Macroscopic quantum measurements of noncommuting observables

Assuming a well-behaving quantum-to-classical transition, measuring large quantum systems should be highly informative with low measurement-induced disturbance, while the coupling between system and measurement apparatus is fairly simple and weak. Here, we show that this is indeed possible within the formalism of quantum mechanics. We discuss an example of estimating the collective magnetization of a spin ensemble by simultaneous measuring three orthogonal spin directions. For the task of estimating the direction of a spin-coherent state, we find that the average guessing fidelity and the system disturbance are nonmonotonic functions of the coupling strength. Strikingly, we discover an intermediate regime for the coupling strength where the guessing fidelity is quasioptimal, while the measured state is almost not disturbed.

Figure 1

Average fidelity ${\mathcal{F}}_{\mathrm{av}}$ for guessing the unknown state as a function of the inverse coupling strength $\mathrm{\Delta }$ for $N=1,\cdots ,4$ (from bottom to top). The dashed lines correspond to the optimal value ${\mathcal{F}}_{\mathrm{opt}}$. For $N\ge 2$, the global maximum is reached by some nonzero $\mathrm{\Delta }$ and is close to the optimal value.

Figure 2

Scaling factor ${\epsilon }_N\lesssim N(1-{\mathcal{F}}_{\mathrm{av}})$ from the approximate lower bound on ${\mathcal{F}}_{\mathrm{av}}$ (upper, blue curve) compared to the optimal scaling factor $N(1-{\mathcal{F}}_{\mathrm{opt}})={(1+2/N)}^{-1}$ (lower, orange curve). For large $N,{\epsilon }_N$ seems to stay slightly above one, which indicates a nonoptimal lower bound. For $N<150$, the approximation clearly does not hold.

Figure 3

Disturbance measured by the quantum fidelity between the pre- and postmeasured states as function of $\mathrm{\Delta }$ for $N=1,2,3$ (from bottom to top). The solid lines correspond to the exact expression, Eq. (7), while the dashed curves are the lowest-order approximations for large $N,D\approx {\left(1+8{\mathrm{\Delta }}^2/N\right)}^{-1}$. The points on the curves indicate where $\mathrm{\Delta }={\mathrm{\Delta }}_{\mathrm{opt}}$.