Quantum-limited Euler angle measurements using anticoherent states

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.

Figure 1

Example of the Majorana representation of a quantum state that can be used to achieve Heisenberg-limited sensitivities $O\left(1/N\right)$ for simultaneously estimating all three parameters of a rotation. The image shows two intersecting tetrahedra that are duals to each other, one colored in purple and the other in orange. The vertices of the respective tetrahedra are on the surface of the sphere, colored as purple (small) and orange (large) points. Also shown as green cubes are the vertices of a truncated tetrahedron aligned with the purple tetrahedron. A state whose Majorana representation has $m$ degenerate points at each of the (purple, small, spherical) vertices of one tetrahedron, $n$ degenerate points at the (orange, large, spherical) vertices of the other tetrahedron, and $k$ degenerate points at the (green, cubic) vertices of the truncated tetrahedron is a 2-anticoherent state, with $N=4(m+n+3k)$.

Figure 2

The Archimedean solids, other than the truncated tetrahedron. These shapes can all be circumscribed by a sphere that intersects every one of their vertices. All of the faces of the Archimedean solids are regular polygons, and every vertex of an Archimedean solid is symmetric with every other vertex of the solid (in the sense that they are connected by global isometries). States whose Majorana representations correspond to the vertices of the Archimedean solids are 2-anticoherent and can be used to attain quantum enhancements in measuring rotations.