Heisenberg scaling of imaging resolution by coherent enhancement

Abstract

Classical imaging works by scattering photons from an object to be imaged, and achieves resolution scaling as 1/ √ t, with t the imaging time. By contrast, the laws of quantum mechanics allow one to utilize quantum coherence to obtain imaging resolution that can scale as quickly as 1/t -the so-called "Heisenberg limit." However, ambiguities in the obtained signal often preclude taking full advantage of this quantum enhancement, while imaging techniques designed to be unambiguous often lose this optimal Heisenberg scaling. Here we demonstrate an imaging technique which combines unambiguous detection of the target with Heisenberg scaling of the resolution. We also demonstrate a binary search algorithm which can efficiently locate a coherent target using the technique, resolving a target trapped ion to within 0.3% of the 1/e 2 diameter of the excitation beam.

Trapped-ion quantum computing: Progress and challenges

“…Higher-order pulse sequences can provide even greater error suppression, but at some point the increase in overall gate length is not justified by the error reduction. In addition to gate errors, composite pulse sequences have also been developed and demonstrated to reduce crosstalk errors arising in optical-qubit control [286][287][288].…”

Heisenberg scaling precision in the estimation of functions of parameters

“…A thorough analysis of the matrix I in (10) (see appendix D) shows that only f (ϕ), or functions of f (ϕ), admit estimators with finite variances. In particular, any unbiased estimator f of f (ϕ) is characterized by a variance which satisfies (see appendix D)…”

“…. , l. It Appendix C: Derivation of the Fisher information matrix in (10) In this appendix we will obtain the expression of the Fisher information matrix (10) from the general Fisher information matrix for a Gaussian distribution (7) when condition ( 8) and ( 9) hold.…”

“…In this appendix we will show that the Fisher information matrix I shown in (10) implies that the only functions of the parameter α(ϕ) which admit unbiased estimators with finite variance are of the form α(ϕ) ≡ g(f (ϕ)), with g(•) a smooth function.…”

“…In particular, it is possible to conceive quantum estimation strategies which lead to errors that scale as fast as 1/N , bound usually called Heisenberg Limit (HL), whereas the error of any classical strategy is bounded by the so-called Shot-Noise Limit (SNL) and cannot scale faster than 1/ √ N . Since these initial works, considerable effort has been put in the development of quantum estimation protocols reaching the HL [5][6][7][8][9], with applications in imaging [10,11], thermometry [12,13], magnetic field [14,15] and gravitational waves detection [16], among others.…”