{
  "id": "2310.06767",
  "version": "v1",
  "published": "2023-10-10T16:46:24.000Z",
  "updated": "2023-10-10T16:46:24.000Z",
  "title": "Optimal estimation of pure states with displaced-null measurements",
  "authors": [
    "Federico Girotti",
    "Alfred Godley",
    "Mădălin Guţă"
  ],
  "comment": "Comments or suggestions are more than welcome",
  "categories": [
    "quant-ph",
    "math-ph",
    "math.MP",
    "math.ST",
    "stat.TH"
  ],
  "abstract": "We revisit the problem of estimating an unknown parameter of a pure quantum state, and investigate `null-measurement' strategies in which the experimenter aims to measure in a basis that contains a vector close to the true system state. Such strategies are known to approach the quantum Fisher information for models where the quantum Cram\\'{e}r-Rao bound is achievable but a detailed adaptive strategy for achieving the bound in the multi-copy setting has been lacking. We first show that the following naive null-measurement implementation fails to attain even the standard estimation scaling: estimate the parameter on a small sub-sample, and apply the null-measurement corresponding to the estimated value on the rest of the systems. This is due to non-identifiability issues specific to null-measurements, which arise when the true and reference parameters are close to each other. To avoid this, we propose the alternative displaced-null measurement strategy in which the reference parameter is altered by a small amount which is sufficient to ensure parameter identifiability. We use this strategy to devise asymptotically optimal measurements for models where the quantum Cram\\'{e}r-Rao bound is achievable. More generally, we extend the method to arbitrary multi-parameter models and prove the asymptotic achievability of the the Holevo bound. An important tool in our analysis is the theory of quantum local asymptotic normality which provides a clear intuition about the design of the proposed estimators, and shows that they have asymptotically normal distributions.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2023-10-10T16:46:24.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "81P50"
    ],
    "keywords": [
      "optimal estimation",
      "pure states",
      "null-measurement",
      "quantum local asymptotic normality",
      "reference parameter"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}