Beyond the Swap Test: Optimal Estimation of Quantum State Overlap

Area of Science:

• Quantum Information Science
• Quantum Metrology
• Quantum State Estimation

Background:

• Estimating the overlap between unknown quantum states is crucial for quantum information processing.
• The conventional method uses N swap tests, a joint measurement on one copy of each state.

Purpose of the Study:

• To develop a more precise estimation strategy for quantum state overlap.
• To compare the efficiency of new methods against the standard swap test.

Main Methods:

• Derivation of optimal measurement statistics for collective measurements on multiple copies of quantum states.
• Computation of optimal mean squared error in asymptotic pointwise and finite Bayesian settings.
• Analysis of state estimation strategies and their performance relative to the swap test.

Main Results:

• Collective measurements yield more precise overlap estimates than individual swap tests.
• New strategies outperform the swap test, particularly for small overlap values.
• Optimal measurements are less invasive than the swap test and show robustness to depolarizing noise.

Conclusions:

• General collective measurements offer a superior approach to quantum state overlap estimation.
• The findings have implications for improving quantum information processing tasks and understanding measurement invasiveness.