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Most informative Cramér-Rao bound for quantum two-parameter estimation with pure-state probes

Simon K. Yung1,*, C. M. Yung, Lorcán O. Conlon2, and Syed M. Assad3

  • *Contact author: sksyung@gmail.com
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Phys. Rev. Applied 26, 024034 – Published 13 August, 2026

DOI: https://doi.org/10.1103/2nx2-k97n

Abstract

Optimal measurements for quantum multiparameter estimation are complicated by the uncertainty principle. Generally, there is a trade-off between the precision with which different parameters can be simultaneously estimated. The task of determining the minimum achievable estimation error is a central task of multiparameter quantum metrology. For estimating parameters encoded in pure quantum states, the ultimate limit is known but is given by the solution of a nontrivial minimization problem. We present an alternative expression for the achievable bound for two-parameter estimation with pure states that is considerably simpler. We also determine the optimal measurements, completing the problem of two-parameter estimation with pure-state probes. To demonstrate the utility of our result, we determine the precision limit for estimating displacements using grid states.

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Outline

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Phys. Rev. Applied 26, 024034– Published 13 August, 2026
  • Received 19 November 2025
  • Revised 20 March 2026
  • Accepted 9 June 2026
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