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.
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.
One key aspect of quantum metrology, measurement incompatibility, is evident only through the simultaneous estimation of multiple parameters. The symmetric logarithmic derivative Cramér-Rao bound (SLDCRB), gives the attainable precision, if the optimal measurements for estimating each individual parameter commute. When the optimal measurements do not commute, the SLDCRB is not necessarily attainable. In this regard, the Holevo Cramér-Rao bound (HCRB) plays a fundamental role, providing the ultimate attainable precisions when one allows simultaneous measurements on infinitely many copies of a quantum state. For practical purposes, the Nagaoka Cramér-Rao bound (NCRB) is more relevant, applying when restricted to measuring quantum states individually. The interplay between these three bounds dictates how rapidly the ultimate metrological precisions can be approached through collective measurements on finite copies of the probe state. We first consider two parameter estimation and prove that if the HCRB cannot be saturated with a single copy of the probe state, then it cannot be saturated for any finite number of copies of the probe state. With this, we show that it is impossible to saturate the HCRB for several physically motivated problems. For estimating any number of parameters, we provide necessary and sufficient conditions for the attainability of the SLDCRB with separable measurements. We further prove that if the SLDCRB cannot be reached with a single copy of the probe state, it cannot be reached with collective measurements on any finite number of copies of the probe state. These results together provide necessary and sufficient conditions for the attainability of the SLDCRB for any finite number of copies of the probe state. This solves a significant generalisation of one of the five problems recently highlighted by [P.Horodecki et al, Phys. Rev. X Quantum 3, 010101 (2022)].
Only with the simultaneous estimation of multiple parameters are the quantum aspects of metrology fully revealed. This is due to the incompatibility of observables. The fundamental bound for multiparameter quantum estimation is the Holevo Cramér-Rao bound (HCRB) whose evaluation has so far remained elusive. For finite-dimensional systems we recast its evaluation as a semidefinite program, with reduced size for rank-deficient states. We show that it also satisfies strong duality. We use this result to study phase and loss estimation in optical interferometry and three-dimensional magnetometry with noisy multiqubit systems. For the former, we show that, in some regimes, it is possible to attain the HCRB with the optimal (single-copy) measurement for phase estimation. For the latter, we show a nontrivial interplay between the HCRB and incompatibility and provide numerical evidence that projective single-copy measurements attain the HCRB in the noiseless 2-qubit case.
The quantum Cramér-Rao bound constrains the precision of parameter estimation through the symmetric logarithmic derivative quantum Fisher information (SLD QFI). It is asymptotically achievable for regular single-parameter estimation models, but generally not in the multiparameter setting, where optimal measurements for different parameters may be incompatible. For multiparameter estimation, allowing collective measurements, the corresponding asymptotically achievable precision limit is the Holevo bound, whose standard formulation is an optimization over auxiliary Hermitian operators. We bridge this apparent difference in formulation by giving an exact characterization of the Holevo bound in terms of a family of QFIs interpolating between the SLD and right logarithmic derivative (RLD) QFIs. Specifically, for any locally identifiable finite-dimensional estimation problem, we prove that the operator-feasible region in the standard variational definition of the Holevo bound coincides with the intersection of the regions defined by the corresponding Cramér-Rao-type matrix constraints for the entire QFI family. This characterization also applies to rank-deficient states and does not require any prior choice of weight matrix. Consequently, the conventional weight-dependent Holevo bound is recovered by minimizing the corresponding weighted cost over the resulting common feasible region.
Abstract Finding the optimal attainable precisions in quantum multiparameter metrology is a non-trivial problem. One approach to tackling this problem involves the computation of bounds which impose limits on how accurately we can estimate certain physical quantities. One such bound is the Holevo Cramér–Rao bound on the trace of the mean squared error matrix. The Holevo bound is an asymptotically achievable bound when one allows for any measurement strategy, including collective measurements on many copies of the probe. In this work, we introduce a tighter bound for estimating multiple parameters simultaneously when performing separable measurements on a finite number of copies of the probe. This makes it more relevant in terms of experimental accessibility. We show that this bound can be efficiently computed by casting it as a semidefinite programme. We illustrate our bound with several examples of collective measurements on finite copies of the probe. These results have implications for the necessary requirements to saturate the Holevo bound.
We study quantum-limited 3D magnetometry using two qubits. Two qubits form the smallest multi-qubit system for 3D magnetometry, the simultaneous estimation of three phases, as it is impossible with a single qubit. We provide an analytical expression for the Holevo-Cramér-Rao bound (HCRB),the fundamental attainable quantum bound of multiparameter estimation, for 3D magnetometry using two-qubit pure states and show its attainability by rank-1 projective measurements. We also examine the attainability of the HCRB in the presence of dephasing noise using numerical methods. While attaining the HCRB may require collective measurements over infinitely many copies, we find that for high noise the HCRB is practically saturated by two copies only. In the low noise regime, up to three copies are unable to attain the HCRB. More generally, we introduce new multiparameter channel bounds to compare quantum-classical and classical-quantum strategies where multiple independent copies of the state are entangled before or after recording the parameters respectively. We find that their relative performance depends on the noise strength, with theclassical-quantum strategy performing better for high noise. We end with shallow quantum circuits that approach the fundamental quantum limit set by the HCRB for two-qubit 3D magnetometry using up to three copies.
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.