Testing identity of collections of quantum states: sample complexity analysis

Marco Fanizza; Raffaele Salvia; Vittorio Giovannetti

doi:10.22331/q-2023-09-11-1105

Testing identity of collections of quantum states: sample complexity analysis

Marco Fanizza¹, Raffaele Salvia², and Vittorio Giovannetti³

¹Física Teòrica: Informació i Fenòmens Quàntics, Departament de Física, Universitat Autònoma de Barcelona, 08193 Bellaterra, Spain.
²Scuola Normale Superiore, I-56127 Pisa, Italy.
³NEST, Scuola Normale Superiore and Istituto Nanoscienze-CNR, I-56127 Pisa, Italy.

Published:	2023-09-11, volume 7, page 1105
Eprint:	arXiv:2103.14511v5
Doi:	https://doi.org/10.22331/q-2023-09-11-1105
Citation:	Quantum 7, 1105 (2023).

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Abstract

We study the problem of testing identity of a collection of unknown quantum states given sample access to this collection, each state appearing with some known probability. We show that for a collection of $d$-dimensional quantum states of cardinality $N$, the sample complexity is $O(\sqrt{N}d/\epsilon^2)$, with a matching lower bound, up to a multiplicative constant. The test is obtained by estimating the mean squared Hilbert-Schmidt distance between the states, thanks to a suitable generalization of the estimator of the Hilbert-Schmidt distance between two unknown states by Bădescu, O'Donnell, and Wright [13].

Featured image: Two settings to which our model applies. In a) independent sources produce a random number, Poisson distributed, of copies of a state in a time $T$. In b), a measurement procedure prepares labeled states, each label appearing with some probability.

► BibTeX data

@article{Fanizza2023testingidentityof,
  doi = {10.22331/q-2023-09-11-1105},
  url = {https://doi.org/10.22331/q-2023-09-11-1105},
  title = {Testing identity of collections of quantum states: sample complexity analysis},
  author = {Fanizza, Marco and Salvia, Raffaele and Giovannetti, Vittorio},
  journal = {{Quantum}},
  issn = {2521-327X},
  publisher = {{Verein zur F{\"{o}}rderung des Open Access Publizierens in den Quantenwissenschaften}},
  volume = {7},
  pages = {1105},
  month = sep,
  year = {2023}
}

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Cited by

[1] Li Gao and Nengkun Yu, "Sample optimal tomography of quantum Markov chains", arXiv:2209.02240, (2022).

[2] Marco Fanizza, Michalis Skotiniotis, John Calsamiglia, Ramon Muñoz-Tapia, and Gael Sentís, "Universal algorithms for quantum data learning", EPL (Europhysics Letters) 140 2, 28001 (2022).

The above citations are from SAO/NASA ADS (last updated successfully 2023-09-22 12:56:06). The list may be incomplete as not all publishers provide suitable and complete citation data.

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