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  • Open Access

Sample complexity of purity and inner product estimation

Weiyuan Gong1,*, Jonas Haferkamp1,†, Qi Ye2,‡, and Zhihan Zhang2,§

  • *Contact author: wgong@g.harvard.edu
  • †Contact author: jhaferkamp42@gmail.com
  • ‡Contact author: yeq22@mails.tsinghua.edu.cn
  • §Contact author: zhihan-z21@mails.tsinghua.edu.cn
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Phys. Rev. Research 8, 043077 – Published 20 October, 2026

DOI: https://doi.org/10.1103/kh2j-jkn6

Abstract

We study the sample complexity of the prototypical tasks quantum purity estimation and quantum inner product estimation. The former has been a key subroutine for experimental benchmarking and entropy estimation, while the latter is a core task for cross-platform validation that compares the unknown quantum states produced by different physical platforms. In purity estimation, we are to estimate tr(ρ2) of an unknown quantum state ρ to additive error ɛ. Meanwhile, for quantum inner product estimation, Alice and Bob are to estimate tr(ρσ) to additive error ɛ, given copies of the unknown quantum states ρ and σ using classical communication and restricted quantum communication. We propose a protocol that solves quantum inner product estimation with k-qubit one-way quantum communication and unentangled local measurements using O(median{1/ɛ2,2n/2/ɛ,2n−k/ɛ2}) copies of ρ and σ. Our protocol can be modified to estimate the purity of an unknown quantum state ρ using k-qubit quantum memory with the same sample complexity. We also prove that arbitrary protocols with k-qubit quantum memory that estimate purity to error ɛ require Ω(median{1/ɛ2,2n/2/ɛ,2n−k/ɛ2}) copies of ρ. This indicates the same sample complexity lower bound for quantum inner product estimation with one-way k-qubit quantum communication and classical communication, and unentangled local measurements. We further improve the sample complexity lower bound for any purity estimation protocols using an identical single-copy projection-valued measurement. Last, we investigate a decisional variant of quantum distributed inner product estimation without quantum communication for mixed states and provide a sample complexity lower bound.

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Phys. Rev. Research 8, 043077– Published 20 October, 2026
  • Received 25 February 2025
  • Accepted 8 January 2026
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Published by the American Physical Society under the terms of the Creative Commons Attribution 4.0 International license. Further distribution of this work must maintain attribution to the author(s) and the published article's title, journal citation, and DOI.

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