Detection-efficiency loophole and the Pusey-Barrett-Rudolph theorem

The detection-efficiency loophole poses a significant problem for experimental tests of Bell inequalities. The recently discovered Pusey-Barrett-Rudolph (PBR) theorem suffers from the same vulnerability. In this paper we calculate the critical detection efficiency, below which the PBR argument for the ontic nature of quantum state is inconclusive. This is done for the maximally $\psi$-epistemic models. We use two different definitions of this property. The optimal number of parties for which the critical detection efficiency is the lowest is given. We also approach the problem from the opposite direction. We provide a function that enables us to specify which epistemic models are ruled out by the results of an experiment with a given detection efficiency.

Figure 1

The PBR argument is based on a joint measurement in a quantum circuit of n qubits followed by a measurement on each qubit in the computational basis. The single qubit gates are defined as $P_{\varphi }=|0\rangle \langle 0|+e^{i\varphi }|1\rangle \langle 1|$ and the Hadamard gate, $|+\rangle \langle 0|+|-\rangle \langle 1|$. The entangling gate in the middle rotates only $|00\cdots 00\rangle ,R_{\xi }|00\cdots 00\rangle =e^{i\xi }|00\cdots 00\rangle .$ Each of the outputs is observed by a single detector station.

Figure 2

A schematic depiction of a maximally $\psi$-epistemic model. Different probability distributions are plotted on the hidden variable space $\lambda$. Supports of ${\rho }_{{\psi }_2}$ and ${\rho }_{{\psi }_2^{\perp }}$ do not overlap due to orthogonality of $|{\psi }_2\rangle$ and $|{\psi }_2^{\perp }\rangle$. $R_{{\psi }_1{\psi }_2}$ is the overlap region of ${\rho }_{{\psi }_2}$ and ${\rho }_{{\psi }_1}.$ In this class of models every $\lambda$ from the support of ${\rho }_{{\psi }_1}$ belongs either to the support of ${\rho }_{{\psi }_2}$ or ${\rho }_{{\psi }_2^{\perp }}$.

Figure 3

Dependence of critical detection efficiency $\eta$ as a function of $\theta$ measured in radians for different values of $n^{*}\in \{2,3,...,7\}$ for tests of maximally ${\psi }_{\Omega }$-epistemic models.

Figure 4

Dependence of critical detection efficiency $\eta$ as a function of $\theta$ measured in radians for different values of $n^{*}\in \{2,3,...,7\}$ for tests of maximally ${\psi }_k$-epistemic models.

Figure 5

Critical value of $\Omega$ as a function of detection efficiency $\eta$. For this plot optimal values of the parameters $n^{*}$ and ${\theta }_{\text{min}}$ have been used, ie. $n^{*}=4$ and ${\theta }_{\text{min}}=0.374$.

Figure 6

Critical value of $k$ as a function of detection efficiency $\eta$. For this plot optimal values of the parameters $n^{*}$ and ${\theta }_{\text{min}}$ have been used, ie. $n^{*}=3$ and ${\theta }_{\text{min}}=0.509$.