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arXiv:2609.35975

Deciding the Attainability of the Multiparameter Quantum Fisher Information is NP-Hard

Lorcan O. Conlon, Laura Shou, V Vijendran, Kishor Bharti, Aritra Das, Simon K. Yung, Jacob M. Taylor, Alexey V. Gorshkov, Jun Suzuki, Syed M. Assad

quant-phSubmitted Sep 28, 2026• Announced Sep 30, 2026• 0 citations
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Abstract

The quantum Fisher information (QFI) sets a fundamental bound on the attainable precision when estimating multiple parameters simultaneously. Incompatibility among the individually optimal measurements can, in some cases, imply that the precision limit set by the QFI is not attainable. In certain special cases, including pure states and full-rank states, the exact conditions for when the precision limit set by the QFI can be saturated are known. However, general conditions for the attainability of the QFI with measurements on individual copies of the quantum state have long been sought. Indeed, this was recently stated as one of the five problems in quantum information theory highlighted by [P. Horodecki et al, PRX Quantum 3, 010101 (2022)]. In this work we prove that exact attainability with individual measurements is NP-hard to decide, even for a restricted set of real, constant-rank quantum models. The source problem for our proof is the NP-hard problem of deciding whether a given bipartite density matrix is separable or not. Our result shows that the longstanding difficulty in obtaining general conditions for QFI attainability reflects a fundamental computational obstruction, rather than merely a limitation of existing mathematical techniques: unless P=NP, no efficiently computable necessary-and-sufficient criterion can exist in general.

