Elementary test for nonclassicality based on measurements of position and momentum

We generalize a nonclassicality test described by Kot et al. [Phys. Rev. Lett. 108, 233601 (2012)], which can be used to rule out any classical description of a physical system. The test is based on measurements of quadrature operators and works by proving a contradiction with the classical description in terms of a probability distribution in phase space. As opposed to the previous work, we generalize the test to include states without rotational symmetry in phase space. Furthermore, we compare the performance of the nonclassicality test with classical tomography methods based on the inverse Radon transform, which can also be used to establish the quantum nature of a physical system. In particular, we consider a nonclassicality test based on the so-called filtered back-projection formula. We show that the general nonclassicality test is conceptually simpler, requires less assumptions on the system, and is statistically more reliable than the tests based on the filtered back-projection formula. As a specific example, we derive the optimal test for quadrature squeezed single-photon states and show that the efficiency of the test does not change with the degree of squeezing.

Figure 1

(a) Wigner function $W(x,p)$ for the pure single-photon state of a single mode of light [compare with Eq. (21)]. It is not a probability distribution as it should be in the classical framework since it takes on negative values. (b) Top view of the same function. The color gradient allows to identify the negative region in the center. The cut represented by the line is the quadrature $Q_{\theta }$. The marginal distribution corresponding to the probability distribution for measurement outcomes for $Q_{\theta }$ is obtained by integration of $W(x,p)$ along the perpendicular directions to the cut, as shown by the arrows.

Figure 2

Violation of nonclassicality for noisy single-photon states. For different efficiencies $\eta$, the figure shows the optimal expectation value of the test function relative to its standard deviation ($\mathcal{G}$) as a function of the order of the polynomial in Eq. (15). Experimentally, this may result in a violation of classicality by $-\sqrt{M-m}\mathcal{G}$ standard deviations, with $M$ being the total number of measurements and $m\ge N+1$ the number of measured quadratures.

Figure 3

Radial profile of $W_{\eta }$ for noisy single-photon states (solid line $\eta =1$, dashed line $\eta =0.7$) and of the optimal test functions ${\mathcal{F}}_{\eta }$ (dashed-dotted line $\eta =1$, dotted line $\eta =0.7$) for such states, at $N=28$. The inset shows the 3D plot of the optimal test function for $\eta =1, N=28$.

Figure 4

Sketch (not to scale) of $W(x,p)$ (left) and $W_{\lambda }(x,p)$ (right), together with a representation of the optimal distribution of phase cuts. The cuts are uniformly distributed for the single-photon state and stretched along the $x$ axis for the squeezed state. Note that the cuts stretch along the opposite direction of stretching for $W_{\lambda }(x,p)$. This reflects that most cuts are placed at angles where the distribution varies the fastest. The inner (blue) region is where the distribution takes on negative values.

Figure 5

Action of the operators $\mathbf{R}$ and ${\mathbf{R}}^{†}$. The inversion cannot be achieved directly through the application of these two operators, i.e., there is a final step missing represented by the question mark. Since the latter is hard to achieve, the idea of the filtered back-projection is to rely on ${\mathbf{R}}^{†}$ for making statements about $W(x,p)$.

Figure 6

Plot of the ratio $\mathcal{R}\left(M\right)$ of the statistical significance of the elementary test and the inverse Radon method for noisy single-photon states. The comparison is with respect to the best elementary test for $N=28,m=29$. The value $\mathcal{R}\left(M\right)>1$ indicates that the elementary test is statistically more reliable. The almost linear part of the graphs shows a nearly logarithmic growth for high $M$; the low-$M$ region deviates from this behavior due to the $\sqrt{M-m}$ dependence in ${\sigma }_{\mathcal{F}}$. The markers on the graphs loosely show the threshold for a violation by at least two standard deviations for the inverse Radon method.

Figure 7

Logarithmic plot of $\frac{1}{u^2(\theta ,\lambda )}$ as a function of $\theta$ for some values of the squeezing parameter $\lambda$. The region that contributes the most to the integral in Eq. (45) gets narrower as lambda increases.

Figure 8

Optimal postion for 12 cuts in the middle-point quadrature. Here, $\lambda =5$ and the precision of the numerical quadrature is within the $3\%$ of error.

Figure 9

The numerical error $\mathcal{E}$ as a function of the number of cuts $m$ for various values of the squeezing parameter $\lambda$. The plot has been truncated at the onset of numerical instabilities that cause the error to oscillate around ${10}^{-5}$. Notice the nearly exponential decrease for each curve and that for larger $\lambda$ more cuts are needed to reach a given error.