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arXiv:2609.08745

Uniqueness and Cramér-Rao Efficiency of Quantum U-Statistics

Ayanava Dasgupta, Naqueeb Ahmad Warsi, Premanshu Chatterjee

quant-phmath.STSubmitted Sep 8, 2026• Announced Sep 9, 2026• 0 citations
v1 · Sep 8, 2026v2 · Sep 30, 2026
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Abstract

We study unbiased estimation of scalar-valued polynomial functionals of quantum states from independent copies. We establish an equivalence between the first-order marginal of a permutation-invariant finite-copy observable and the functional gradient. We then prove that, among unbiased permutation-invariant estimators, the quantum U-statistic is the unique extension to an arbitrary number of copies. We further derive a universal variance expansion in which the leading $1/n$ term is determined by the variance of the functional gradient, while higher-order contributions are of order $O(1/n^2)$. This leading variance coincides with the multiparameter quantum Cramér-Rao limit, establishing asymptotic efficiency of quantum U-statistics. We also characterize the higher-order scaling at points where the variance of the first-order gradient vanishes. As an application, we analyze the Bures $χ^2$-divergence and show that a spectral lower bound on the reference state is sufficient but not necessary for bounded-variance estimation.

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We study unbiased estimation of scalar-valued polynomial functionals of quantum states from independent copies. We establish an equivalence between the first-order marginal of a permutation-invariant finite-copy observable and the functional gradient. We then prove that, among unbiased permutation-invariant estimators, the quantum U-statistic is the unique extension to an arbitrary number of copies. We further derive a universal variance expansion in which the leading 1/n1/n1/n term is determined by the variance of the functional gradient, while higher-order contributions are of order O(1/n2)O(1/n^2)O(1/n2). This leading variance coincides with the multiparameter quantum Cramér-Rao limit, establishing asymptotic efficiency of quantum U-statistics. We also characterize the higher-order scaling at points where the variance of the first-order gradient vanishes. As an application, we analyze the Bures χ2χ^2χ2-divergence and show that a spectral lower bound on the reference state is sufficient but not necessary for bounded-variance estimation.
The quantum Cramér-Rao bound (QCRB) as the ultimate lower bound for precision in quantum parameter estimation is only known to be saturable in the multiparameter setting in special cases and under conditions such as full or average commutavity of the symmetric logarithmic derivatives (SLDs) associated with the parameters. Moreover, for general mixed states, collective measurements over infinitely many identical copies of the quantum state are generally required to attain the QCRB. In the important and experimentally relevant single-copy scenario, a necessary condition for saturating the QCRB in the multiparameter setting for general mixed states is the so-called partial commutativity condition on the SLDs. However, it is not known if this condition is also sufficient. This paper establishes necessary and sufficient conditions for saturability of the multiparameter QCRB in the single-copy setting in terms of the commutativity of a set of projected SLDs and the existence of a unitary solution to a system of nonlinear partial differential equations. New necessary conditions that imply partial commutativity are also obtained, which together with another condition become sufficient. Moreover, when the sufficient conditions are satisfied an optimal measurement saturating the QCRB can be chosen to be projective and explicitly characterized. An example is developed to illustrate the case of a multiparameter quantum state where the conditions derived herein are satisfied and can be explicitly verified.
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Quantum parameter estimation theory is an important component of quantum information theory and provides the statistical foundation that underpins important topics such as quantum system identification and quantum waveform estimation. When there is more than one parameter the ultimate precision in the mean square error given by the quantum Cramér-Rao bound is not necessarily achievable. For non-full rank quantum states, it was not known when this bound can be saturated (achieved) when only a single copy of the quantum state encoding the unknown parameters is available. This single-copy scenario is important because of its experimental/practical tractability. Recently, necessary and sufficient conditions for saturability of the quantum Cramér-Rao bound in the multiparameter single-copy scenario have been established in terms of i) the commutativity of a set of projected symmetric logarithmic derivatives and ii) the