Multiparameter Estimation with Two-Qubit Probes in Noisy Channels

Abstract

This work compares the performance of single- and two-qubit probes for estimating several phase rotations simultaneously under the action of different noisy channels. We compute the quantum limits for this simultaneous estimation using collective and individual measurements by evaluating the Holevo and Nagaoka–Hayashi Cramér-Rao bounds, respectively. Several quantum noise channels are considered, namely the decohering channel, the amplitude damping channel, and the phase damping channel. For each channel, we find the optimal single- and two-qubit probes. Where possible we demonstrate an explicit measurement strategy that saturates the appropriate bound and we investigate how closely the Holevo bound can be approached through collective measurements on multiple copies of the same probe. We find that under the action of the considered channels, two-qubit probes show enhanced parameter estimation capabilities over single-qubit probes for almost all non-identity channels, i.e., the achievable precision with a single-qubit probe degrades faster with increasing exposure to the noisy environment than that of the two-qubit probe. However, in sufficiently noisy channels, we show that it is possible for single-qubit probes to outperform maximally entangled two-qubit probes. This work shows that, in order to reach the ultimate precision limits allowed by quantum mechanics, entanglement is required in both the state preparation and state measurement stages. It is hoped the tutorial-esque nature of this paper will make it easily accessible.

Multi-parameter quantum estimation of single- and two-mode pure Gaussian states

“…Quantum estimation theory provides the tools to establish the ultimate limits on the precision of parameter estimation in the quantum domain, and aims to identify potential advantages with respect to classical protocols by leveraging quantum resources, including entanglement and squeezing [1][2][3][4][5][6][7][8][9]. Multiparameter quantum metrology [10][11][12][13] has received much attention in the last years, ranging from the joint estimation of unitary parameters [14][15][16][17][18][19][20][21][22], of unitary and loss parameters [23][24][25][26][27][28], and for both spatial and time superresolution imaging [29][30][31][32][33][34][35]. From the theoretical point of view, the derivations of the ultimate bounds on the estimation precision relies on the seminal works by Helstrom [36] and Holevo [37]; by inspecting these derivations it is immediate clear how in the quantum realm the multiparameter bounds are not a trivial generalization of the single-parameter ones, as it is indeed the case in the classical scenario.…”