multiparameter-hamiltonian-estimation

name	multiparameter-hamiltonian-estimation
description	Optimal multiparameter estimation for quantum systems using the unified Cramér-Rao bound framework. Use when: (1) estimating functions of multiple parameters in quantum Hamiltonians, (2) designing quantum sensing protocols, (3) analyzing precision limits for non-commuting generators, (4) optimizing quantum metrology with multiple parameters. Based on arXiv:2605.04136.

Multiparameter Hamiltonian Estimation

Ultimate quantum limit and estimation protocol for functions of multiple parameters in general Hamiltonians (arXiv:2605.04136).

Core Result

The multiparameter estimation problem reduces to an optimized single-parameter quantum Cramér-Rao bound, even for arbitrary (possibly non-commuting) generators.

Protocol Design

1. Identify generators: H = sum(theta_i * G_i)
2. Compute QFI matrix: F_Q[ij] = 4 * Cov(G_i, G_j) in optimal probe state
3. Gradient projection: Project gradient of target function onto QFI-inverse
4. Optimal measurement: Choose measurement saturating the single-parameter bound

Application to Quantum Sensing

• Multiple field components (magnetic, electric, gravitational)
• Non-commuting observables (different spin directions)
• Function estimation (field magnitude, direction, gradients)

Pitfalls

• Non-commuting generators introduce fundamental trade-offs
• Optimal probe state may be entangled across multiple probes
• Saturation requires collective measurements on all probes