Evaluating the Holevo Cramér-Rao Bound for Multiparameter Quantum Metrology

Area of Science:

• Quantum Metrology
• Quantum Information Theory
• Parameter Estimation

Background:

• Simultaneous estimation of multiple parameters is key to fully understanding quantum metrology.
• The incompatibility of observables poses a challenge in multiparameter quantum estimation.
• The Holevo Cramér-Rao bound (HCRB) is the fundamental limit, but its evaluation has been difficult.

Purpose of the Study:

• To develop a tractable method for evaluating the Holevo Cramér-Rao bound (HCRB) for multiparameter quantum estimation.
• To apply the HCRB evaluation to practical quantum sensing scenarios like optical interferometry and magnetometry.
• To investigate the attainability of the HCRB with optimal measurements in specific quantum systems.

Main Methods:

• Recasting the HCRB evaluation as a semidefinite program for finite-dimensional systems.
• Reducing the semidefinite program size for rank-deficient quantum states.
• Analyzing phase and loss estimation in optical interferometry.
• Studying three-dimensional magnetometry with noisy multiqubit systems.

Main Results:

• The HCRB evaluation is reformulated as a semidefinite program with strong duality.
• For phase estimation in optical interferometry, the HCRB can be attained with optimal single-copy measurements in certain regimes.
• In three-dimensional magnetometry, a complex relationship between HCRB and incompatibility was observed.
• Numerical evidence suggests projective single-copy measurements achieve the HCRB in noiseless 2-qubit systems.

Conclusions:

• A novel semidefinite programming approach simplifies HCRB evaluation for multiparameter quantum estimation.
• The study demonstrates achievable HCRB in specific quantum sensing applications, advancing quantum metrology.
• Understanding the interplay between HCRB and quantum incompatibility is crucial for designing optimal quantum estimators.