Experimental refutation of a class of $\psi$-epistemic models

The quantum state $\psi$ is a mathematical object used to determine the outcome probabilities of measurements on physical systems. Its fundamental nature has been the subject of discussions since the origin of the theory: Is it ontic, that is, does it correspond to a real property of the physical system? Or is it epistemic, that is, does it merely represent our knowledge about the system? Recent advances in the foundations of quantum theory show that epistemic models that obey a simple continuity condition are in conflict with quantum theory already at the level of a single system. Here we report an experimental test of continuous epistemic models using high-dimensional attenuated coherent states of light traveling in an optical fiber. Due to nonideal state preparation (of coherent states with imperfectly known phase) and nonideal measurements (arising from losses and inefficient detection), this experiment tests only epistemic models that satisfy additional constraints which we discuss in detail. Our experimental results are in agreement with the predictions of quantum theory and provide constraints on a class of $\psi$-epistemic models.

Figure 1

Illustration of $\psi$-ontic and $\delta$-continuous $\psi$-epistemic models. Depicted (left and center) is the space $\Lambda$ of ontic states, as well as the support of the probability distribution $P\left(\lambda \right|Q_k)$ for preparation $Q_k$ associated with distinct pure states ${\psi }_k$, $k=1,...,5$. In $\psi$-ontic models (left) distinct quantum states give rise to probability distribution $P\left(\lambda \right|Q_k)$ with no overlap. In $\delta$-continuous $\psi$-epistemic models (center), states that are close to each other (such as $\{{\psi }_1,{\psi }_2,{\psi }_3\}$ and $\{{\psi }_3,{\psi }_4,{\psi }_5\}$) all share common ontic states. However, states that are further from each other (such as ${\psi }_1$ and $\{{\psi }_4,{\psi }_5\}$) do not necessarily have common ontic states $\lambda$. On the right we represent the relationship between the states in Hilbert space. We have depicted by dots the positions of the states on the Bloch sphere, and in gray around each state the sphere of radius $\delta$. For instance, the sphere of radius $\delta$ around state ${\psi }_1$ includes states ${\psi }_2$ and ${\psi }_3$, but not states ${\psi }_4$ and ${\psi }_5$. In $\delta$-continuous epistemic models all states in the ball of radius $\delta$ share at least one ontic state.

Figure 2

Top panel: Experimental setup. Red (light gray) links are optical fibers and blue (dark gray) links are radio-frequency cables. A continuous laser source (S) emits a coherent state whose intensity is modulated by an acousto-optic modulator (AOM) driven by a pattern generator (PG), yielding a train of $d$ pulses with one missing. The train of pulses is then attenuated to the single-photon regime by passing through an optical attenuator (A). The complete pulse train is stored in a 5 km fiber spool (FS), and then sent to a superconducting single-photon detector (SSPD). A fiber polarization controller (FPC) is used to ensure maximum sensitivity of the SSPD (which is polarization sensitive). When a photon is detected, an electric signal is sent to a data acquisition system (DAQ) which stores the time at which the detection event has taken place relative to the time at which the state preparation began. Bottom panel: Example of experimental results for $d=10$, and states $|{\alpha }_1\rangle$, $|{\alpha }_5\rangle$, and $|{\alpha }_{10}\rangle$. Horizontal axis: time at which a detection is registered. Vertical axis: number of detections in each 10 ns time window. Continuous $\psi$-epistemic models would predict a nonzero count rate in the bins which should be empty. The small number of counts which do occur due to detector dark counts and finite extinction ratio of the AOM are not visible on this scale.

Figure 3

Schematic depiction of the structure of the space $\Lambda$ of ontic states in the case of inefficient detectors. We have represented the supports Sup${}_{{\psi }_k}$ of the probability distributions $p\left(\lambda \right|{\psi }_k)$ for three states $|{\psi }_1\rangle ,|{\psi }_2\rangle$, and $|{\psi }_3\rangle$. The subset of ontic states which never give rise to a click is denoted by ${\Lambda }_0$. The experiment described in the main text cannot rule out the existence of a nonempty intersection of the supports Sup${}_{{\psi }_k}$ and ${\Lambda }_0$, since all ontic states in this intersection always give rise to no-click events. The reported experiment can rule out the existence of a nonempty intersection of the supports Sup${}_{{\psi }_k}$ and the complementary space ${\Lambda }_{\mathrm{clk}}=\Lambda \setminus {\Lambda }_0$.

Figure 4

Experimental bound on $\psi$-epistemic models as a function of dimension $d$ of the quantum state space. The vertical axis gives the measured value of ${\epsilon }_{\mathrm{expt}}={\sum }_{k}N(k,Q_{{\alpha }_k})/{\sum }_{j}N(j,Q_{{\alpha }_k})$, where $N(j,Q_{{\alpha }_k})$ is the number of clicks registered in bin $j$ when one prepares state ${\alpha }_k$ (error bars are statistical). These values are also given in Table . The curve gives the dependency of ${\epsilon }_{\mathrm{expt}}$ on $d$ as predicted by quantum theory, taking into account the values of measured experimental parameters. The gray area gives the range in which this theoretical prediction could vary, given the uncertainty on dark-count rate, extinction ratio, and overall detection probability $\eta \langle n\rangle$. The main uncertainty comes from the dark-count rate which depends on the exact temperature of the detectors and the amount of ambient light, both of which can vary during the experiment. Positive deviation from the curve would signal a breakdown of quantum theory. The absence of such deviation rules out a large class of $\delta$-continuous models.

Figure 5

(a) Experimentally excluded region in the ${\delta }_0,\epsilon$ plane. For each choice of the parameter $q$ we compute the values ${\Delta }_0(d,q)$ and ${\epsilon }_{\mathrm{expt}}\left(d\right)/q$ which are then plotted in the figure, and connected by a line. For curves from top to bottom, $q=$ 0.09, 0.10, 0.11, 0.12, 0.13, 0.14, 0.15, 0.16, 0.17, 0.18, 0.19, 0.20, 0.22, 0.24, 0.26, 0.28, 0.30, 0.35, 0.40, 0.45, 0.50, 0.60, 0.70, 0.80, and 0.90. The couples $({\delta }_0,{\epsilon }_{\mathrm{expt}}\left(d\right))$ that would be obtained if one did not take into account the phase fluctuations of the laser are plotted in light gray (lowest curve). For $d=3,10$, taking into account the phase fluctuations of the laser does not modify significantly the results, while for $d=30,50,80$ the effect is important. In fact the excluded region for $d=80$ is practically the same as for $d=50$ when the phase fluctuations are taken into account. (b) Here we take only the points that are most constraining. We plot them with their error bars. The gray zone is the area in the ${\delta }_0,\epsilon$ plane that is excluded by our experiment.