How Does Pilot-Wave Theory Explain a Quantum Eraser?

A quantum eraser combines a two-path interference experiment with a detector that can record which path the particle took. The apparent puzzle is that interference can be recovered by measuring that detector in a different way, sometimes after the particle has reached the screen. Pilot-wave theory, also called Bohmian mechanics, describes this without changing earlier detection records or sending information into the past.

The key is to calculate the joint state of the particle and marker. A simulation must likewise include the marker, rather than treating the particle as a classical point that simply picks a slit and ignores the rest of the apparatus.

How the marker changes interference

Let $\psi_1(x)$ and $\psi_2(x)$ be the particle’s amplitudes at screen position $x$ for the two paths. Let $|m_1\rangle$ and $|m_2\rangle$ be the marker states associated with those paths. For equal path amplitudes, the joint state is

$$|\Psi\rangle=\frac{1}{\sqrt{2}}\bigl(|\psi_1\rangle|m_1\rangle+|\psi_2\rangle|m_2\rangle\bigr).$$

If you do not read the marker, the particle’s screen distribution is

$$p(x)=\frac{1}{2}\bigl(|\psi_1(x)|^2+|\psi_2(x)|^2\bigr)+\operatorname{Re}\!\left[\psi_1^*(x)\psi_2(x)\langle m_1|m_2\rangle\right].$$

The overlap $\langle m_1|m_2\rangle$ determines how much interference remains. Identical marker states have overlap 1 and preserve full interference. Orthogonal states have overlap 0: in principle, they carry perfectly distinguishable which-way information, and the unsorted screen pattern has no cross term. Partial overlap gives partial visibility.

For two paths with equal screen intensity and relative phase $\phi(x)$, the fringe visibility is $|\langle m_1|m_2\rangle|$. For example, overlap 0.6 gives a visibility of 0.6. In an ideal sinusoidal pattern, the bright-to-dark intensity ratio is $(1+0.6)/(1-0.6)=4$.

What erasing the marker does

Suppose the marker states are orthogonal, but you measure the marker in the basis

$$|m_\pm\rangle=\frac{|m_1\rangle\pm|m_2\rangle}{\sqrt{2}}.$$

Sorting detections by the two marker outcomes gives particle amplitudes proportional to $\psi_1+\psi_2$ and $\psi_1-\psi_2$. Each selected subset can therefore show interference. The minus outcome has fringes shifted relative to the plus outcome: its bright regions line up with the other’s dark regions.

When you combine both subsets, the cross terms cancel. The total, unsorted distribution still has no fringes. The eraser does not restore interference in the full dataset; it reveals correlations between the marker outcome and screen position. A later choice of marker measurement changes which correlations you inspect, not the particle detections already recorded.

The pilot-wave account

In Bohmian mechanics, the wavefunction evolves according to the quantum dynamics, while the particle has a definite configuration guided by that wavefunction. For a spinless particle with no vector potential, the guidance law is

$$\frac{d\mathbf{x}}{dt}=\frac{\hbar}{m}\operatorname{Im}\left(\frac{\nabla\Psi}{\Psi}\right).$$

With an entangled marker, the relevant wavefunction is over the full configuration, such as the particle position and the detector’s degrees of freedom. The particle’s velocity depends on the full configuration, not on a separate wave assigned to its chosen slit. In models with internal marker states, the corresponding Bohmian guidance law uses the appropriate spinor or internal-state probability current.

After the marker becomes correlated with the paths, the two wave branches can guide the particle differently. A marker measurement in a superposition basis involves further interactions that recombine or compare those branches. Bohmian trajectories respond to that actual interaction. They do not need to travel backward in time, and the theory does not predict a visible unsorted fringe pattern when the marker states are perfectly distinguishable.

A practical simulation plan

A useful pilot-wave simulation needs both wave evolution and particle trajectories:

1. Choose a two-slit wavefunction and a model for the marker, such as two orthogonal detector states.
2. Evolve the joint state through the marker interaction and onward to the screen.
3. Sample initial configurations from the quantum-equilibrium distribution, $|\Psi|^2$.
4. Integrate the guidance equation for the particle and the modeled detector variables.
5. Record screen positions and marker outcomes, then plot both the combined data and the outcome-sorted subsets.

The expected result is a fringe-free combined histogram for orthogonal which-way states, with fringes and shifted fringes in the two eraser-selected subsets. A wave-only calculation can predict the detection probabilities; adding the guidance equation produces Bohmian trajectories as well. The trajectories depend on the modeled apparatus, so a plot of paths is not by itself a universal picture of what happens in every quantum-eraser experiment.

Common mistakes

• Looking only at a postselected subset: A fringe pattern in one subset does not mean the complete screen data show fringes.
• Treating the marker as irrelevant: Once it is entangled with the particle, the joint state determines the probabilities and, in Bohmian mechanics, the guidance.
• Calling the eraser retrocausal: Later sorting cannot alter the earlier screen record. It separates recorded events according to their correlations with marker outcomes.
• Assuming trajectories are observable tracks: Bohmian trajectories are part of the theory’s model. Experiments test its statistical predictions, not a direct photograph of an individual trajectory.