Metrological symmetries in singular quantum multi-parameter estimation

TL;DR

The paper analyzes why quantum Fisher information matrices can become singular in multi-parameter quantum estimation, linking singularities to metrological symmetries and over-parametrization. It shows that Bayesian estimation reveals the true effective parameters and their encodings as persistent lines or contour structures in the posterior, even when the standard CRB fails. By introducing re-parametrizations to define an effective single-parameter CRB, the work derives conditions and techniques to extract meaningful bounds and estimators for singular problems. It also demonstrates how thermal fluctuations can lift singularities, turning otherwise degenerate estimation tasks into well-posed ones, with practical implications for calibrating and interpreting quantum sensors. Overall, the framework provides a rigorous route to identify, quantify, and overcome singularities in multi-parameter quantum metrology via Bayesian analysis and adaptive re-parameterization.

Abstract

The theoretical foundation of quantum sensing is rooted in the Cramér-Rao formalism, which establishes quantitative precision bounds for a given quantum probe. In many practical scenarios, where more than one parameter is unknown, the multi-parameter Cramér-Rao bound (CRB) applies. Since this is a matrix inequality involving the inverse of the quantum Fisher information matrix (QFIM), the formalism breaks down when the QFIM is singular. In this paper, we examine the physical origins of such singularities, showing that they result from an over-parametrization on the metrological level. This is itself caused by emergent metrological symmetries, whereby the same set of measurement outcomes are obtained for different combinations of system parameters. Although the number of effective parameters is equal to the number of non-zero QFIM eigenvalues, the Cramér-Rao formalism typically does not provide information about the effective parameter encoding. Instead, we demonstrate through a series of concrete examples that Bayesian estimation can provide deep insights. In particular, the metrological symmetries appear in the Bayesian posterior distribution as lines of persistent likelihood running through the space of unknown parameters. These lines are contour lines of the effective parameters which, through suitable parameter transformations, can be estimated and follow their own effective CRBs.

Figures (8)

• Figure 1: Illustration contrasting Bayesian estimation for singular vs non-singular multi-parameter metrological problems. We show typical posterior distributions after $\mathcal{M}=100$, $300$ and $1000$ measurements, with the non-singular case (top panels) converging to a point in the space of unknown parameters $\theta_1$ and $\theta_2$; whereas in the singular case (bottom panels) the distribution converges instead to a line, encoding an effective metrological symmetry. The example shown is for the Heisenberg trimer, see Appendix \ref{['app:triangle']} for further details.
• Figure 2: Multi-parameter Bayesian estimation for the non-singular case. We consider joint estimation of $\lambda$ and $\gamma$ using the ground state of the $XY$ model on the ring geometry with $N=4$ sites, assuming that the field $h=1$ is known. Multinomial posterior distributions after $\mathcal{M} = 200$ and $10000$ measurements are shown in (a,b), with the white dotted lines denoting the true parameter values at $\left( \lambda^{\rm tr}, \gamma^{\rm tr} \right) = \left( 0.5,0.6\right)$. Line-plots show the corresponding marginalized univariate distributions for each parameter. (c) Average estimate $\hat{\lambda}$ (black line) and its error (blue shaded region) as a function of $\mathcal{M}$, averaged over $500$ numerical experiments. Red line denotes the true value $\lambda^{\rm tr} = 0.5$, to which the estimator correctly converges. (d) Inverse variance of the posterior distribution for $\hat{\lambda}$ (black line) vs $\mathcal{M}$ compared with the scaling prediction from the CRB Eq. \ref{['eq:QCRB_Def']} (orange dashed line) using the QFIM from Eq. \ref{['eq:xy_gam_lam_qfim']}. (e) Full sensitivity phase diagram of posterior variance $\mathcal{M}\text{Var}\left[ \hat{\lambda} \right]$ vs $\lambda^{\rm tr}$ and $\gamma^{\rm tr}$. Results shown for $\mathcal{M} = 10000$. (f) Analogous sensitivity phase diagram obtained from the CRB. The star point corresponds to the values used in (a-d).
• Figure 3: Multinomial posterior distributions for Bayesian estimation of $\lambda$ and $h$ in the $XY$ model. Results shown for estimation using the ground state of the $N=4$ system in the ring geometry, whose QFIM is singular. The true values of the parameters to be estimated (dotted white lines) is $\vec{\theta}^{\rm \:tr}\equiv (\lambda^{\rm tr}, h^{\rm tr})=(0.4, 0.55)$ and we set $\gamma=1$. Panels (a) and (b) compare the results after $\mathcal{M}=200$ and $10000$ measurements, respectively. Associated line plots show the individual (marginalized) posteriors for each parameter.
• Figure 4: Multinomial posterior distributions for Bayesian estimation of $T$ and $K$ in the Heisenberg model. For dual estimation of the temperature $T$ and spin coupling $K$, we use either the full thermal state of the entire system (top panels: a,b) or the reduced state of the central two spins (bottom panels: c,d). The QFIM is singular in the case of the reduced state. We compare $\mathcal{M}=200$ vs $10000$ measurements in the left and right panels, respectively. The true parameter values are $\vec{\theta}^{\rm \:tr}=\left( K^{\rm tr}, T^{\rm tr} \right) = \left( 0.52, 0.35 \right)$, shown as the white dotted line. Line plots show the individual (marginalized) distributions for each parameter. We use $N=4$ and $J=1$.
• Figure 5: Multinomial posterior distributions for Bayesian estimation of $\lambda$ and $\gamma$ from the ground state of the $XY$ model with all-to-all coupling. Shown for $\mathcal{M} = 200$ and $10000$ measurements in panels (a) and (b) respectively. White dotted lines show the true parameter values at $\left( \lambda^{\rm tr}, \gamma^{\rm tr} \right) = \left( 0.5,0.6\right)$. Line-plots show the corresponding marginalized univariate distributions for each parameter. Results shown for $N=4$ spins and known constant field $h=1$.