Testing identity of collections of quantum states: sample complexity analysis

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 (https://dl.acm.org/doi/10.1145/3313276.3316344).

23+6 pages, 1 figure

References in corpus (12)

• Efficient Quantum Circuits for Schur and Clebsch-Gordan Transforms
• Theory of quantum system certification: a tutorial
• The Structure of Bipartite Quantum States - Insights from Group Theory and Cryptography
• The quantum capacity with symmetric side channels
• Optimal measurements for relative quantum information
• Exact Identification of a Quantum Change Point
• Changepoint Problem in Quantumn Setting
• Estimating Quantum and Private capacities of Gaussian channels via degradable extensions
• Hypothesis testing for an entangled state produced by spontaneous parametric down conversion
• Ultimate limits for quickest quantum change-point detection
• Quantum Causal Unravelling
• Entanglement is Necessary for Optimal Quantum Property Testing