Copying an unknown quantum state into many approximate copies is known to become equivalent to estimating it and re-preparing: the best measure-and-prepare copier approaches the best quantum copier as the number of copies grows. We determine how fast. For equatorial qubit states, one input copy and m output copies, judged by the global fidelity, we prove F_EB/F_coh = 1 − 1/√(2m) + O(1/m). For phase states on d levels we prove, with explicit constants, the law 1 − F_EB/F_coh = (d − 1)/√(2m) + O(1/m), confirmed by exact computations up to m = 400; for qubits with T input copies the gap is √2·σ_T/√m, proved with explicit constants for T = 2, 3 (the upper bound conditionally on the optimality of the flat seed, R0(T); the lower bound unconditionally). The measure-and-prepare copier's deficit is the ground-state energy of a harmonic oscillator, one quantum per unknown phase. The exact quantum fidelity has plateaus at m = kd and kd + 1 (m = kd + 1 is the economical point). Numerical results suggest a period-d modulation of the ratio at order 1/m; a proof of its next-order coefficient remains open. This complements the known universality of the cloning fidelity itself, which depends only on the number of free parameters: within the phase-state family, the leading gap coefficient counts the independent phase directions. The classical values rest on the flat-seed theorem of the companion Paper 5, and the several-copy quantum values on the reduced coherent dual of the companion Paper 5. Core lemmas of the qubit proof and its explicit upper-bound constants are machine-checked in Lean 4; the remaining steps are proved on paper and independently refereed.
Copying an unknown quantum state into many approximate copies is known to become equivalent to estimating it and re-preparing: the best measure-and-prepare copier approaches the best quantum copier as the number of copies grows. We determine how fast. For equatorial qubit states, one input copy and m output copies, judged by the global fidelity, we prove F_EB/F_coh = 1 − 1/√(2m) + O(1/m). For phase states on d levels we prove, with explicit constants, the law 1 − F_EB/F_coh = (d − 1)/√(2m) + O(1/m), confirmed by exact computations up to m = 400; for qubits with T input copies the gap is √2·σ_T/√m, proved with explicit constants for T = 2, 3 (the upper bound conditionally on the optimality of the flat seed, R0(T); the lower bound unconditionally). The measure-and-prepare copier's deficit is the ground-state energy of a harmonic oscillator, one quantum per unknown phase. The exact quantum fidelity has plateaus at m = kd and kd + 1 (m = kd + 1 is the economical point). Numerical results suggest a period-d modulation of the ratio at order 1/m; a proof of its next-order coefficient remains open. This complements the known universality of the cloning fidelity itself, which depends only on the number of free parameters: within the phase-state family, the leading gap coefficient counts the independent phase directions. The classical values rest on the flat-seed theorem of the companion Paper 5, and the several-copy quantum values on the reduced coherent dual of the companion Paper 5. Core lemmas of the qubit proof and its explicit upper-bound constants are machine-checked in Lean 4; the remaining steps are proved on paper and independently refereed.
