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.
In quantum purity amplification, one is given n copies of a noisy quantum state ρ∈Cd×d and asked to prepare k copies of its principal eigenstate ∣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 n tends to infinity. In this paper, we establish the following nonasymptotic guarantee: if ρ's eigenvalues are sorted p1≤⋯≤pd and pd−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−δ with ∣vd⊗k⟩. 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.
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 Dk,min, achieving worst-case all-site error ε scales as O(m/(εDk,min2)). 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.
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 d-dimensional states of rank at most r: producing M additional copies with worst-case fidelity at least 1−ε requires and is achievable with 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=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.
We study the following quantum state identity problem: given access to copies of n unknown pure quantum states, the goal is to determine whether every pair has fidelity at least 1−ε or some pair has fidelity at most ε. This problem relaxes the identical-or-orthogonal promise considered in previous work. For any fixed overlap parameter ε∈(0,1/4), we present two quantum algorithms with complementary guarantees on circuit depth and sample complexity. The first runs in constant depth using parallel swap tests and takes O(log(n/δ)log(2/δ)) copies of each input state to succeed with probability at least 1−δ. The second uses the Schur transform to reduce the sample complexity to O(log(n/δ)), through an analysis relating Jucys--Murphy elements to Schur sampling. We also prove a matching sample complexity lower bound of Ω(log(n/δ)) for any quantum algorithm with failure probability at most δ≤1/2−c, for any constant c>0, establishing that the second algorithm achieves the optimal dependence on both n and δ for fixed ε.
Given many copies of an unknown quantum state ρ, we consider the task of learning a classical description of its principal eigenstate. Namely, assuming that ρ has an eigenstate ∣ϕ⟩ with (unknown) eigenvalue λ>1/2, the goal is to learn a (classical shadows style) classical description of ∣ϕ⟩ which can later be used to estimate expectation values ⟨ϕ∣O∣ϕ⟩ for any O in some class of observables. We consider the sample-complexity setting in which generating a copy of ρ is expensive, but joint measurements on many copies of the state are possible. We present a protocol for this task scaling with the principal eigenvalue λ and show that it is optimal within a space of natural approaches, e.g., applying quantum state purification followed by a single-copy classical shadows scheme. Furthermore, when λ is sufficiently close to 1, the performance of our algorithm is optimal--matching the sample complexity for pure state classical shadows.
Standard quantum inference converts quantum data into classical outputs. We study an alternative inference setting in which the desired output is quantum. Such settings include quantum purity amplification (QPA), random purification and approximate cloning, and density matrix exponentiation. We show that such protocols can achieve exponentially lower sample complexity than incoherent, measurement-mediated protocols, establishing, to the best of our knowledge, the first nontrivial separations between dimension-independent coherent and dimension-dependent incoherent sample complexity. For QPA with principal-eigenstate targets and d-dimensional inputs, at fixed output number and spectral gap, coherent processing achieves error ε using O(1/ε) copies, whereas any incoherent protocol requires Ω(d/ε) copies. Together, these separations motivate a unified picture of coherent quantum inference, in which an entanglement-breaking limit identifies the optimal incoherent counterpart.
The Uhlmann fidelity F(ρ0,ρ1)=tr∣ρ0ρ1∣ is one of the most fundamental quantities in quantum information theory for quantifying the closeness between two quantum states. Estimating the Uhlmann fidelity to within additive error ε requires a number of copies of the states, or queries to their state-preparation circuits, that depends at least linearly on the smaller of the ranks of ρ0 and ρ1. Consequently, this rank dependence disappears when either state is pure, in which case the query and sample complexities depend only polynomially on 1/ε. However, the known optimal estimator for F(ρ,∣ψ⟩⟨ψ∣) due to Fang and Wang (ESA 2025) requires prior knowledge of which state is pure.
In this work, we remove this mathematically unnecessary prior-knowledge requirement and establish an optimal estimator for F(ρ,∣ψ⟩⟨ψ∣) under the sole promise that one of the two states is pure, without knowing which one. Our estimator is obtained by specializing the refined algorithmic Uhlmann transform of Utsumi, Nakata, Wang, and Takagi (2025) to the case where one state is pure. In this setting, the Uhlmann fidelity can be recovered as follows: apply a unitary dilation of trA(∣ψ0⟩⟨ψ1∣) (or its inverse) to the reference register R of the purification ∣ψ1⟩ (or ∣ψ0⟩) on the registers A and R, estimate the corresponding square-root amplitude in each case, and take the maximum of the resulting two estimates.