Holevo Cramér-Rao bound: How close can we get without entangling measurements?

TL;DR

The paper investigates how much precision gain entangling (collective) measurements can provide in multi-parameter quantum estimation compared with separable strategies. By introducing the enhancement measures $\\mathcal{R}^{\mathrm{NH}}$ and $\\mathcal{R}^{\mathrm{MI}}$ and analyzing both model-independent and GMM-based qudit tomography, it proves a universal bound $\\mathcal{R}^{\mathrm{NH}}_n \le n$ and shows the collective advantage scales linearly with dimension, yielding $\\mathcal{R}^{\mathrm{NH}} = O(d)$ (specifically $d+1$ for ONB tomography of the maximally mixed state, with a $d+2$ upper bound for arbitrary states). The work also bounds the true ratio $\\mathcal{R}^{\mathrm{MI}}$ to be between $d+1$ and $d+2$, and it demonstrates that, while SIC POVMs can attain the NHCRB in some cases, the finite-copy advantage of entangling measurements remains limited, underscoring a modest practical benefit for collective measurements. The results, complemented by SDP-based analysis and numerical studies, suggest using the NHCRB as a robust finite-copy benchmark and provide insights into the structure of optimal measurements for high-dimensional quantum tomography.

Abstract

In multi-parameter quantum metrology, the resource of entanglement can lead to an increase in efficiency of the estimation process. Entanglement can be used in the state preparation stage, or the measurement stage, or both, to harness this advantage; here we focus on the role of entangling measurements. Specifically, entangling or collective measurements over multiple identical copies of a probe state are known to be superior to measuring each probe individually, but the extent of this improvement is an open problem. It is also known that such entangling measurements, though resource-intensive, are required to attain the ultimate limits in multi-parameter quantum metrology and quantum information processing tasks. In this work we investigate the maximum precision improvement that collective quantum measurements can offer over individual measurements for estimating parameters of qudit states, calling this the 'collective quantum enhancement'. We show that, whereas the maximum enhancement can, in principle, be a factor of $n$ for estimating $n$ parameters, this bound is not tight for large $n$. Instead, our results prove an enhancement linear in dimension of the qudit is possible using collective measurements and lead us to conjecture that this is the maximum collective quantum enhancement in any local estimation scenario.

Figures (11)

• Figure 1: Summary of our main results on the maximum ratio quantities $\mathcal{R}^{\mathrm{NH}}_n$ & $\mathcal{R}^{\mathrm{NH}}$. Our $\mathcal{R}^{\mathrm{NH}}_n \leq n$ (blue), $\mathcal{R}^{\mathrm{NH}}\leq d+1$ (green) and $\mathcal{R}^{\mathrm{NH}}_n\leq\min(n,d+1)$ (red) bounds are plotted against numerically and analytically found maximum collective enhancement values (bar chart) for $n$-parameter estimation from qutrits ($d=3$).
• Figure 2: Comparison of the collective enhancements specified by the GMCRB ($\mathcal{R}^{\mathrm{GM}}_n$) and by the NHCRB ($\mathcal{R}^{\mathrm{NH}}_n$) for $n$-parameter models ($1\leq n\leq n_\mathrm{max}$). The maximum GMCRB ratios (dark gray bar chart) satisfy $\mathcal{R}^{\mathrm{GM}}_n = n/(d-1)$ (blue line). The maximum NHCRB ratios (light gray bar chart) satisfy $\mathcal{R}^{\mathrm{NH}}_n \leq \min(n,d+1)$ (red line). The ratios are plotted against the number of parameters, $n$, and include numerically found maximum ratios from random-sampling experiments (1300 samples for each $n$ for $d=3$) as well as analytically found ratios. The NHCRB ratio $\mathcal{R}^{\mathrm{NH}}_n$ generally predicts a larger collective enhancement than the GMCRB ratio $\mathcal{R}^{\mathrm{GM}}_n$, except at the maximum number of parameters ($n=n_\mathrm{max}$).
• Figure 3: Ratio between the NHCRB and the HCRB versus purity for estimating all $d^2-1$ GMMs from arbitrary states. For qubits (a), we find a one-to-one dependence between ratio and purity (10,000 samples). However, for qutrits (b) and quartets (c), there is a region of allowed ratios at any given purity (15,000 and 25,000 samples, respectively). The ratio at any fixed purity is maximised by the state $\rho_\mathrm{max}$, which is a depolarised pure state, and minimised by the states $\rho_\mathrm{min}^{(2)}, \rho_\mathrm{min}^{(3)}$ (and $\rho_\mathrm{min}^{(4)}$ in (c)), which are rank-deficient classical states
• Figure 4: Maximum ratio $\mathcal{R}^{\mathrm{NH}}_n$ between the NHCRB and the HCRB over 10,000 random models for each dimension, $d$, from three to eight and for each number of parameters, $n$, from two to eight. (See Fig. \ref{['fig:GridPlotDN']} in Appendix \ref{['sec:appGridPlot']} for the distribution of ratios for each $d$ and $n$.) The bar chart (with black callouts) on the back panel depicts the maximum ratio for estimating $n$ GMM coefficients from the maximally-mixed qutrit (Table \ref{['tab:Ratios']} in Appendix \ref{['sec:AppEstimatingFewGGMMs']}). The red line on the bar chart (with red callouts) represents the maximum ratio for each $n$ from known analytic models, applicable for all $d\geq 3$.
• Figure 5: Comparison of the HCRB and the NHCRB to their lower and upper bounds, respectively. (a) HCRB and its lower bound $d-\mathrm{P}(\rho_{\theta})$ (from Eq. \ref{['eq:lowerboundHCRB']}). The lower solid parabolic curves show the lower bound and the upper triangular curves (beginning and ending with dots) show the numerically-computed HCRB. (b) NHCRB and its upper bound $d^2+d-1-\mathrm{P}(\rho_{\theta})$ (from Eq. \ref{['eq:upperboundNHCRB']}). The lower dotted curves show the numerically-computed NHCRB and the upper solid curves show the upper bound. The state chosen in both (a) and (b) is a mixed qutrit $\rho_{\theta}=\mathds{1}_d/d + \theta_1 \lambda_2 + \theta_2 \lambda_4$.

Theorems & Definitions (36)

• Theorem 1
• Theorem 2
• Theorem 3
• Theorem 4
• Theorem 5
• Theorem 1
• proof
• Lemma 1
• proof
• Lemma 2