TY - JOUR
T1 - Beyond the Swap Test
T2 - Optimal Estimation of Quantum State Overlap
AU - Fanizza, M.
AU - Rosati, M.
AU - Skotiniotis, M.
AU - Calsamiglia, J.
AU - Giovannetti, V.
N1 - Funding Information:
M. F. and V. G. acknowledge support by MIUR via PRIN 2017 (Progetto di Ricerca di Interesse Nazionale): project QUSHIP (2017SRNBRK). M. R., M. S., and J. C. acknowledge support from the Spanish MINECO, Project No. FIS2016-80681-P with the support of AEI/FEDER funds, and from the Generalitat de Catalunya, Project No. CIRIT 2017-SGR-1127. M. R. also acknowledges partial financial support from the Baidu-UAB collaborative project “Learning of quantum hidden Markov models.” M. S. also acknowledges support from Spanish MINECO Project No. IJCI-2015-24643. J. C. also acknowledges the Catalan Government for the project QuantumCAT 001-P-001644 (RIS3CAT comunitats) co-financed by the European Regional Development Fund (FEDER). This project has received funding from the European Union’s Horizon 2020 research and innovation programme under the Marie Skłodowska-Curie Grant Agreement No. 845255. M. F. thanks Matthias Christandl for the helpful comments. M. S. and J. C. acknowledge the useful discussion with Nana Liu.
Publisher Copyright:
© 2020 American Physical Society.
Copyright:
Copyright 2020 Elsevier B.V., All rights reserved.
PY - 2020/2/14
Y1 - 2020/2/14
N2 - 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.
AB - 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.
UR - http://www.scopus.com/inward/record.url?scp=85080934777&partnerID=8YFLogxK
U2 - 10.1103/PhysRevLett.124.060503
DO - 10.1103/PhysRevLett.124.060503
M3 - Article
C2 - 32109123
AN - SCOPUS:85080934777
VL - 124
IS - 6
M1 - 060503
ER -