Discretized continuous quantum-mechanical observables that are neither continuous nor discrete

Most of the fundamental characteristics of quantum mechanics, such as nonlocality and contextuality, are manifest in discrete, finite-dimensional systems. However, many quantum information tasks that exploit these properties cannot be directly adapted to continuous variable systems. To access these quantum features, continuous quantum variables can be made discrete by binning together their different values, resulting in observables with a finite number, $d$, of outcomes. While direct measurement indeed confirms their manifestly discrete character, here we employ a salient feature of quantum physics known as mutual unbiasedness to show that such coarse-grained observables are in a sense neither continuous nor discrete. Depending on $d$, the observables can reproduce either the discrete or the continuous behavior, or neither. To illustrate these results, we present an example for the construction of such measurements and employ it in an optical experiment confirming the existence of four mutually unbiased measurements with $d=3$ outcomes in a continuous variable system, surpassing the number of mutually unbiased continuous variable observables.

Figure 1

Pictorial illustration of mutual unbiasedness in phase space. (a) Complementary position and momentum eigenstates are distributions denoted by the vertical and horizontal lines, respectively. The overlap between any position eigenstate and any momentum eigenstate is the same, represented by the intersection of the distributions. (b) The nonphysical eigenstates maintaining complementarity are approximated by physical states that are given by positive distributions centered around the eigenstates, with finite widths in both phase-space directions. The overlap between physical states is represented graphically by the overlap of the distributions. The physical states are no longer mutually unbiased. (c) Eigenstates of rotated observables ${\widehat{q}}_j$ and ${\widehat{q}}_k$, illustrating that any two nonparallel observables are mutually unbiased, as per Eq. (1). (d) Physical states corresponding to the eigenstates in (c) also lose their mutual unbiasedness.

Figure 2

Examples of phase-space variables for (a) $R=6$ and (b) the $R=4$ implemented experimentally. Here, $q_0$ is taken to be the position variable $x$. (c) The coarse-graining bin functions (experimentally implemented as amplitude masks) $M_{j,0}$ for $d=3$, represented by the shaded regions.

Figure 3

(a) Schematic of the experiment. Fractional Fourier Transforms (FrFTs) and periodic amplitude masks M are used to prepare and measure the transverse spatial profile of the laser beam. All output light is incident on the full-field single-photon detector. (b) Experimental setup using spatial light modulators (SLMs) for preparation and measurement. The output of a 632.8-$\mathrm{nm}$ He-Ne laser is enlarged and collimated using two lenses. The FrFTs are performed modulating the phase according to the quadratic phase of the lenses with the right focal distances. The use of SLMs allows us to synthesize any order FrFT, which would be challenging with actual lenses. The amplitude masks are also implemented using phase-only modulators applying diffraction gratings and collecting only the first diffraction order. Finally, the beam is attenuated with a neutral density filter (NDF) and detected by a single-photon detector [avalanche photodiode (APD)].

Figure 4

Experimental results for the entropy for $R=4$ PCG measurements with a dimension parameter $d=3$, using angle $\theta =\pi /4$. Entropy is plotted as a function of the physical period $T_2^{\prime }$ in units of pixels (px) of the measurement mask, with preparation fixed at position “0” and period that satisfies the mutual unbiasedness condition (8) for $m_{j0}=1$. The measurement direction is fixed at $k=2$, and the entropy is determined for preparation directions (a) $j=0$, (b) $j=1$, and (c) $j=3$. The blue curves correspond to theoretical predictions for the experimental initial state. The red dots represent the experimental data; the error bars are smaller than the dots. The vertical dashed lines indicate the special values of the period satisfying (5) for different values of $m_{j2}$.

Figure 5

Experimental intensity profiles (red solid curves) at the output SLM obtained by scanning the SLM with a narrow slit aperture while measuring the number of photocounts. The intensity profile is proportional to the probability density of $q_k$. The localized state prepared in direction $j=0$ is represented in directions (a) $k=1$ and (b) $k=2$. The experimental result is compared with the theoretical curves obtained by numerically propagating the initial state (dashed black curves), i.e., by applying the periodic masks and FrFTs numerically onto a Gaussian wave function representing the initial profile of the optical field.

Figure 6

Experimental outcome probabilities used to calculate the Shannon entropies shown in Table 2 for preparation directions (a) $j=0$, (b) $j=1$, (c) $j=2$, and (d) $j=3$. Only the cases when the measurement direction $k$ is different from that used to prepare the localized state are presented. The dashed horizontal line corresponds to $p=\frac{1}{d}=\frac{1}{3}$.