Multiparameter estimation for qubit states with collective measurements

Yink Loong Len

Published 2021-09-15Version 1

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.