Angle and angular momentum: Uncertainty relations, simultaneous measurement, and phase-space representation

Reaching ultimate performance of quantum technologies requires the use of detection at quantum limits and access to all resources of the underlying physical system. We establish a full quantum analogy between the pair of angular momentum and exponential angular variable, and the structure of canonically conjugate position and momentum. This includes the notion of optimal simultaneous measurement of the angular momentum and angular variable, the identification of Einstein-Podolsky-Rosen-like variables and states, and, finally, a phase-space representation of quantum states. Our construction is based on close interconnection of the three concepts and may serve as a template for the treatment of other observables. This theory also provides a test bed for implementation of quantum technologies combining discrete and continuous quantum variables.

Figure 1

Phase-space-representation of von Mises states $|n,\alpha \rangle , n\in \mathbb{Z}$, Eq. (2). The phase space consists of parallel equidistant rings (black rings), which are orthogonal to $z$ axis and their centers possess the $z\mathrm{th}$ coordinate $n$. The von Mises state $|n,\alpha \rangle$ is represented by a noise ellipsis (red ellipsis) centered around a point on the circle with $z\mathrm{th}$ coordinate $n$ and polar angle $\alpha$ (positive angle between blue line segment and positive $x$ axis). The shape of the ellipsis depends on the value of the spread parameter $\kappa$, which is chosen to grow from the bottom to the top. Accordingly, the uncertainties $\langle {\left(\mathrm{\Delta }L\right)}^2\rangle$ (${\omega }^2$), Eq. (4), grow (decrease) from the bottom to the top. The red ring represents von Mises state with $n=-2$ and $\kappa =0$, which is an angular momentum eigenstate, so the other phase-space rings are images of the respective angular momentum eigenstates. The red vertical line represents the von Mises state with $n=2$ in the limit of $\kappa \to \infty$. The red circle represents the von Mises state with $n=0$ and symmetrical uncertainties $\langle {\left(\mathrm{\Delta }L\right)}^2\rangle ={\omega }^2=1/2$ for $\kappa \doteq 1.292$.

Figure 2

Uncertainties and uncertainty products for angular momentum and angular variable for optimal states and measurements versus the signal-state spread parameter ${\kappa }_s$. Uncertainties $\langle {\left(\mathrm{\Delta }\mathcal{L}\right)}^2\rangle$ (green crosses) and ${\mathrm{\Omega }}^2=\langle {\left(\mathrm{\Delta }\mathcal{S}\right)}^2\rangle /|\langle E_s\rangle {|}^2{\left|\langle E_a\rangle \right|}^2$ (black stars), and uncertainty product $\langle {\left(\mathrm{\Delta }\mathcal{L}\right)}^2\rangle {\mathrm{\Omega }}^2$ (solid red line) for optimal simultaneous measurement with von Mises signal and ancilla states with different spread parameters satisfying the optimal matching condition (9) whose inverse is depicted in the inset. The same uncertainty product for suboptimal simultaneous measurement with von Mises signal and ancilla states with the same spread parameters ${\kappa }_s={\kappa }_a$ (dotted magenta line). The product mean $\langle {\left(\mathrm{\Delta }\mathcal{L}\right)}^2{\left(\mathrm{\Delta }\mathcal{S}\right)}^2\rangle /|\langle E_s\rangle {|}^2{\left|\langle E_a\rangle \right|}^2$ for von Mises signal and ancilla states satisfying optimal matching condition (9) (dashed blue line). The uncertainty product for optimal simultaneous measurement approaches, asymptotically its lower bound of 1, which is four times larger than the lower bound of 1/4 for uncertainty relations (3). The product mean always lies below the uncertainty product for optimal measurement and it may even lie below 1. Equality $\langle {\left(\mathrm{\Delta }\mathcal{L}\right)}^2\rangle ={\mathrm{\Omega }}^2\doteq 1.099$ is achieved for ${\kappa }_s\doteq 1.146$ and ${\kappa }_a\doteq 1.632$.