Role of phase of optimal probe in noncommutativity vs coherence in quantum multiparameter estimation

TL;DR

This paper investigates the structure of optimal quantum probes for simultaneous estimation of two phases, revealing that a fixed-phase superposition state is universally optimal regardless of the weight matrix, and explores the impact of noncommutativity on estimation precision.

Contribution

It identifies a universally optimal probe state for two-parameter quantum phase estimation and analyzes how noncommutativity affects estimation accuracy.

Findings

Optimal probe is a superposition of eigenstates with fixed relative phase.

Optimal probe maximizes the determinant of the quantum Fisher information matrix.

High noncommutativity degrades estimation precision, but maximal noncommutativity is not always optimal.

Abstract

Quantum multiparameter estimation offers a framework for the simultaneous estimation of multiple parameters, pertaining to possibly noncommutating observables. While the optimal probe for estimating a single unitary phase is well understood - being a pure state that is an equal superposition of the eigenvectors of the encoding Hamiltonian corresponding to its maximum and minimum eigenvalues - the structure of optimal probes in the multiparameter setting remains more intricate. We investigate the simultaneous estimation of two phases, each encoded through arbitrary qubit Hamiltonians, using arbitrary weight matrices, and considering single-qubit probes. We also consider single-qutrit probes, for which the encoding Hamiltonians are chosen as SU(2) generators. We find that in both the qubit and qutrit scenarios, the optimal probe is a coherent superposition of the eigenstates corresponding to the largest and smallest eigenvalues of the total encoding Hamiltonian, with a fixed - and not arbitrary - relative phase. Remarkably, this optimal probe is independent of the specific choice of weight matrix, making it a universally optimal input state for the estimation of any pair of SU(2) parameters, which can be reparameterized to the phase estimation problem. Furthermore, we show that this probe also maximizes the determinant of the quantum Fisher information matrix, providing a handy tool for identifying the optimal probe state. We also examine the role of commutativity between the generators of the unitary encodings. Our results demonstrate that a high degree of commutativity degrades the achievable precision, rendering estimation infeasible in the extreme case. However, maximal noncommutativity does not necessarily yield optimal precision.