Lower Bounds for Quantum Parameter Estimation

Michael Walter; Joseph M. Renes

doi:10.1109/tit.2014.2365174

Home/Archive/arXiv

arXiv · 1310.2155

Lower Bounds for Quantum Parameter Estimation

Michael Walter·Joseph M. Renes

View original source ↗Read source document ↗Export metadata ↓

Abstract

The laws of quantum mechanics place fundamental limits on the accuracy of measurements and therefore on the estimation of unknown parameters of a quantum system. In this work, we prove lower bounds on the size of confidence regions reported by any region estimator for a given ensemble of probe states and probability of success. Our bounds are derived from a previously unnoticed connection between the size of confidence regions and the error probabilities of a corresponding binary hypothesis test. In group-covariant scenarios, we find that there is an ultimate bound for any estimation scheme which depends only on the representation-theoretic data of the probe system, and we evaluate its asymptotics in the limit of many systems, establishing a general "Heisenberg limit" for region estimation. We apply our results to several examples, in particular to phase estimation, where our bounds allow us to recover the well-known Heisenberg and shot-noise scaling.

Explore related subjects

quant-ph cs.IT math-ph math.IT math.MP

Keep this discovery

Explore connections, maps & timelines

BibTeX RIS

Michael Walter, Joseph M. Renes. 2014-11-19. Lower Bounds for Quantum Parameter Estimation. https://doi.org/10.1109/tit.2014.2365174

Cite the original work for its findings. Save a collection to share your selection of sources.

KEEP EXPLORING

Related papers

arXiv2026

Enhanced measurements on quantum computers via the simultaneous probing of non-commuting Pauli operators

Measuring the state of quantum computers is a highly non-trivial task, with implications for virtually all quantum algorithms. A promising avenue is multi-copy schemes, where identical copies of a quantum state are measured jointly so that all Pauli operators within the considered observable can be simultaneously assessed. Here, we present a first implementation of such a two-copy scheme in a measurement protocol. Based on Bayesian statistics, it accurately estimates not only the average of the desired observable but also the error en route. This enables an adaptive shot-allocation algorithm that preferentially samples the most uncertain Pauli terms. In regimes with many non-commuting Pauli operators, this ``double'' scheme can outperform the state-of-the-art measurement protocol in minimizing total shots for a given precision. We also numerically confirm the finding in previous theoretical works that the two-copy scheme incurs an overhead due to the square-root relationship between the variance of measured quantities and the number of measurement shots.

quant-ph↗

arXiv2026

Thermodynamics of a phaseonium-driven optomechanical Otto engine

We study an optomechanical Otto engine whose working medium is a single-mode cavity driven by beams of coherently prepared three-level phaseonium atoms. The atoms are not thermal reservoirs in the Gibbs sense; rather, their populations and ground-state coherence set the detailed-balance ratio of the cavity collision map, so that the field relaxes to a Gibbs state at an operational apparent temperature. We combine the finite-time collision-model dynamics with radiation-pressure work extraction and compare three reservoir preparations: a thermal reference at the same apparent temperatures, an incoherent atomic beam with the same populations, and the coherent phaseonium beam. We show that the phaseonium isochore charges the cavity passively: the cavity ergotropy and energy-basis coherence remain zero up to numerical precision, while the state converges to the Gibbs fixed point selected by the apparent detailed balance. We further estimate lower bounds on the cost of preparing the atomic populations and coherence, showing that the relevant advantage of phaseonium is a resource-preparation tradeoff rather than a cost-free enhancement over a thermal bath at the same temperature. Finally, we assess the finite-time performance of a two-cavity cascade with additive mechanical work accounting. Over the investigated coherence-phase range, the cascade produces approximately $47\%$--$52\%$ more power than the single-cavity engine while requiring only $65\%$--$68\%$ of the hot and cold phaseonium atoms needed by two independent engines, resulting in a $9\%$--$15\%$ enhancement of power per injected atom over a complete cycle.

quant-ph↗

arXiv2026

Dynamical Correlation of the Post-quench Non-thermal Equilibrium State

After a quantum quench, the integrable system is expected to relax to a non-thermal equilibrium state (NTES) whose local properties are believed to be governed by a generalized Gibbs ensemble (GGE). Combining quench action and the form factor approach, we compute the field-field correlation in the NTES produced by an interaction quench of the Lieb-Liniger model. The spectral distribution is shown to be qualitatively different from that of a thermal equilibrium state (TES): a new dispersion branch appears whose microscopic mechanism can be traced to the algebraic decaying tail for the root density distribution function, and indicates the existence of a broader family of NTES featuring similar spectral property.

quant-ph↗