Mutual Unbiasedness in Coarse-Grained Continuous Variables

The notion of mutual unbiasedness for coarse-grained measurements of quantum continuous variable systems is considered. It is shown that while the procedure of “standard” coarse graining breaks the mutual unbiasedness between conjugate variables, this desired feature can be theoretically established and experimentally observed in periodic coarse graining. We illustrate our results in an optics experiment implementing Fraunhofer diffraction through a periodic diffraction grating, finding excellent agreement with the derived theory. Our results are an important step in developing a formal connection between discrete and continuous variable quantum mechanics.

Figure 1

Periodic coarse-graining scheme for $d\equiv T/s=4$. A diagram of the mask function (4) for $j=0$ is also represented.

Figure 2

Sketch of the experimental setup used for the demonstration of unbiased coarse-grained measurements. The transverse field distribution of a laser beam is prepared and measured using periodic spatial masks displayed on SLMs. Preparation and measurement sites are connected via optical Fourier transform. The light power transmitted through preparation and measurement spatial masks is monitored with an optical power meter.

Figure 3

(a) Entropy plot associated with outcomes’ probabilities of PCG measurements with $d=4$ and $T_x=192\mu \mathrm{m}$, as a function of $T_p$. (b) Examples of measured outcome distributions $p_{l|0}$ for the selected data points shown in plot (a). Unbiasedness ($p_{l|0}\approx 1/d$) is shown for data points $\mathrm{I}$ and III.

Figure 4

Measured optimal periodicity in momentum domain when using mask functions with constant bin width $s_x=48\mu \mathrm{m}$ for the preparation of $d$-dimensional PCG. The theoretically predicted value is ${\mathcal{T}}_p^{\text{opt}}=f_e\lambda /s_x\approx 1319\mu \mathrm{m}$.