Multiparameter estimation for qubit states with collective measurements: a case study

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 two parameters that are respectively associate with 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 that when the state is nearly pure, for sufficiently but not arbitrarily large $N$, most information will be captured in the largest three $j$-subspaces. Then, we study 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. We also obtain numerical results that suggest 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.