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

Abstract

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.

Optimal Estimation of Quantum Coherence by Bell State Measurement: A Case Study

“…If the estimation of parameter C x is an unbiased estimator, the mean squared error is lower bounded by the Cramér-Rao bound [48]:…”

“…If the BSM is performed N times on N identically prepared quantum systems, outcome j occurs n j times with ∑ j n j = N. Now, we aim to infer the quantifier of coherence C x from the measurement data D = {n 1 , n 2 , ..., n m }. Maximum likelihood estimation searches for the C ML x that maximizes the likelihood function [48] are…”

Multiparameter Estimation with Two-Qubit Probes in Noisy Channels

“…Additionally, quantum super-resolution for resolving two incoherent point sources of light [ 11 , 49 , 50 ] and many estimation problems concerning Gaussian quantum states [ 51 , 52 , 53 , 54 , 55 , 56 , 57 , 58 ] can be cast as multiparameter estimation problems. Given this motivation, there has been significant experimental [ 59 , 60 , 61 , 62 , 63 ] and theoretical [ 64 , 65 , 66 , 67 , 68 , 69 , 70 , 71 , 72 , 73 , 74 , 75 , 76 , 77 , 78 , 79 , 80 , 81 , 82 , 83 , 84 , 85 ] interest in quantum multiparameter estimation. See Refs.…”

Quantum optics at its best: from quantum interferometry to quantum metrology

“…Even though superresolution had large advance both theoretically [14][15][16][17][18][19][20][21][22][23][24][25] as well experimentally [26][27][28][29][30][31][32][33] the current measurement strategies, e.g. SPADE [2] require exact knowledge of the centroid of the distribution.…”

“…The results of Tsang, solve the multi-parameter estimation issue for the separation and centroid of two point sources. This work resulted in a plethora of theoretical [14][15][16][17][18][19][20][21][22][23][24][25] as well experimental [26][27][28][29][30][31][32][33] results.…”