Beyond the Swap Test: Optimal Estimation of Quantum State Overlap

M. Fanizza, M. Rosati, M. Skotiniotis, J. Calsamiglia, and V. Giovannetti
Phys. Rev. Lett. 124, 060503 – Published 14 February 2020
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Abstract

We study the estimation of the overlap between two unknown pure quantum states of a finite-dimensional system, given M and N copies of each type. This is a fundamental primitive in quantum information processing that is commonly accomplished from the outcomes of N swap tests, a joint measurement on one copy of each type whose outcome probability is a linear function of the squared overlap. We show that a more precise estimate can be obtained by allowing for general collective measurements on all copies. We derive the statistics of the optimal measurement and compute the optimal mean square error in the asymptotic pointwise and finite Bayesian estimation settings. Besides, we consider two strategies relying on the estimation of one or both states and show that, although they are suboptimal, they outperform the swap test. In particular, the swap test is extremely inefficient for small values of the overlap, which become exponentially more likely as the dimension increases. Finally, we show that the optimal measurement is less invasive than the swap test and study the robustness to depolarizing noise for qubit states.

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  • Received 27 June 2019
  • Revised 11 December 2019
  • Accepted 12 December 2019

DOI:https://doi.org/10.1103/PhysRevLett.124.060503

© 2020 American Physical Society

Physics Subject Headings (PhySH)

Quantum Information, Science & Technology

Authors & Affiliations

M. Fanizza1,*, M. Rosati2,†, M. Skotiniotis2, J. Calsamiglia2, and V. Giovannetti1

  • 1NEST, Scuola Normale Superiore and Istituto Nanoscienze—CNR, I-56126 Pisa, Italy
  • 2Física Teòrica: Informació i Fenòmens Quàntics, Departament de Física, Universitat Autònoma de Barcelona, 08193 Bellaterra (Barcelona), Spain

  • *marco.fanizza@sns.it
  • Matteo.Rosati@uab.cat

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Issue

Vol. 124, Iss. 6 — 14 February 2020

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