The Twin-World road to reality in quantum mechanics

Abstract

I introduce a novel realistic, stochastic approach to quantum mechanics by extending the recently proposed grabit formalism \cite{braun_stochastic_2022} to two Twin Worlds. According to the picture developed, we live at the intersection of two worlds with identical stochastic laws of evolution. Our World is limited to that intersection, and only coincidence events from the two Twin Worlds, post-selected automatically by our restriction to the intersection, have physical reality in Our World. This fully reproduces standard non-relativistic quantum mechanics, including Born's rule and the violation of Bell's inequality. I derive the stochastic evolution equation in each Twin World that fully reproduces Schrödinger's equation for an arbitrary number of particles with arbitrary interactions, and demonstrate that hall-mark quantum effects such as tunnling are correctly reproduced.

Figures (9)

• Figure 1: Quantum circuit for the creation of a single-qubit quantum superposition $|\psi(\varphi)\rangle=(|0\rangle+e^{i\varphi}|1\rangle)/\sqrt{2}$ in the second qubit and the subsequent measurement of $X$ with the help of the first (ancilla) qubit measured in the computational basis.
• Figure 2: As FIG. \ref{['fig:qcSingQubit']} but for the realified quantum state $|\Phi\rangle$. The third qubit works as ReIm-qubit, i.e. codes the real or imaginary part of the state. The complex $U(\varphi)$ gate on the second qubit becomes a controlled rotation gate $R(\varphi)$ on the ReIm qubit.
• Figure 3: Two identical grabit quantum circuits in the two Twin Worlds (${\text{I}},{\text{II}}$) corresponding to FIG. \ref{['fig:qcSingQubitReal']}. In each Twin World, the realified quantum state $|\Phi\rangle$ is encoded in three gradient bits ("grabits"), each realized with two classical stochastic bits, initialized in grabit state $|0\rangle\rangle$ (the double kets denote grabit states). Unitary gates become stochastic maps acting on those classical bits, and the complex single-qubit $U(\varphi)$ becomes a controlled real rotation gate on the ReIm grabit. The refresh gates $\mathcal{R}$ make the final state interference free and assures that the physical probability distributions $\tilde{p}^{\text{I}}_{i}$ and $\tilde{p}^{\text{II}}_{i}$ over the outcomes $i=0,\ldots,7$ satisfy $\tilde{p}^{\text{I}}_{i}=\tilde{p}^{\text{II}}_{i}=|\Phi_i|/||\Phi||_1$, where $\Phi$ is the true, realified, quantum mechanical state. Post-selected coincident outcomes of the blvs from the two Twin Worlds marginalized over the second and third grabit define the observed outcomes and their statistics in Our World, and satisfy the standard Born-2 rule, $\accentset{\approx}{p}_{i_1}=\sum_{i_2,i_3=0,1}{\accentset{\approx}{p}}_{i_1i_2i_3}=\sum_{i_2,i_3=0,1}\tilde{p}^{\text{I}}_{i_1i_2i_3}\cdot\tilde{p}^{\text{II}}_{i_1i_2i_3}=\sum_{i_2,i_3=0,1}\Phi_{i_1i_2i_3}^2=\sum_{i_2}|\Psi_{i_1i_2}|^2$, where $i_3$ labels the states of the ReIm grabit.
• Figure 4: Rotation of the Bloch vector of a qubit in the $xy$-plane (probability of finding 0 in the computational basis measurement of the first qubit, $\accentset{\approx}{p}_0(\varphi)=(1+\langle X(\varphi)\rangle)/2=(1+\cos\varphi)/2$ in the state $|\Psi(\varphi)\rangle=(|0\rangle+e^{i\varphi}|1\rangle)/\sqrt{2}$) calculated from the stochastic emulation with grabits in the two Twin Worlds and coincidence detection of the measurement outcomes 0 and 1. Blue continuous line: quantum mechanical prediction. Orange dots: results from the stochastic emulation (see FIG. \ref{['fig:qcSingGrabit']}), ${N_\text{smpl}}=100\,000$
• Figure 5: Quantum circuit for the realization of the CHSH experiment. The first four gates prepare a singlet state between qubits 2,4, on which $Q(\theta_1)$ and $Q(\theta_2)$ are then measured via a measurement of the ancilla qubits 1,3 in the computational basis, respectively. Four different combinations of $\theta_1,\theta_2$ are randomly chosen in a large number of runs, and empirical averages computed that estimate the expectation values in \ref{['eq:CHSH']}. The grabit emulation of the quantum circuit is shown in FIG.\ref{['fig:CHSHgrabit']} and the results are in FIG.\ref{['fig.chshResult']}.

Theorems & Definitions (3)

• Definition 5.1
• Lemma 7.1
• proof