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Hierarchical incompatibility measures in multi-parameter quantum estimation

Hongzhen Chen, Yu Chen, Haidong Yuan

Published Sep 13, 2021• 3 citations
OpenAlex

Abstract

When collective measurements on an infinite number of copies of identical quantum states can be performed, the precision limit of multi-parameter quantum estimation is quantified by the Holevo bound. In practice, however, the collective measurements are always restricted to a finite number of quantum states, under which the precision limit is still poorly understood. Here we provide an approach to study the multi-parameter quantum estimation with general $p$-local measurement where the collective measurements are restricted to at most $p$ copies of quantum states. We demonstrate the power of the approach by providing a hierarchy of nontrivial tradeoff relations for multi-parameter quantum estimation which quantify the incompatibilities of general $p$-local measurement. These tradeoff relations also provide a necessary condition for the saturation of the quantum Cramer-Rao bound under $p$-local measurement, which is shown reducing to the weak commutative condition when $p\rightarrow \infty$. To further demonstrate the versatility of the approach, we also derive another set of tradeoff relations in terms of the right logarithmic operators(RLD).

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When collective measurements on an infinite number of copies of identical quantum states can be performed, the precision limit of multi-parameter quantum estimation is quantified by the Holevo bound. In practice, however, the collective measurements are always restricted to a finite number of quantum states, under which the precision limit is still poorly understood. Here we provide an approach to study the multi-parameter quantum estimation with general ppp-local measurement where the collective measurements are restricted to at most ppp copies of quantum states. We demonstrate the power of the approach by providing a hierarchy of nontrivial tradeoff relations for multi-parameter quantum estimation which quantify the incompatibilities of general ppp-local measurement. These tradeoff relations also provide a necessary condition for the saturation of the quantum Cramer-Rao bound under ppp-local measurement, which is shown reducing to the weak commutative condition when p→∞p\rightarrow \inftyp→∞. To further demonstrate the versatility of the approach, we also derive another set of tradeoff relations in terms of the right logarithmic operators(RLD).
The incompatibility of the measurements constrains the achievable precisions in multiparameter quantum estimation. Understanding the tradeoff induced by such incompatibility is a central topic in quantum metrology. Here we provide an approach to study the incompatibility under general ppp-local measurements, which are the measurements that can be performed collectively on at most ppp copies of quantum states. We demonstrate the power of the approach by presenting a hierarchy of analytical bounds on the tradeoff among the precisions of different parameters. These bounds lead to a necessary condition for the saturation of the quantum Cram\'er-Rao bound under ppp-local measurements, which recovers the partial commutative condition at p=1p=1p=1 and the weak commutative condition at $p=\ensuremath{\infty}$. As a further demonstration of the power of the framework, we present another set of tradeoff relations with the right logarithmic operators.
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.
Multiparameter quantum estimation faces a fundamental challenge due to the inherent incompatibility of optimal measurements for different parameters, a direct consequence of quantum non-commutativity. This incompatibility is quantified by the gap between the symmetric logarithmic derivative (SLD) quantum Cramér-Rao bound, which is not always attainable, and the asymptotically achievable Holevo bound. This work provides a comprehensive analysis of this gap by introducing and contrasting two scalar measures. The first is the weight-independent quantumness measure RRR, which captures the intrinsic incompatibility of the estimation model. The second is a tighter, weight-dependent measure T[W]T[W]T[W] which explicitly incorporates the cost matrix WWW assigning relative importance to different parameters. We establish a hierarchy of bounds based on these two measures and derive necessary and sufficient conditions for their saturation. Through analytical and numerical studies of tunable qubit and qutrit models with SU(2) unitary encoding, we demonstrate that the weight-dependent bound CT[W]C_{T}[W]CT​[W] often provides a significantly tighter approximation to the Holevo bound CH[W]C_{H}[W]CH​[W] than the RRR-dependent bound, especially in higher-dimensional systems. We also develop an approach based on CT[W]C_{T}[W]CT​[W] to compute the Holevo bound CH[W]C_{H}[W]CH​[W] analytically. Our results highlight the critical role of the weight matrix's structure in determining the precision limits of multiparameter quantum metrology.
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.
Quantum Cramér--Rao theory is intrinsically local: it bounds precision near a specified parameter value, and its saturating measurement generally depends on that value. Barankin-type bounds use finite parameter displacements, but remain anchored to a chosen reference value. This leaves open a basic global-estimation problem: when the parameter is known only within a broad domain, what precision can be guaranteed by a single estimator and a single measurement strategy fixed before the true value is localized? We answer this question by introducing global score functions tied to a weighted variance over the whole parameter domain. Their correlations generate a hierarchy of precision bounds: global Cramér--Rao and Barankin-type bounds arise as restricted levels, whereas unrestricted score correlations yield a fully global bound for the prescribed weighted variance. The hierarchy recovers local Cramér--Rao theory in the many-repetition limit and reveals genuinely global precision limits for finite data over broad domains. In the quantum setting, the construction identifies when this fully global bound can be realized by a single parameter-independent measurement. The same framework extends to Bayesian estimation, recovering the Van Trees bound in the local limit while yielding stronger finite-width lower bounds on the Bayesian mean-square error beyond this limit.
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)].