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 determine exactly how well a measure-and-prepare ("measure-first") forger can copy qudit phase states. For one input copy and every number m of output copies, in every dimension d, we prove that the flat seed is the optimal measure-first strategy for the continuous family of phase states, the optimum being the top eigenvalue of an explicit matrix on the symmetric subspace (for example (7d − 8 + √(25d² − 64d + 48))/(2d³) at 1→2). The proof reduces the question to a negative-correlation inequality for closed lattice walks in symmetric time-dependent windows, which we prove by summation by parts; this combinatorial core is verified in the Lean 4 proof assistant. Consequently, for cyclic phase-state codebooks ψ_x = d^(−1/2) Σ_j ω^(x·k_j) |j⟩, the states used in quantum fingerprinting and quantum hashing, in every dimension and every 1→m task the measure-first optimum equals the continuous one if and only if the keys form a B_{m+1} set, whereas at 1→2 the coherent optimum does if and only if they form a Sidon set. Around this law we compare, for every copying task T→T+k, the optimal coherent copier (any quantum channel) and the optimal measure-first copier on a finite codebook with the continuous family: both finite-codebook optima are at least the continuous ones; if the keys have no additive collisions among multisets of size 2T+k, both are equal for every dimension; and for codebooks that are collision-free at size T, every collision at size 2T+k strictly increases the value of the natural measure-first strategy. For cyclic qubit codebooks of q states and every 1→m task (m ≥ 2) we prove an exact leak-order theorem: the measure-first optimum equals the continuous one if and only if q ≥ m+2, while the coherent optimum already equals it when q ≥ m+1, so q = m+1 separates the two attackers for every m. Pre-registered experiments on IBM superconducting processors, in which the strategies are emulated by classically recombining two circuit stages, confirm the predicted qubit gaps at 1→2 and 1→3. We give an explicit polynomial-phase hash with a guaranteed (sufficient) copy budget and prove a separation for it. Whether measure-first attackers leak no later than coherent ones for every codebook remains open; the stronger statement that the measure-first gap always exceeds the coherent gap is false for general finite phase codebooks.
Phase-state codebooks ψ_x = d^{-1/2} Σ_j ω^{x k_j} |j⟩ are the states used in quantum fingerprinting and quantum hashing. We compare, for every copying task T→T+k, the optimal coherent copier (any quantum channel) and the optimal measure-first copier (entanglement-breaking: measure, then prepare) on a finite codebook with the same quantities for the continuous family of all phase states. Both finite-codebook optima are at least the continuous ones; if the keys have no additive collisions among multisets of size 2T+k, both are equal for every dimension; and for codebooks that are collision-free at size T, every collision at size 2T+k strictly increases the value of the natural measure-first strategy. For cyclic qubit codebooks of q states and every 1→m task (m ≥ 2) we prove an exact leak-order theorem: the measure-first optimum equals the continuous one if and only if q ≥ m+2, while the coherent optimum already equals it when q ≥ m+1, so q = m+1 separates the two attackers for every m, beginning with three qubit states at 1→2. For one input copy and two or three output copies we determine the measure-first optimum of the continuous family in every dimension d ≥ 3—for example (7d−8+√(25d²−64d+48))/(2d³) at 1→2—so that for cyclic codebooks in every dimension the coherent 1→2 optimum equals the continuous one if and only if the keys form a Sidon set, while the measure-first optimum does if and only if they form a B_3 set. We give an explicit polynomial-phase hash with a guaranteed (sufficient) copy budget and prove a separation for it. Whether measure-first attackers leak no later than coherent ones for every codebook remains open; the stronger statement that the measure-first gap always exceeds the coherent gap is false for general finite phase codebooks. Files: the paper (13 pages); hsep_qhn_certificates.zip — every certificate, independent checkers from three systems, expected logs and a SHA-256 manifest (SHA-256 2d9d96c45902b036bd1495569a0d972f5199567dc15fa22eb01611c633708413); paper4_source.zip — LaTeX source.
