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Physical Review Applied

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

Simon K. Yung, C. M. Yung, Lorcán O. Conlon, Syed M. Assad

Published Jun 9, 2026• 0 citations
Publisher
DOI
10.1103/2nx2-k97n

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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 non-trivial minimisation problem. We present a new 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.
Distributed quantum sensing network has the potential of enhancing the precision in estimating a global function of local parameters by utilizing an entangled probe, compared with that achieved with separable probes. This advantage is often characterized as a quadratic improvement of the quantum Cramér-Rao bound (QCRB). This argument is incomplete in that QCRB assumes a team of all-powerful sensors that can perform arbitrary joint measurements allowed by quantum mechanics. An immediate question arises as to whether such an advantage persists for isolated sensors with physically motivated constraints in their measurement strategies. In this paper, we first consider local operations and classical communication (LOCC) strategies and prove that the QCRB is indeed asymptotically attainable for arbitrary pure probe states, by extending previous work on single-parameter estimation [1]. We further numerically analyze a more restricted scenario where the sensors can only make independent local measurements, and provide evidence that the QCRB is not informative enough for comparing different probe states.
One key aspect of quantum metrology, measurement incompatibility, is evident only through the simultaneous estimation of multiple parameters. The symmetric logarithmic derivative Cramér-Rao bound (SLDCRB), gives the attainable precision, if the optimal measurements for estimating each individual parameter commute. When the optimal measurements do not commute, the SLDCRB is not necessarily attainable. In this regard, the Holevo Cramér-Rao bound (HCRB) plays a fundamental role, providing the ultimate attainable precisions when one allows simultaneous measurements on infinitely many copies of a quantum state. For practical purposes, the Nagaoka Cramér-Rao bound (NCRB) is more relevant, applying when restricted to measuring quantum states individually. The interplay between these three bounds dictates how rapidly the ultimate metrological precisions can be approached through collective measurements on finite copies of the probe state. We first consider two parameter estimation and prove that if the HCRB cannot be saturated with a single copy of the probe state, then it cannot be saturated for any finite number of copies of the probe state. With this, we show that it is impossible to saturate the HCRB for several physically motivated problems. For estimating any number of parameters, we provide necessary and sufficient conditions for the attainability of the SLDCRB with separable measurements. We further prove that if the SLDCRB cannot be reached with a single copy of the probe state, it cannot be reached with collective measurements on any finite number of copies of the probe state. These results together provide necessary and sufficient conditions for the attainability of the SLDCRB for any finite number of copies of the probe state. This solves a significant generalisation of one of the five problems recently highlighted by [P.Horodecki et al, Phys. Rev. X Quantum 3, 010101 (2022)].