Quantum-limited Euler angle measurements using anticoherent states

Abstract

Many protocols require precise rotation measurement. Here we present a general class of states that surpass the shot noise limit for measuring rotation around arbitrary axes. We then derive a quantum Cramér-Rao bound for simultaneously estimating all three parameters of a rotation (e.g., the Euler angles), and discuss states that achieve Heisenberg-limited sensitivities for all parameters; the bound is saturated by "anticoherent" states [Zimba, Electron. J. Theor. Phys. 3, 143 (2006)] (we are reluctant to use "anticoherent" to describe the states, but the name has become commonplace over the last decade). Anticoherent states have garnered much attention in recent years, and we elucidate a geometrical technique for finding new examples of such states. Finally, we discuss the potential for divergences in multiparameter estimation due to singularities in spherical coordinate systems. Our results are useful for a variety of quantum metrology and quantum communication applications.

Optimal Detection of Rotations about Unknown Axes by Coherent and Anticoherent States

“…This problem involves estimating rotations about unknown axes. It has been shown in [13] that spin states with vanishing spin expectation value and isotropic variances of the spin components are valuable for estimating such rotations, as they saturate the quantum Cramér-Rao bound for any axis. Also, recently, the problem of characterizing a rotation about an unknown direction encoded into a spin-j state has been considered in [14].…”

“…For small rotation angles η, the average fidelity is minimized by anticoherent states, which are characterized by the fact that they do not manifest any privileged direction; in this respect, they are as distinct as possible from coherent states [17]. The role of anticoherent states for optimal detection of rotations has also been observed and was subsequently quantified in terms of quantum Fisher information in [13]. Between these two extreme cases of η ∼ 0 and η ∼ π, optimal states are neither coherent nor anticoherent in general.…”

“…It was shown in [13] that minimizing the uncertainty in the measurement of an unknown angle about a known rotation axis is equivalent to identifying the states which minimize the fidelity F |ψ (η, n), assuming small rotation angles and using parameter estimation theory. To see this, first expand the fidelity as…”

Quantum concepts in optical polarization

“…To investigate this point it is advantageous to use the Majorana stellar representation [343], which allows us to uniquely depict a spin state state living in H by 2 points on the unit sphere [344]. Several decades after its conception, this representation has recently attracted a great deal of attention in several fields [345][346][347][348][349][350][351][352][353][354][355][356][357][358].…”

Multiphase estimation without a reference mode

“…However, this focus on the single-parameter case is neither necessary nor advisable. Recent suggestions advise adopting a multiple parameter approach [15,16], thus making quantumenhanced multiparameter estimation [17][18][19][20][21][22][23][24][25][26][27][28] an important component of the next quantum revolution.…”