Testing Identity Of Collections Of Quantum States: Sample Complexity Analysis

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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.

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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}<br /> }

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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 2023-09-13 12:15:38). The list may be incomplete as not all publishers provide suitable and complete citation data.

On Crossref’s cited-by service no data on citing works was found (last attempt 2023-09-13 12:15:37).

This Paper is published in Quantum under the Creative Commons Attribution 4.0 International (CC BY 4.0) license. Copyright remains with the original copyright holders such as the authors or their institutions.

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