Very strong evidence in favor of quantum mechanics and against local hidden variables from a Bayesian analysis

The data of four recent experiments—conducted in Delft, Vienna, Boulder, and Munich with the aim of refuting nonquantum hidden-variables alternatives to the quantum-mechanical description—are evaluated from a Bayesian perspective of what constitutes evidence in statistical data. We find that each of the experiments provides strong, or very strong, evidence in favor of quantum mechanics and against the nonquantum alternatives. This Bayesian analysis supplements the previous non-Bayesian ones, which refuted the alternatives on the basis of small $p$ values but could not support quantum mechanics.

Figure 1

Symbolic sketch of the eight-dimensional set of permissible probabilities. The points inside the ellipse symbolize probabilities accessible by quantum mechanics (QM); the triangle encloses the probabilities permitted by local hidden variables (LHV). There are no QM probabilities in the blue portion of the LHV set and no LHV probabilities in the red part of the QM set. The green overlap region contains the probabilities that are possible both for QM and LHV.

Figure 2

Upon receiving a trigger signal, the source of qubit pairs equips Alice and Bob with one qubit each; the success probability for this initial step is $\gamma$. The qubits are detected with the respective efficiencies ${\eta }_{\mathrm{A}}$ and ${\eta }_{\mathrm{B}}$ if they pass a selection process, specified by $\mathit{a}$ or ${\mathit{a}}^{\prime }$ for Alice's qubit and by $\mathit{b}$ or ${\mathit{b}}^{\prime }$ for Bob's qubit. For each trigger signal, we record whether there was a coincidence event, or only Alice or only Bob observed a detector click, or both did not. Data are collected for all four settings: $\mathit{a}$ or ${\mathit{a}}^{\prime }$ together with $\mathit{b}$ or ${\mathit{b}}^{\prime }$.

Figure 3

Boulder data: For $0.0005\le \gamma \le 0.0009$, the solid curve in black graphs the maximum value of $L\left(D\right|p)$ over the QM-permitted probabilities (inside the symbolic ellipse in Fig. 1); the dashed black line marks the maximum value of $L\left(D\right|p)$ over the LHV-permitted probabilities (inside the symbolic triangle in Fig. 1); and the solid curve in blue is the graph of the Bhattacharyya angle ${\varphi }_{\mathrm{B}}^{}$ between the $p_{\alpha \beta }^{\left(S\right)}\mathrm{s}$ of the target state and those of the QM-MLE.

Figure 4

Vienna data: (a) For $0.0025\le \gamma \le 0.0034$, the solid curves graph the maximum values of $L\left(D\right|p)$ over the QM-permitted probabilities (inside the symbolic ellipse in Fig. 1) for dataset 6 (purple), dataset 7 (orange), and dataset 8 (green); the dashed lines mark the corresponding maximum values of $L\left(D\right|p)$ over the LHV-permitted probabilities (inside the symbolic triangle in Fig. 1). The values for $\gamma =0.0035$ are far outside the range of the plot: $3.9\times {10}^{-1286}$ for dataset 6, $5.5\times {10}^{-1875}$ for dataset 7, and $2.8\times {10}^{-13060}$ for dataset 8; see also Table 11. The horizontal bars show the respective ranges of $\gamma$ values for which the probabilities of the QM-MLE violate a Bell inequality of the Eberhard kind; these ranges correspond to the grey strip in Fig. 3. (b) For the same $\gamma$ range, the curves graph the Bhattacharyya angle between the $p_{\alpha \beta }^{\left(S\right)}\mathrm{s}$ of the target state and those of the QM-MLE. The angle is smallest for $\gamma =0.00307$, $0.00296$, and $0.00274$ for dataset 6, dataset 7, and dataset 8, respectively.

Figure 5

Delft and Munich data: Probability-space triangles. The corners of the triangles indicate the sets of probabilities for the respective target state (circle), QM-MLE (square), and the relative frequencies observed (unmarked corner). The black triangles at the top are for runs 1, 2, and 1 and 2 combined of the Delft experiment (left to right) and the blue triangles below are for the two runs of the Munich experiment. The lengths of the sides of the triangles are the Bhattacharyya angles between the probabilities at the corners, on the indicated scale.