∈Cd×d and asked to prepare kkk copies of its principal eigenstate ∣vd⟩|v_d\rangle∣vd​⟩. 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 nnn tends to infinity. In this paper, we establish the following nonasymptotic guarantee: if ρρρ's eigenvalues are sorted p1≤⋯≤pdp_1 \leq \cdots \leq p_dp1​≤⋯≤pd​ and pd−1<pdp_{d-1} < p_dpd−1​<pd​, 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−δ1-δ1−δ with ∣vd⊗k⟩|v_d^{\otimes k}\rangle∣vd⊗k​⟩. Our bound holds for arbitrary spectra, and is independent of the dimension ddd. 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.
Quantum state cloning is the task of approximately producing additional copies of an unknown quantum state from a finite number of input copies. The optimal cloning fidelity is known exactly for pure states, but no comparable characterization is known for general mixed states. We determine the optimal asymptotic copy complexity for ddd-dimensional states of rank at most rrr: producing MMM additional copies with worst-case fidelity at least 1−ε1-\varepsilon1−ε requires and is achievable with N=Θ(Mrd/ε)N=Θ(Mrd/\varepsilon)N=Θ(Mrd/ε) input copies. Remarkably, the lower bound already holds for a family of states with a fixed flat spectrum, while the matching upper bound is achieved by random purification followed by optimal pure-state cloning. For M=1M=1M=1, we further show that high-fidelity tomography can be coherently converted into cloning with comparable error, revealing an operational origin of the matching cloning and tomography complexities.
Quantum purity amplification (QPA) provides a novel approach to counteracting the pervasive noise that degrades quantum states. We present the optimal QPA protocol for general quantum systems and global noise, resolving a two-decade open problem. Under strong depolarization, our protocol achieves an exponential reduction in sample complexity over the best-known methods. We provide an efficient implementation of the protocol based on generalized quantum phase estimation. Additionally, we introduce SWAPNET, a sparse and shallow circuit that enables QPA for near-term experiments. Simulations in both digital and analog quantum settings, along with experiments on superconducting quantum processors, confirm the protocol's robustness and practical utility. Our findings suggest that QPA could improve the performance of quantum information processing tasks, particularly in the context of Noisy Intermediate-Scale Quantum (NISQ) devices, where reducing the effect of noise with limited resources is critical.
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We determine asymptotic global fidelities of cloning generic mixed states with simple (non-degenerate) full-rank spectra. In fixed dimension ddd, let nnn inputs and mnm_nmn​ outputs satisfy mn/n→γ>1m_n/n\toγ>1mn​/n→γ>1. For a known simple full-rank spectrum p1>⋯>pd>0p_1>\cdots>p_d>0p1​>⋯>pd​>0, we prove that the optimal root fidelity between the entire output and the product target state is $$ F_\rm{orb}(γ,p) =\prod_{i<j} \frac{\sqrtγ+\sqrt{q_{ij}(γ-1+q_{ij})}} {γ+q_{ij}}, \qquad q_{ij}=\frac{p_j}{p_i}. $$ When the spectrum is unknown, one spectrum-independent cloner attains $$ F_\rm{univ}(γ,p) =\left(\frac{2\sqrtγ}{1+γ}\right)^{(d-1)/2} F_\rm{orb}(γ,p), $$ uniformly on compact subsets of the simple full-rank chamber and is minimax optimal. The two factors quantify spectral and eigenbasis uncertainty. Our channels combine classical processing of Schur--Weyl labels with a covariant Cartan-sector channel. Local asymptotic normality and a Gaussian fidelity bound against arbitrary channels yield the converses. For flat projector states ρ=P/rρ=P/rρ=P/r, with PPP a rank-rrr orthogonal projector, we separately prove the optimum γ−r(d−r)/2γ^{-r(d-r)/2}γ−r(d−r)/2. Furthermore, we estimate the exact asymptotic fidelity of purify--clone--trace (PCT), and show that our cloners strictly outperform PCT at every fixed gain for simple full-rank spectra and rank-rrr projector states with 1<r<d1<r<d1<r<d.
Given multiple copies of a mixed quantum state with an unknown, nondegenerate principal eigenspace, quantum state purification is the task of recovering a quantum state that is closer to the principal eigenstate. A streaming protocol relying on recursive swap tests has been proposed and analysed for noisy depolarized states with arbitrary dimension and noise level. Here, we show that the same algorithm applies much more broadly, enabling the purification of arbitrary mixed states with a nondegenerate principal eigenvalue. We demonstrate this through two approaches. In the first approach, we show that, given the largest two eigenvalues, the depolarized noise is the most difficult noise to purify for the recursive swap tests, thus the desirable bounds on performance and cost follow from prior work. In the second approach, we provide a new and direct analysis for the performance of purification using recursive swap tests for the more general noise. We also derive simple lower bounds on the sample complexity, showing that the recursive swap test algorithm attains optimal sample complexity (up to a constant factor) in the low-noise regime.