Abstract

In the study of quantum limits to parameter estimation, the high dimensionality of the density operator and that of the unknown parameters have long been two of the most difficult challenges. Here we propose a theory of quantum semiparametric estimation that can circumvent both challenges and produce simple analytic bounds for a class of problems in which the dimensions are arbitrarily high, few prior assumptions about the density operator are made, but only a finite number of the unknown parameters are of interest. We also relate our bounds to Holevo's version of the quantum Cram\'er-Rao bound, so that they can inherit the asymptotic attainability of the latter in many cases of interest. The theory is especially relevant to the estimation of a parameter that can be expressed as a function of the density operator, such as the expectation value of an observable, the fidelity to a pure state, the purity, or the von Neumann entropy. Potential applications include quantum state characterization for many-body systems, optical imaging, and interferometry, where full tomography of the quantum state is often infeasible and only a few select properties of the system are of interest.

Highlights

  • The random nature of quantum mechanics has practical implications for the noise in sensing, imaging, and quantum-information applications [1,2,3,4,5,6]

  • We propose a theory of quantum semiparametric estimation that can circumvent both challenges and produce simple analytic bounds for a class of problems in which the dimensions are arbitrarily high, few prior assumptions about the density operator are made, but only a finite number of the unknown parameters are of interest

  • This work addresses a foundational question by Horodecki [18]: “What kind of information can be extracted from an unknown quantum state at a small measurement cost?” Our work shows that quantum metrology—and quantum semiparametric estimation, in particular—offers a viable attack on the question via a statistical notion of efficiency

Read more

Summary

INTRODUCTION

The random nature of quantum mechanics has practical implications for the noise in sensing, imaging, and quantum-information applications [1,2,3,4,5,6]. Our formalism is primarily based on Helstrom’s version of the quantum Cramer-Rao bound [1] While this approach allows us to adapt the classical methods more it is unable to account for the increased errors due to the incompatibility of quantum observables when multiple parameters are involved [7,28]. We address this issue by studying Holevo’s version of the quantum Cramer-Rao bound [7] in the semiparametric setting and proving that the two versions turn out to be close. This result enables our bounds to inherit the asymptotic attainability of Holevo’s bound [28,29,30] in many cases of interest

PREVIEW OF TYPICAL RESULTS
GEOMETRIC PICTURE OF QUANTUM ESTIMATION THEORY
Helstrom bound Let
Hilbert spaces for operators
Generalized Helstrom bound
Influence operator via a functional gradient
Projection into the tangent space
PARAMETRIC SUBMODELS
CONSTRAINED BOUNDS
Entropy estimation in quantum thermodynamics
Philosophy
Looser bounds
Quadrature estimation
Family of classical states
The mother model
Semiparametric measurements and estimators
Lower bounds via parametric submodels
The efficient score operator
Displacement estimation with a constrained family of initial states
NkSeff k2
VIII. VECTORAL PARAMETER OF INTEREST
CONCLUSION
Full Text
Paper version not known

Talk to us

Join us for a 30 min session where you can share your feedback and ask us any queries you have

Schedule a call

Disclaimer: All third-party content on this website/platform is and will remain the property of their respective owners and is provided on "as is" basis without any warranties, express or implied. Use of third-party content does not indicate any affiliation, sponsorship with or endorsement by them. Any references to third-party content is to identify the corresponding services and shall be considered fair use under The CopyrightLaw.