existence of a unitary solution to a system of coupled nonlinear partial differential equations. New sufficient conditions were also obtained that only depend on properties of the symmetric logarithmic derivatives. In this paper, key structural properties of optimal measurements that saturate the quantum Cramér-Rao bound are illuminated. These properties are exploited to i) show that the sufficient conditions are in fact necessary and sufficient for an optimal measurement to be projective, ii) give an alternative proof of previously established necessary conditions, and iii) describe general POVMs, not necessarily projective, that saturate the multiparameter QCRB. Examples are given where a unitary solution to the system of nonlinear partial differential equations can be explicitly calculated when the required conditions are fulfilled.
In multiparameter quantum estimation, incompatible measurements can impose trade-offs among estimation errors. Understanding the precision attainable with a finite number of jointly measured copies is therefore important, but direct optimization becomes costly as the tensor product space grows exponentially. For independent and identically distributed (i.i.d.) copies, the quantum statistical model is invariant under permutations of the copies. We exploit this symmetry through Schur--Weyl duality to remove permutation multiplicity spaces while using their dimensions as weights. This gives a reduced statistical model equivalent to the original under arbitrary collective measurements. On this reduced model, we derive an exact conic formulation of the tight Cramér--Rao bound and a semidefinite program for its Nagaoka--Hayashi (NH) relaxation. For fixed single-copy dimension, the reduced operator representation grows polynomially rather than exponentially with the number of copies. To approximate the tight bound, we use an exact quadratic identity to correct finite-ray dual solutions and obtain rigorous upper and lower bounds. For fixed copy number and parameter dimension, sufficiently fine meshes admit feasible primal points, and dual solutions with bounded objective gaps have uniformly bounded compact-search radii. These results yield quadratic convergence in the projective covering radius. Numerical examples illustrate the quadratic scaling of the mesh correction term and the computational reach of the reduced NH formulation.
The Holevo Cramér-Rao bound sets the asymptotically attainable precision of quantum multiparameter estimation, but its attainability with finite quantum state copies remains unresolved in general. While a gap between the single-copy optimum and the Holevo bound persists at any finite copy number for two-parameter estimation models, the higher-parameter case has remained open. Here we disprove universal finite-copy gap persistence beyond two parameters by constructing a three-parameter rank-two model whose Holevo bound is unattainable with one copy but attained exactly with two. We further prove finite-copy persistence for full-rank models with any number of parameters. We then experimentally implement the two-copy collective measurement for the constructed model using superconducting qubits. We observe a two-copy estimation error below the single-copy and separate-measurement bounds determined from state tomography. By introducing controlled depolarization, we show that this precision advantage persists over a finite noise range. Our results connect the attainability of ultimate quantum precision to the structure of the multiparameter estimation model and the resources available for collective measurements.
We consider estimation of a single unknown parameter embedded in a quantum state. Quantum Cramér-Rao bound (QCRB) is the ultimate limit of the mean squared error for any unbiased estimator. While it can be achieved asymptotically for a large number of quantum state copies, the measurement required often depends on the true value of the parameter of interest. Prior work addresses this paradox using a two-stage approach: in the first stage, a preliminary estimate is obtained by applying, on a vanishing fraction of quantum state copies, a sub-optimal measurement that does not depend on the parameter of interest. In the second stage, the preliminary estimate is used to construct the QCRB-achieving measurement that is applied to the remaining quantum state copies. This is akin to two-step estimators for classical problems with nuisance parameters. Unfortunately, the original analysis imposes conditions that severely restrict the class of classical estimators applied to the quantum measurement outcomes, hindering applications of this method. We relax these conditions to substantially broaden the class of usable estimators for single-parameter problems at the cost of slightly weakening the asymptotic properties of the two-stage method. We also account for nuisance parameters. We apply our results to obtain the asymptotics of quantum-enhanced transmittance sensing.