Quantum Metrology Achieves Heisenberg Scaling with Body Permutation-Invariant Hamiltonians

The pursuit of enhanced precision in measurement, known as quantum metrology, often demands complex quantum states exhibiting strong correlations between multiple particles. Majid Hassani, Mengyao Hu, and Guillem Müller-Rigat, alongside colleagues from Instituut-Lorentz, Universiteit Leiden, Jagiellonian University, and ETH Zurich, investigate how well relatively simple quantum states can perform in this arena. Their work addresses a key challenge, namely that creating states with full-body correlations is experimentally difficult, while practical systems often involve interactions between only a few particles at a time. The team demonstrates that randomly generated quantum states, originating from specific types of Hamiltonians, surprisingly achieve the theoretical limit of precision known as Heisenberg scaling, and they reveal a fundamental relationship between the ease of creating these states and their ability to enhance measurement accuracy. This discovery broadens the scope of accessible quantum metrology schemes and offers a pathway towards building more practical quantum sensors.

Multipartite quantum states achieving the Heisenberg limit of sensitivity generally require fully correlated states for preparation. However, experimentally feasible Hamiltonians frequently involve only few-body correlations. This work investigates metrological performance under this constraint, employing techniques derived from the quantum Fisher information. Results demonstrate that typical random symmetric ground states of k-body permutation-invariant Hamiltonians exhibit Heisenberg scaling, and establish a tradeoff between the Hamiltonian’s energy gap, which quantifies preparation difficulty, and achievable precision.

Calculating Upper Bounds on Quantum Fisher Information

This research focuses on determining how precisely a parameter can be estimated by observing a quantum system. The quantum Fisher information (QFI) serves as a fundamental limit on estimation precision, with higher values indicating better performance. Scientists calculated the QFI for a system where a probe state evolves under the influence of a Hamiltonian dependent on the parameter being measured, and sought to find an upper bound on this value. The method involves examining a probe state as it evolves unitarily. To establish an upper bound, the team utilized the concept of an operator seminorm and derived a key result demonstrating that the QFI cannot exceed a value proportional to the number of qubits in the system.

Results show that the upper bound on the QFI scales linearly with the number of qubits, suggesting that adding more qubits can improve measurement precision. The team also identified specific probe states that can saturate this upper bound, achieving the maximum possible precision. The structure of the Hamiltonian plays a crucial role in determining the QFI and the optimal probe states. Numerical simulations confirmed the scaling of the QFI and the saturation of the upper bound. These states are less susceptible to noise, allowing for more precise parameter estimation.

Ground State Sensitivity Nears Heisenberg Limit

Scientists investigated the metrological properties of ground states arising from physically realistic Hamiltonians, focusing on systems with limited interactions between particles. Their work demonstrates that these ground states can achieve sensitivity approaching the Heisenberg limit. The team developed tools based on the quantum Fisher information to quantify how well these states can estimate unknown parameters, revealing that even with limited interactions, high precision is attainable. The research establishes a relationship between the energy gap of the Hamiltonian and the quantum Fisher information, revealing a crucial tradeoff for designing practical quantum sensors.

The team’s methods extend to scenarios where the parameter being measured influences the encoding process itself. To analyze complex systems, scientists focused on states within a symmetric subspace, drastically reducing computational demands without sacrificing metrological utility. Results demonstrate that ground states of these Hamiltonians can achieve a quantum Fisher information close to the number of particles, surpassing the standard quantum limit and indicating the presence of entanglement enhancing measurement sensitivity. This versatile computational approach has potential applications beyond quantum metrology, including the construction of Bell inequalities based on few-body correlations.

Heisenberg Scaling From Simple Quantum States

This work investigates the potential of many-body quantum states for enhancing precision measurements, focusing on systems governed by relatively simple interactions. Researchers demonstrate that randomly generated ground states of few-body permutation-invariant Hamiltonians surprisingly exhibit Heisenberg scaling, a hallmark of quantum enhancement in metrology. This finding indicates that these states can achieve measurement precision comparable to those requiring much more complex, fully-correlated states. The team also identified minimal interaction sets capable of simulating these advantageous states and characterized the properties of Hamiltonians that produce them. Importantly, they established a fundamental tradeoff between the ease of preparing a ground state, quantified by the Hamiltonian’s gap, and the resulting precision of the measurement. As a practical outcome of this research, scientists developed an efficient algorithm for evaluating spin Hamiltonians within specific symmetry constraints.