Tight Cramér-Rao type bounds for multiparameter quantum metrology through conic programming

Masahito Hayashi1,2,3 and Yingkai Ouyang4

1School of Data Science, The Chinese University of Hong Kong, Shenzhen, Longgang District, Shenzhen, 518172, China
2International Quantum Academy (SIQA), Futian District, Shenzhen 518048, China
3Graduate School of Mathematics, Nagoya University, Nagoya, 464-8602, Japan
4Department of Physics & Astronomy, University of Sheffield, Sheffield, S3 7RH, United Kingdom

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In the quest to unlock the maximum potential of quantum sensors, it is of paramount importance to have practical measurement strategies that can estimate incompatible parameters with best precisions possible. However, it is still not known how to find practical measurements with optimal precisions, even for uncorrelated measurements over probe states. Here, we give a concrete way to find uncorrelated measurement strategies with optimal precisions. We solve this fundamental problem by introducing a framework of conic programming that unifies the theory of precision bounds for multiparameter estimates for uncorrelated and correlated measurement strategies under a common umbrella. Namely, we give precision bounds that arise from linear programs on various cones defined on a tensor product space of matrices, including a particular cone of separable matrices. Subsequently, our theory allows us to develop an efficient algorithm that calculates both upper and lower bounds for the ultimate precision bound for uncorrelated measurement strategies, where these bounds can be tight. In particular, the uncorrelated measurement strategy that arises from our theory saturates the upper bound to the ultimate precision bound. Also, we show numerically that there is a strict gap between the previous efficiently computable bounds and the ultimate precision bound.

In the quest to unlock the maximum potential of quantum sensors, it is of paramount importance to have practical measurement strategies that can estimate incompatible parameters with best precisions possible. However, it is still not known how to find practical measurements with optimal precisions, even for uncorrelated measurements over probe states. Here, we give a concrete way to find uncorrelated measurement strategies with optimal precisions.

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[1] Trung Kien Le, Hung Q. Nguyen, and Le Bin Ho, "Variational quantum metrology for multiparameter estimation under dephasing noise", Scientific Reports 13 1, 17775 (2023).

[2] Arshag Danageozian, "Recovery With Incomplete Knowledge: Fundamental Bounds on Real-Time Quantum Memories", Quantum 7, 1195 (2023).

[3] Lorcán O. Conlon, Jun Suzuki, Ping Koy Lam, and Syed M. Assad, "The gap persistence theorem for quantum multiparameter estimation", arXiv:2208.07386, (2022).

[4] Yingkai Ouyang and Gavin K. Brennen, "Finite-round quantum error correction on symmetric quantum sensors", arXiv:2212.06285, (2022).

[5] Yingkai Ouyang and Narayanan Rengaswamy, "Describing quantum metrology with erasure errors using weight distributions of classical codes", Physical Review A 107 2, 022620 (2023).

[6] Lorcán O. Conlon, Ping Koy Lam, and Syed M. Assad, "Multiparameter Estimation with Two-Qubit Probes in Noisy Channels", Entropy 25 8, 1122 (2023).

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