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arXiv:quant-ph/0609179 · Physical Review Letters

Generalized Limits for Single-Parameter Quantum Estimation

Sergio Boixo, Steven T. Flammia, Carlton M. Caves, J. M. Geremia

Published Feb 28, 2007• 403 citations
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Abstract

We develop generalized bounds for quantum single-parameter estimation problems for which the coupling to the parameter is described by intrinsic multisystem interactions. For a Hamiltonian with k-system parameter-sensitive terms, the quantum limit scales as 1/Nk, where N is the number of systems. These quantum limits remain valid when the Hamiltonian is augmented by any parameter-independent interaction among the systems and when adaptive measurements via parameter-independent coupling to ancillas are allowed.

DOI
10.1103/physrevlett.98.090401

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Quantum metrology enhances the sensitivity of parameter estimation using the distinctive resources of quantum mechanics such as entanglement. It has been shown that the precision of estimating an overall multiplicative factor of a Hamiltonian can be increased to exceed the classical limit, yet little is known about estimating a general Hamiltonian parameter. In this paper, we study this problem in detail. We find that the scaling of the estimation precision with the number of systems can always be optimized to the Heisenberg limit, while the time scaling can be quite different from that of estimating an overall multiplicative factor. We derive the generator of local parameter translation on the unitary evolution operator of the Hamiltonian, and use it to evaluate the estimation precision of the parameter and establish a general upper bound on the quantum Fisher information. The results indicate that the quantum Fisher information generally can be divided into two parts: one is quadratic in time, while the other oscillates with time. When the eigenvalues of the Hamiltonian do not depend on the parameter, the quadratic term vanishes, and the quantum Fisher information will be bounded in this case. To illustrate the results, we give an example of estimating a parameter of a magnetic field by measuring a spin-12\frac{1}{2}21​ particle and compare the results for estimating the amplitude and the direction of the magnetic field.
We present a quantum algorithm to estimate parameters at the quantum metrology limit using deterministic quantum computation with one bit. When the interactions occurring in a quantum system are described by a Hamiltonian $H=\ensuremath{\theta}{H}_{0}$, we estimate $\ensuremath{\theta}$ by zooming in on previous estimations and by implementing an adaptive Bayesian procedure. The final result of the algorithm is an updated estimation of $\ensuremath{\theta}$ whose variance has been decreased in proportion to the time of evolution under HHH. For the problem of estimating several parameters, we implement dynamical-decoupling techniques and use the results of single parameter estimation. The cases of discrete-time evolution and reference frame alignment are also studied within the adaptive approach.