We determine the optimal average global fidelity of T→T+k approximate cloning for the uniform ensemble of n-qubit stabilizer states, for every number of qubits n, every k ≥ 1 and every 1 ≤ T ≤ 3. With D = 2^n, the optimum is P_stab(n;T,k) = C(D+T−1, T) · F_{T+k−κ(k)}, where F_s = ∏_{i=0}^{s−2}(2^i+1) / (D ∏_{i=0}^{s−2}(D+2^i)) is the stabilizer frame potential and κ(k) is the largest dimension of a doubly-even binary code of length k. The result rests on an exact formula for the largest eigenvalue of the mixed stabilizer moment operators: for all a+b ≥ 1 and all n it equals F_{a+b−w}, where w = min(a,b) + κ(|a−b|) is the Witt index of an associated Z4-valued quadratic form. We then ask what is lost by measuring first. For measure-and-prepare (entanglement-breaking) cloners we show that, after the pretty-good measurement, re-preparing a better output than copies of the guess helps exactly from four output copies on, because fourth powers erase the phases ±1, ±i of stabilizer amplitudes; that for 1→4 the optimum is 3(D+48+√(D²+16D+1664)) / ((D+1)(D+2)(D+4)(D+8)) for every n, so coherent cloning wins by less than a factor 5; and that for 2→4 there are two regimes: the pretty-good-measurement strategy is optimal for n ≤ 3, while for every n ≥ 4 Bell sampling is optimal, with fidelity 3/(D(D+2)) and a coherent advantage of exactly 5D/(D+4). The upper bounds are explicit dual certificates for the PPT relaxation, covering every n. Version 2 adds Part II (measure-and-prepare cloning), Part III (open problems), a certificates appendix, and citation fixes. Part I results and theorem numbering are unchanged from version 1. Files:• Weldon_2026_Exact_optimal_cloning_of_stabilizer_states_v2.pdf — the paper (15 pages).• paper3_v2_certificates.zip — every Part II certificate with standalone checkers, run scripts, expected logs and a SHA-256 manifest (SHA-256 236f7e4c5405e8e4030d476958e51aae5fb7b2fd2349c3ec9ddcb882cddf054f).• paper3_code.zip — Part I code and certificates, unchanged from version 1 (SHA-256 de2fc42b8bc4fa710b8506de2001d475a8d8b4a011ed0df72c693110a5d417e0).• paper3_v2_source.zip — LaTeX source.
Since the initial discovery of the Wootters-Zurek no-cloning theorem, a wide variety of quantum cloning machines have been proposed aiming at imperfect but optimal cloning of quantum states within its own context. Remarkably, most previous studies have employed the Bures fidelity or the Hilbert-Schmidt norm as the figure of merit to characterize the quality of the corresponding cloning scenarios. However, in many situations, what we truly care about is the relevant information about certain parameters encoded in quantum states. In this work, we investigate the multiple phase estimation problem in the framework of quantum cloning machines, from the perspective of quantum Fisher information matrix (QFIM). Focusing on the generalized d-dimensional equatorial states, we obtain the analytical formulas of QFIM for both universal quantum cloning machine (UQCM) and phase-covariant quantum cloning machine (PQCM), and prove that PQCM indeed performs better than UQCM in terms of QFIM. We highlight that our method can be generalized to arbitrary cloning schemes where the fidelity between the single-copy input and output states is input-state independent. Furthermore, the attainability of the quantum Cram\'er-Rao bound is also explicitly discussed.
Topological phases of matter are characterized by global properties of quantum states beyond conventional local order parameters. The Chern number is a central invariant in this characterization, linking the geometry of quantum states to robust physical observables and making its determination key to understanding quantum matter. We construct a nearly optimal quantum algorithm for estimating the Chern number of a smooth Hamiltonian family over a two-dimensional parameter torus with a unique, gapped ground state, including interacting many-body systems. The algorithm transports and reuses two ground-state copies prepared at a single reference point and uses generalized quantum signal processing to coherently accumulate geometric phases. It achieves a Hamiltonian-oracle evolution time of O~(LxLy/Δmin3), where Lx,Ly bound the parameter derivatives of the Hamiltonian and Δmin is a known spectral-gap lower bound. We prove a matching worst-case oracle lower bound, even when the Chern number is promised to be zero or one, establishing optimality up to polylogarithmic factors. For local Hamiltonian families with an inverse-polynomial gap and a supplied guiding state, exact Chern-number computation is in FBQP and is BQP-hard, and merely deciding whether the Chern number is zero or one is already BQP-complete. These results establish how efficiently quantum computers can estimate a quantized topological invariant and provide evidence for quantum advantage in characterizing topological phases.