Quantum probabilistic sampling of multipartite 60-qubit Bell-inequality violations

We show that violation of genuine multipartite Bell inequalities can be obtained with sampled, probabilistic phase-space methods. These genuine Bell violations cannot be replicated if any part of the system is described by a local hidden variable theory. The Bell violations are simulated probabilistically using quantum phase-space representations. We treat mesoscopically large Greenberger-Horne-Zeilinger (GHZ) states having up to 60 qubits, using both a multipartite $\text{SU(2)-}Q$ representation and the positive-$P$ representation. Surprisingly, we find that sampling with phase-space distributions can be exponentially faster than experiment. This is due to the classical parallelism inherent in the simulation of quantum measurements using phase-space methods. Our probabilistic sampling method predicts a contradiction with local realism of “Schrödinger-cat” states that can be realized as a GHZ spin state, either in ion traps or with photonic qubits. We also present a quantum simulation of the observed superdecoherence of the ion-trap “cat” state, using a phenomenological noise model.

Figure 1

Correlations for the different parts of the quantity (43) in the positive-$P$ representation, with the number-state method and $2^{26}$ samples.

Figure 2

Correlations for the different parts of the quantity (61) in the SU(2)-$Q$ representation, $2^{26}$ samples.

Figure 3

Scaling properties for sampled correlations of multiparticle GHZ states. Relative errors are plotted for high-order ($V$) correlations (blue solid line) and first-order correlations, or total number of “spin-ups” (green dash-dotted line) using the SU(2)-$Q$ representation with $2^{40}$ samples. The dotted reference line shows the point at which the sampling errors would give scaling properties of an experimental measurement. The red dashed line shows the scaling of the $V$ correlations using the less efficient positive-$P$ representation.

Figure 4

Violations for multiparticle GHZ states. Simulated Mermin violation using the SU(2)-$Q$ representation with $2^{43}$ samples. The values of expectations and errors are normalized by the quantum mechanical prediction for the corresponding $M$. The horizontal grey dashed line gives the quantum prediction. The error bars show the sampled result and estimated sampling errors at each value of $M$. The dash-dotted line is the LHV prediction, which gives a Bell violation when above this line. Genuine multipartite Bell violations occur for even $M$ when $V/V_{\mathrm{QM}}>1/\sqrt{2}$.

Figure 5

Decay of the sampled quantity $V$ using the model of superdecoherence of Eq. (67), for two to six particles, with decoherence rate $\epsilon =0.1$. The horizontal axis is the dimensionless time $\tau =t/\Delta t$.