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The quantum Fisher information (QFI) sets a fundamental bound on the attainable precision when estimating multiple parameters simultaneously. Incompatibility among the individually optimal measurements can, in some cases, imply that the precision limit set by the QFI is not attainable. In certain special cases, including pure states and full-rank states, the exact conditions for when the precision limit set by the QFI can be saturated are known. However, general conditions for the attainability of the QFI with measurements on individual copies of the quantum state have long been sought. Indeed, this was recently stated as one of the five problems in quantum information theory highlighted by [P. Horodecki et al, PRX Quantum 3, 010101 (2022)]. In this work we prove that exact attainability with individual measurements is NP-hard to decide, even for a restricted set of real, constant-rank quantum models. The source problem for our proof is the NP-hard problem of deciding whether a given bipartite density matrix is separable or not. Our result shows that the longstanding difficulty in obtaining general conditions for QFI attainability reflects a fundamental computational obstruction, rather than merely a limitation of existing mathematical techniques: unless P=NP, no efficiently computable necessary-and-sufficient criterion can exist in general.
The Heisenberg limit provides a fundamental bound on the achievable estimation precision with a limited number of NNN resources used (e.g., atoms, photons, etc.). Using entangled quantum states makes it possible to scale the precision with NNN better than when resources would be used independently. Consequently, the optimal use of all resources involves accumulating them in a single execution of the experiment. Unfortunately, that implies that the most common theoretical tool used to analyze metrological protocols - quantum Fisher information (QFI) - does not allow for a reliable description of this problem, as it becomes operationally meaningful only with multiple repetitions of the experiment. In this thesis, using the formalism of Bayesian estimation and the minimax estimator, I derive asymptotically saturable bounds on the precision of the estimation for the case of noiseless unitary evolution. For the case where the number of resources NNN is strictly constrained, I show that the final measurement uncertainty is πππ times larger than would be implied by a naive use of QFI. I also analyze the case where a constraint is imposed only on the average amount of resources, the exact value of which may fluctuate (in which case QFI does not provide any universal bound for precision). In both cases, I study the asymptotic saturability and the rate of convergence of these bounds. In the following part, I analyze the problem of the Heisenberg limit when multiple parameters are measured simultaneously on the same physical system. In particular, I investigate the existence of a gain from measuring all parameters simultaneously compared to distributing the same amount of resources to measure them independently. I focus on two examples - the measurement of multiple phase shifts in a multi-arm interferometer and the measurement of three magnetic field components.
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.
A quantum theory of multiphase estimation is crucial for quantum-enhanced sensing and imaging and may link quantum metrology to more complex quantum computation and communication protocols. In this Letter, we tackle one of the key difficulties of multiphase estimation: obtaining a measurement which saturates the fundamental sensitivity bounds. We derive necessary and sufficient conditions for projective measurements acting on pure states to saturate the ultimate theoretical bound on precision given by the quantum Fisher information matrix. We apply our theory to the specific example of interferometric phase estimation using photon number measurements, a convenient choice in the laboratory. Our results thus introduce concepts and methods relevant to the future theoretical and experimental development of multiparameter estimation.
Multiparameter quantum estimation faces two fundamental obstacles: sloppiness, i.e., anisotropy of the quantum Fisher information matrix (QFIM) that renders some parameter directions insensitive, and incompatibility, the non-commutativity of optimal measurements for different parameters. The trade-off bound CTC_TCT​ captures their joint impact on precision, but it has remained unclear how the distribution of incompatibility across parameter planes affects its overall cost. Here we separate the total amount of incompatibility from its location. We introduce a dimensionless quantity Gn(F)G_n^{(F)}Gn(F)​ that measures the alignment between the incompatibility distribution and the eigenvalues of the QFIM, and show how the Frobenius scale of the incompatibility contribution factorizes. We obtain a bound and prove the incompatibility cost lies between this bound and a rank-dependent multiple thereof. We also prove that at fixed sloppiness, or equivalently fixed Fisher volume, concentrating incompatibility into a single parameter plane reduces the optimized trade-off cost because the Fisher geometry can then be reshaped to allocate more Fisher area to that plane. A qutrit SU(2)SU(2)SU(2) encoding numerically confirms that states with larger incompatibility strength can nevertheless incur a smaller cost if the matching factor GGG is sufficiently small. Our results establish that the distribution of incompatibility relative to the Fisher eigenbasis is a central diagnostic for multiparameter estimation, beyond the total incompatibility strength.
We present a comprehensive and pedagogical formulation of Bayesian multiparameter quantum estimation. Within this framework, we analyse the role of measurement incompatibility and establish its quantitative effect on attainable precision. We achieve this by deriving upper bounds based on the pretty good measurement -- a notion from hypothesis testing -- combined with the evaluation of the Nagaoka-Hayashi lower bound. In general, we prove that, as in the many-copy regime of local estimation theory, incompatibility can at most double the minimum loss relative to the idealised scenario in which individually optimal measurements are assumed jointly implementable. Therefore, in practical situations, the latter may provide a sufficient and computationally efficient benchmark without solving the full optimisation problem. Our results, which we illustrate through applications of discrete phase imaging, phase and dephasing estimation, and qubit sensing, provide analytical and numerical tools for assessing ultimate precision limits and the role of measurement incompatibility in Bayesian multiparameter quantum metrology, including an open-source package for all the bounds discussed here.
We derive a universal upper bound on the speed of multipartite entanglement generation. For arbitrary differentiable pure multipartite states undergoing parameter-dependent evolution, we prove that the rate of change of generalized concurrence is bounded by the square root of the quantum Fisher information (QFI). The proof follows directly from the symmetric logarithmic derivative formalism, revealing the bound as a consequence of quantum estimation theory, while a complementary Schmidt-spectrum analysis identifies the conditions for saturation and yields a tighter bound for rank-two bipartitions. We illustrate these results in interacting many-body spin models and investigate their behavior across the quantum phase transition of the transverse-field Ising model. At the critical point, we find that the QFI can be strongly enhanced while the generalized concurrence is nearly stationary and its sensitivity to the coupling is suppressed. This demonstrates that the enhanced parameter sensitivity associated with quantum criticality need not arise from rapid changes in the magnitude of multipartite entanglement.