Estimation of Hamiltonian parameters from thermal states

Abstract

We upper- and lower-bound the optimal precision with which one can estimate an unknown Hamiltonian parameter via measurements of Gibbs thermal states with a known temperature. The bounds depend on the uncertainty in the Hamiltonian term that contains the parameter and on the term's degree of noncommutativity with the full Hamiltonian: higher uncertainty and commuting operators lead to better precision. We apply the bounds to show that there exist entangled thermal states such that the parameter can be estimated with an error that decreases faster than $1/\sqrt{n}$, beating the standard quantum limit. This result governs Hamiltonians where an unknown scalar parameter (e.g. a component of a magnetic field) is coupled locally and identically to $n$ qubit sensors. In the high-temperature regime, our bounds allow for pinpointing the optimal estimation error, up to a constant prefactor. Our bounds generalize to joint estimations of multiple parameters. In this setting, we recover the high-temperature sample scaling derived previously via techniques based on quantum state discrimination and coding theory. In an application, we show that noncommuting conserved quantities hinder the estimation of chemical potentials.

5 pages, 2 figures + 13 pages, 1 figure, 1 table

References in corpus (13)

• Observable measure of quantum coherence in finite dimensional systems
• Extending Noether's theorem by quantifying the asymmetry of quantum states
• Quantum criticality as a resource for quantum estimation
• Individual quantum probes for optimal thermometry
• Bures metric over thermal state manifolds and quantum criticality
• Quantum-limited metrology with product states
• Geometric lower bound for a quantum coherence measure
• Learning many-body Hamiltonians with Heisenberg-limited scaling
• Noncommuting conserved charges in quantum thermodynamics and beyond
• Non-Abelian symmetry can increase entanglement entropy
• Quantum many-body systems in thermal equilibrium
• Practical Black Box Hamiltonian Learning
• Universal shot-noise limit for quantum metrology with local Hamiltonians

Cited by in corpus (9)

• Fundamental limits of metrology at thermal equilibrium
• Noncommuting charges can remove non-stationary quantum many-body dynamics
• Gibbs Sampling gives Quantum Advantage at Constant Temperatures with O(1)-Local Hamiltonians
• Critical Quantum Sensing: a tutorial on parameter estimation near quantum phase transitions
• Low-temperature Quantum Metrology Enhanced by Strong Couplings
• Evolved Quantum Boltzmann Machines
• Information geometry of bosonic Gaussian thermal states
• Bounds on quantum Fisher information and uncertainty relations for thermodynamically conjugate variables
• Natural gradient and parameter estimation for quantum Boltzmann machines