Quantum parameter estimation holds the promise of quantum technologies, in which physical parameters can be measured with much greater precision than what is achieved with classical technologies. However, how to obtain the best precision when the optimal measurement is not accessible is still an open problem. In this paper, we present a theoretical framework to explore the parameter estimation with limited access of measurements by analyzing the effect of nonoptimal measurement on the estimation precision. We define a quantity $\mathrm{\ensuremath{\Lambda}}$ to characterize the effect and illustrate how to optimize observables to attain a bound with limited accessibility of observables. In addition, we introduce the minimum Euclidean distance to quantify the difference between an observable and the optimal ones in terms of the Frobenius norm and find that the measurement with a shorter distance to the optimal ones benefits the estimation. Two examples are presented to show our theory. In the first, we analyze the effect of nonoptimal measurement on the estimation precision of the transition frequency for a driven qubit. In the second example, we consider a bipartite system, in which one of them is measurement inaccessible. To be specific, we take a toy model, the nitrogen-vacancy (NV) center in diamond, as the bipartite system, where the NV-center electronic spin interacts with a single nucleus via the dipole-dipole interaction. We achieve a precise estimation for the nuclear Larmor frequency by optimizing only the observables of the electronic spin. In these two examples, the minimum Euclidean distance between an observable and the optimal ones is analyzed and the results show that the closer the observable to the optimal ones the better the estimation precision. This paper establishes a relation between the estimation precision and the distance of the nonoptimal observable to the optimal ones, which could be helpful for experiments seeking the best observable in parameter estimation.
Quantum estimation involving multiple parameters remains an important problem of both theoretical and practical interest. In this work, we study the problem of simultaneous estimation of the length and direction of the Bloch vector for identically prepared qubit states that is confined to a plane, where in order to obtain the optimal estimation precision for both parameters, collective measurements on multiple qubits are necessary. Upon treating
N qubits as an ensemble of spin-1/2 systems, we show that simultaneous optimal estimation for both parameters can be attained asymptotically with a simple collective measurement strategy -- First, we estimate the length parameter by measuring the populations in spaces corresponding to different total angular momentum values
j, then we estimate the direction parameter by performing a spin projection onto an optimal basis. Furthermore, we show how the total angular-momentum measurement can be realized by observing output signatures from a Bell multiport setup, either exactly for
N=2,3, or approximately when the qubits are nearly pure for other
N values. Our results point to the conjecture that using a Bell multiport setup, one can distinguish between projection onto the
j=N/2 and
j=N/2−1 subspaces from their respective interference signatures at the output.
Quantum parameter estimation holds the promise of quantum technologies, in which physical parameters can be measured with much greater precision than what is achieved with classical technologies. However, how to obtain a best precision when the optimal measurement is not accessible is still an open problem. In this work, we present a theoretical framework to explore the parameter estimation with limited access of measurements by analyzing the effect of non-optimal measurement on the estimation precision. We define a quantity to characterize the effect and illustrate how to optimize observables to attain a bound with limited accessibility of observables. On the other side, we introduce the minimum Euclidean distance to quantify the difference between an observable and the optimal ones in terms of Frobenius norm and find that the measurement with a shorter distance to the optimal ones benefits the estimation. Two examples are presented to show our theory. In the first, we analyze the effect of non-optimal measurement on the estimation precision of the transition frequency for a driven qubit. While in the second example, we consider a bipartite system, in which one of them is measurement inaccessible. To be specific, we take a toy model, the NV-center in diamond as the bipartite system, where the NV-center electronic spin interacts with a single nucleus via the dipole-dipole interaction. We achieve a precise estimation for the nuclear Larmor frequency by optimizing only the observables of the electronic spin. In these two examples, the minimum Euclidean distance between an observable and the optimal ones is analyzed and the results show that the observable closed to the optimal ones better the estimation precision.