Optimal measurements in phase estimation: simple examples

TL;DR

This paper investigates optimal measurement strategies for phase estimation across various quantum states, revealing how measurement choices depend on state purity, number of qubits, and symmetry, with specific optimal strategies identified.

Contribution

It provides a comprehensive analysis of optimal measurements for phase estimation in different quantum scenarios, including pure and mixed states, and multi-qubit systems.

Findings

Optimal measurements for pure single-qubit states form a continuous set.

For mixed states, the optimal measurement set reduces to two options.

Estimation from population imbalance is optimal only for pure states.

Abstract

We identify optimal measurement strategies for phase estimation in different scenarios. For pure states of a single qubit, we show that optimal measurements form a broad set parametrized with a continuous variable. When the state is mixed this set reduces to merely two possible measurements. For two qubits, we focus on the symmetric Werner state. We find an optimal measurement and show that estimation from the population imbalance is optimal only if the state is pure. Finally, for a pure state of $N$ qubits, we finds under which conditions the estimation from the full $N$-body correlation and from the population imbalance are optimal.