Beyond the Swap Test: Optimal Estimation of Quantum State Overlap

M. Fanizza*, M. Rosati, M. Skotiniotis, J. Calsamiglia, V. Giovannetti

*Corresponding author for this work

Research output: Contribution to journalArticleResearchpeer-review

13 Citations (Scopus)


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.

Original languageEnglish
Article number060503
JournalPhysical Review Letters
Issue number6
Publication statusPublished - 14 Feb 2020


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