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Measurement-based estimation of causal conditional variances and its applications to macroscopic quantum phenomena

Kosei Hatakeyama1,*, Ryotaro Fukuzumi1,†, Akira Matsumura1,2,‡, Daisuke Miki3,§, and Kazuhiro Yamamoto1,2

  • *Contact author: koseihtkym998@gmail.com
  • †Contact author: fukuzumi.ryotaro.709@s.kyushu-u.ac.jp
  • ‡Contact author: matsumura.akira@phys.kyushu-u.ac.jp
  • §Contact author: dmiki@caltech.edu
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Phys. Rev. A 114, 032403 – Published 1 September, 2026

DOI: https://doi.org/10.1103/wp55-8kx3

Abstract

Building on the framework developed by Meng et al. [Sci. Adv. 8, eabm7585 (2022)], we present an analytical study of a measurement-record-based state estimation method for a mechanical oscillator serving as one end mirror of a detuned-cavity system. Unlike conventional approaches, this method enables quantum-state verification without requiring the unconditional covariance matrix as an externally provided theoretical input. We construct a relative-estimate operator from causal and anticausal quantum Wiener filters and calculate its variance. The deviation from the causal conditional variance is defined as a reconstruction bias, whose magnitude is evaluated analytically. We show that, within experimentally relevant parameter regimes for typical quantum-state preparation, the reconstruction bias is sufficiently small to be neglected. As applications to state verification, we apply the method to proposals for macroscopic quantum entanglement mediated by electromagnetic interactions and for conditional momentum-squeezed states generated by homodyne detection and clarify the conditions under which the bias remains negligible and when the reconstruction bias becomes significant.

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Outline

Information

Phys. Rev. A 114, 032403– Published 1 September, 2026
  • Received 3 April 2026
  • Accepted 31 July 2026
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