Encoding parameters by measurement: Forgetting can be better in quantum metrology

TL;DR

This work investigates quantum metrology where the parameter is encoded by a quantum measurement, comparing outcome-remembered and outcome-forgotten strategies. It shows that in a broad class of two-outcome qubit measurements, forgetting outcomes often yields higher precision and derives a necessary condition for when retaining outcomes could help, with extensions to two-parameter estimation via a complete criterion for QCRB achievability. The authors analyze four two-parameter scenarios with qubit probes and derive when the quantum Fisher information matrix is invertible under reachable bounds, revealing that including a measurement-direction parameter can prevent singularity and enable meaningful multiparameter bounds. These results sharpen our understanding of QCRB applicability in measurement-encoded metrology and provide practical guidance for when to preserve or discard measurement outcomes in quantum sensing experiments.

Abstract

We introduce quantum parameter estimation with the encoding being via a quantum measurement. We quantify the precision for estimating parameters characterizing a general two-outcome qubit measurement, considering two cases: when the outcomes of the encoding measurement are recorded and when the same are ignored. We find that in a large variety of such estimation scenarios, forgetting the outcomes yields higher precision. We derive a necessary criterion under which remembering the measurement outcomes provides better precision in comparison to the outcome-forgotten strategy. Furthermore, we establish a necessary and sufficient criterion for the simultaneous estimation of two parameters encoded by an arbitrary quantum process, including those involving measurements, using qubit probes, and find when the quantum Cramér$-$Rao bound is valid and achievable. For simultaneous estimation of two parameters characterizing the measurement, we find that the achievable quantum Cramér$-$Rao bound can be a valid precision bound only when the measurement direction depends on the parameters of interest.

Figures (2)

• Figure 1: (a) The left plot shows $(\bar{\Delta}_{\textbf{OR}}-\bar{\Delta}_{\textbf{OF}})$ along the vertical axis while the ratio $\alpha/\beta$ is varied along the horizontal axis for estimation of the parameter of $\beta$. The $(\bar{\Delta}_{\textbf{OR}}-\bar{\Delta}_{\textbf{OF}})$ is always nonnegative in this case. The non-existence of any negative value of $(\bar{\Delta}_{\textbf{OR}}-\bar{\Delta}_{\textbf{OF}})$ implies that forgetting measurement outcomes always gives better precision than remembering whatever the choice of the values of parameters $\alpha$ and $\beta$ are. Hence, when the parameter to be estimated is in the coefficient of the traceless matrix of two-outcome qubit POVM elements, remembering measurement outcomes is not good for precision at all. (b) The right plot shows $(\bar{\Delta}_{\textbf{OR}}-\bar{\Delta}_{\textbf{OF}})$ along the vertical axis, during estimation of $\alpha$, as the ratio $\beta/\alpha$ is changed along the horizontal axis, for various values of the parameter $\alpha$. There are many values of $\alpha$ and $\beta$ such that $(\bar{\Delta}_{\textbf{OR}}-\bar{\Delta}_{\textbf{OF}})$ have negative values, implying that keeping encoding measurement outcomes can provide better precision. This is altered if the value of $\alpha$ is lowered. Therefore, remembering encoding measurement outcomes can provide better precision in such scenarios only if the parameter of interest is in the coefficient of the nonzero trace part of POVM elements and it is a necessary criterion, not a sufficient one. Both axes, as well as $\alpha$, of both panels, represent dimensionless quantities.
• Figure 2: The quantity $(\bar{\Delta}_{\textbf{ OR}}-\bar{\Delta}_{\textbf{ OF}})$ is plotted along the vertical axis in case of estimating $\theta$, which is varied along the horizontal axis in the unit of $\pi$. Various values of $\alpha$ are considered with $\beta=\alpha\cos\theta$ is taken in this plot. Note that $(\bar{\Delta}_{\textbf{ OR}}-\bar{\Delta}_{\textbf{ OF}})$ never has negative values when eigenstates of the encoded state depend on the parameter to be estimated and hence, it supports the generalization of our Theorem. \ref{['theorem:0']}. Both axes, as well as $\alpha$, of this plot, represent dimensionless quantities.

Theorems & Definitions (7)

• Theorem 1
• proof
• Theorem 2
• Corollary 1
• proof
• Theorem 3
• proof