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arXiv:2202.03668 · Physical Review A

Multiparameter simultaneous optimal estimation with an SU(2) coding unitary evolution

Yu Yang, Shihao Ru, Min An, Yunlong Wang, Feiran Wang, Pei Zhang, Fuli Li

Published Feb 7, 2022• 10 citations
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Abstract

In a ubiquitous SU(2) dynamics, achieving the simultaneous optimal estimation of multiple parameters is significant but difficult. Using quantum control to optimize this SU(2) coding unitary evolution is one of solutions. We propose a method, characterized by the nested cross products of the coefficient vector $\mathbf{X}$ of SU(2) generators and its partial derivative ${\ensuremath{\partial}}_{\ensuremath{\ell}}\mathbf{X}$, to investigate the control-enhanced quantum multiparameter estimation. Our work reveals that quantum control is not always functional in improving the estimation precision, which depends on the characterization of an SU(2) dynamics with respect to the objective parameter. This characterization is quantified by the angle ${\ensuremath{\alpha}}_{\ensuremath{\ell}}$ between $\mathbf{X}$ and ${\ensuremath{\partial}}_{\ensuremath{\ell}}\mathbf{X}$. For an SU(2) dynamics featured by ${\ensuremath{\alpha}}_{\ensuremath{\ell}}=\ensuremath{\pi}/2$, the promotion of the estimation precision can get the most benefits from the controls. When ${\ensuremath{\alpha}}_{\ensuremath{\ell}}$ gradually closes to 0 or $\ensuremath{\pi}$, the precision promotion contributed to by quantum control correspondingly becomes inconspicuous. Until a dynamics with ${\ensuremath{\alpha}}_{\ensuremath{\ell}}=0$ or $\ensuremath{\pi}$, quantum control completely loses its advantage. In addition, we find a set of conditions restricting the simultaneous optimal estimation of all the parameters, but fortunately, which can be removed by using a maximally entangled two-qubit state as the probe state and adding an ancillary channel into the configuration. Lastly, a spin-$1/2$ system is taken as an example to verify the above-mentioned conclusions. Our proposal sufficiently exhibits the hallmark of control-enhancement in fulfilling the multiparameter estimation mission, and it is applicable to an arbitrary SU(2) parametrization process.

DOI
10.1103/physreva.105.022406

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In a ubiquitous SU(2) dynamics, achieving the simultaneous optimal estimation of multiple parameters is significant but difficult. Using quantum control to optimize this SU(2) coding unitary evolution is one of solutions. We propose a method, characterized by the nested cross products of the coefficient vector X\mathbf{X}X of SU(2) generators and its partial derivative \ensuremath∂\ensuremathℓX{\ensuremath{\partial}}_{\ensuremath{\ell}}\mathbf{X}\ensuremath∂\ensuremathℓ​X, to investigate the control-enhanced quantum multiparameter estimation. Our work reveals that quantum control is not always functional in improving the estimation precision, which depends on the characterization of an SU(2) dynamics with respect to the objective parameter. This characterization is quantified by the angle \ensuremathα\ensuremathℓ{\ensuremath{\alpha}}_{\ensuremath{\ell}}\ensuremathα\ensuremathℓ​ between X\mathbf{X}X and \ensuremath∂\ensuremathℓX{\ensuremath{\partial}}_{\ensuremath{\ell}}\mathbf{X}\ensuremath∂\ensuremathℓ​X. For an SU(2) dynamics featured by \ensuremathα\ensuremathℓ=\ensuremathπ/2{\ensuremath{\alpha}}_{\ensuremath{\ell}}=\ensuremath{\pi}/2\ensuremathα\ensuremathℓ​=\ensuremathπ/2, the promotion of the estimation precision can get the most benefits from the controls. When \ensuremathα\ensuremathℓ{\ensuremath{\alpha}}_{\ensuremath{\ell}}\ensuremathα\ensuremathℓ​ gradually closes to 0 or \ensuremathπ\ensuremath{\pi}\ensuremathπ, the precision promotion contributed to by quantum control correspondingly becomes inconspicuous. Until a dynamics with \ensuremathα\ensuremathℓ=0{\ensuremath{\alpha}}_{\ensuremath{\ell}}=0\ensuremathα\ensuremathℓ​=0 or \ensuremathπ\ensuremath{\pi}\ensuremathπ, quantum control completely loses its advantage. In addition, we find a set of conditions restricting the simultaneous optimal estimation of all the parameters, but fortunately, which can be removed by using a maximally entangled two-qubit state as the probe state and adding an ancillary channel into the configuration. Lastly, a spin-1/21/21/2 system is taken as an example to verify the above-mentioned conclusions. Our proposal sufficiently exhibits the hallmark of control-enhancement in fulfilling the multiparameter estimation mission, and it is applicable to an arbitrary SU(2) parametrization process.
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Using quantum systems as sensors or probes has been shown to greatly improve the precision of parameter estimation by exploiting unique quantum features such as entanglement. A major task in quantum sensing is to design the optimal protocol, i.e., the most precise one. It has been solved for some specific instances of the problem, but in general even numerical methods are not known. Here, we focus on the single-shot Bayesian setting, where the goal is to find the optimal initial state of the probe (which can be entangled with an auxiliary system), the optimal measurement, and the optimal estimator function. We leverage the formalism of higher-order operations to develop a method based on semidefinite programming that finds a protocol that is close to the optimal one with arbitrary precision. Crucially, our method is not restricted to any specific quantum evolution, cost function, or prior distribution, and thus can be applied to any estimation problem. Moreover, it can be applied to both single-parameter or multiparameter estimation tasks. We demonstrate our method with three examples, consisting of unitary phase estimation, thermometry in a bosonic bath, and multiparameter estimation of an SU(2) transformation. Exploiting our methods, we extend several results from the literature. For example, in the thermometry case, we find the optimal protocol at any finite time and quantify the usefulness of entanglement. Published by the American Physical Society 2024
Quantum estimation of a single parameter has been studied extensively. Practical applications, however, typically involve multiple parameters, for which the ultimate precision is much less understood. Here, by relating the precision limit directly to the Heisenberg uncertainty relation, we show that to achieve the highest precisions for multiple parameters at the same time requires the saturation of multiple Heisenberg uncertainty relations simultaneously. Guided by this insight, we experimentally demonstrate an optimally controlled multipass scheme, which saturates three Heisenberg uncertainty relations simultaneously and achieves the highest precisions for the estimation of all three parameters in SU(2) operators. With eight controls, we achieve a 13.27-dB improvement in terms of the variance (6.63 dB for the SD) over the classical scheme with the same loss. As an experiment demonstrating the simultaneous achievement of the ultimate precisions for multiple parameters, our work marks an important step in multiparameter quantum metrology with wide implications.
Quantum theory allows the traversing of multiple channels in a superposition of different orders. When the order in which the channels are traversed is controlled by an auxiliary quantum system, various unknown parameters of the channels can be estimated by measuring only the control system, even when the state of the probe alone would be insensitive. Moreover, increasing the dimension of the control system increases the number of simultaneously estimable parameters, which has important metrological ramifications. We demonstrate this capability for simultaneously estimating both unitary and noise parameters, including multiple parameters from the same unitary such as rotation angles and axes and from noise channels such as depolarization, dephasing, and amplitude damping in arbitrary dimensions. We identify regimes of unlimited advantages, taking the form of p2{p}^{2}p2 smaller variances in estimation when the noise probability is $1\ensuremath{-}p$, for both single and multiparameter estimations when using our schemes relative to any comparable scheme whose causal order is definite.