20250418_Quantum Eraser.md

• PDF
• YouTube video

Here’s the extracted OCR text from your PDF, “The Quantum Eraser”:

---

The Quantum Eraser

1. Initial Superposition State

We begin with a photon in a superposition of paths through a double slit:

$$|\psi\rangle = \frac{1}{\sqrt{2}}|\text{left}\rangle + \frac{1}{\sqrt{2}}|\text{right}\rangle$$

The probability of detecting the photon at position ( x ) on the screen:

$$P(x) = |\langle x | \psi \rangle|^2 = |\langle x | \text{left} \rangle + \langle x | \text{right} \rangle|^2$$

Constructive or destructive interference depends on the relative amplitudes and phases of the two components.

---

2. Marking the Path (Which-way Information)

We now add polarization filters to mark the path:

• Vertical polarization for the left slit: ( |V\rangle|\text{left}\rangle )
• Horizontal polarization for the right slit: ( |H\rangle|\text{right}\rangle )

So the new state becomes:

$$|\psi_{\text{marked}}\rangle = |V\rangle|\text{left}\rangle + |H\rangle|\text{right}\rangle$$

Since ( \langle V | H \rangle = 0 ), the two terms are orthogonal, and no interference occurs — the which-path information is now encoded in the polarization.

---

3. Quantum Eraser

We now add a diagonal polarizer (at ( 45^\circ )) which projects both ( |V\rangle ) and ( |H\rangle ) into the same state:

$$|D\rangle = \frac{1}{\sqrt{2}}(|V\rangle + |H\rangle)$$

Applying this projection:

$$|\psi_{\text{erased}}\rangle = |\text{↗}\rangle \left( \frac{1}{\sqrt{2}}|\text{left}\rangle + \frac{1}{\sqrt{2}}|\text{right}\rangle \right) = |\text{↗}\rangle \cdot \frac{1}{\sqrt{2}}(|\text{left}\rangle + |\text{right}\rangle)$$

Now the which-path information is erased, and interference is restored:

$$P(x) = \left| \langle x | \text{left} \rangle + \langle x | \text{right} \rangle \right|^2$$

---

Conclusion

The 45° polarizer erases the which-path information encoded in polarization. This restores coherence between the two paths and brings back the double-slit interference pattern.

$$45^\circ \text{ arrow} \; \rightarrow \text{↗}$$