Does the Future Rewrite the Past? QTT Explains the Delayed-Choice Quantum Eraser

Reference: https://doi.org/10.5281/zenodo.17594186

The delayed-choice quantum eraser is one of those experiments that sounds like pure science fiction: you let a photon hit a screen, then later decide whether it behaved like a wave or a particle. People love to say the future is “changing the past”.

In this post I’ll walk through the experiment in two layers:

1. Plain-language, step-by-step story of the delayed-choice quantum eraser.
2. QTT-style math showing how the Access Law handles “which path” vs “interference” without any retrocausality.

Along the way I’ll label:

• ✅ = standard quantum physics (textbook / lab-confirmed),
• ⭐ / ⭐⭐ = Quantum Traction Theory (QTT) specific structure or interpretation.

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1. The delayed-choice quantum eraser in plain language

1.1 First: the ordinary double-slit (no eraser)

1. You fire very weak light at a screen with two slits.
2. Each photon hits a second screen (or detector) behind the slits.
3. If you let many photons build up, you see an interference pattern (bright and dark fringes), as if each photon “took both paths” like a wave. ✅
4. If instead you add a gadget that tells you which slit each photon went through (which-path info), the interference pattern disappears. ✅
   • You now get a two-blob pattern, one blob behind each slit, like little bullets.

So:

• Wave-like pattern ↔ You cannot know the path.
• Particle-like pattern ↔ You can know the path.

Quantum theory says: it’s not the physical device itself, it’s whether which-path information is available in principle. If the universe “could know” which slit it was, the interference disappears even if you personally don’t check.

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1.2 The quantum eraser idea (still no delay yet)

Now imagine we’re sneaky.

1. We let the photon go through both slits, but we attach a marker that tags the path, like a tiny colored sticker:
   • If it goes through slit A, the marker becomes “red”.
   • If it goes through slit B, the marker becomes “blue”.
2. We don’t look directly at the sticker yet, but the information exists in principle.
3. Because the paths are tagged differently, the two waves can no longer interfere:
   • The photon’s wave is now “red at A” + “blue at B”.
   • Those two tagged pieces do not add coherently; the interference cancels out. ✅

Now comes the eraser part:

4. Before we check the sticker color, we pass the marker through another gadget that mixes red and blue into new colors:
   • “purple = red + blue”
   • “green = red − blue”
5. If we now sort the detection events according to whether the marker ended purple or green, we find:
   • The “purple” subset of screen hits shows a fringes pattern.
   • The “green” subset shows anti-fringes (fringes shifted by half a period).
6. If we ignore the marker and look at all hits together, fringes + anti-fringes add to give no interference overall.

So the “eraser” is simply: we’ve destroyed the raw which-path information (red/blue) by mixing it into a new basis (purple/green). Path information is no longer available, and interference comes back—but only in those carefully sorted subsets.

Nothing spooky yet: everything is local and causal.

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1.3 The delayed-choice twist

The delayed-choice quantum eraser adds one more twist: timing.

The actual experiment uses entangled photon pairs:

• One photon is called the signal (S).
• The other is the idler (I).

Roughly, the steps are:

1. A laser hits a nonlinear crystal and produces entangled photon pairs. ✅
2. The signal photon goes toward a screen (detector D0) through an effective double-slit arrangement.
3. The idler photon takes a longer path through mirrors and beam splitters to one of four detectors, usually labeled D1, D2, D3, D4:
   • D3 and D4: give which-path info (“it was slit A” or “slit B”).
   • D1 and D2: are tuned so the which-path info is erased and you only get interference-type information.

The key timing trick:

• The signal photon hits the screen first, and you register its position.
• Only later does its idler twin hit one of D1–D4, which decides whether the path info is readable (D3/D4) or erased (D1/D2).

What you actually see:

1. If you look at the raw pattern at D0 (all signal hits, ignoring idlers), there is no interference pattern at all—just a broad blob. ✅
2. Later, you take your data and do coincidence sorting:
   • Take all signal hits whose idler partner went to D3 or D4 (which-path detectors) → no interference pattern.
   • Take signal hits whose idler went to D1 → interference fringes.
   • Take signal hits whose idler went to D2 → anti-fringes.
3. Adding all these subsets together gives back the original non-interference blob.

This leads to the naive paradox:

“But the idler measurement happened later! Did we decide in the future whether the signal had interfered in the past?”

Standard quantum mechanics answer: ✅

• No retrocausality.
• The joint state of the pair is fixed at creation.
• The “choice” of idler measurement just determines how you slice the correlations.
• Only the conditional subsets (sorted later) show the patterns; the raw signal data never retro-changes.

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1.4 How QTT tells the story (conceptual)

Quantum Traction Theory (QTT) keeps those predictions but adds a richer picture built around:

• A time plane (absolute time \(T\) plus a Reality direction \(w\)). ⭐⭐
• A universal clock tilt (Time Tilt) between lab time and absolute time. ⭐⭐
• An Access Law that says: “What you can actually access as information changes how quantum fuzz is allowed to show up.”

In QTT terms:

1. The signal + idler pair is created as a single, reality-linked object in the time plane.
2. The experimental choice—“will the idler end up in a which-path channel (D3/D4), or in an eraser channel (D1/D2)?”—is really a choice of which observable becomes accessible for that pair:
   • Path-accessible mode: the setup makes which-path information physically extractable.
   • Path-erased mode: the setup makes only a complementary “phase” property accessible, and which-path info becomes physically inaccessible even in principle.
3. QTT’s Access Law says: ⭐⭐
   • The more sharply you can access “which path?”, the more you weaken the system’s ability to show “interference-type” behavior.
   • The interplay is encoded in modified commutators and uncertainty bounds (more on that below).

So:

• D3/D4 branch: the path observable has been made high-access → interference is killed, and the signal photon behaves as if it came from one slit or the other.
• D1/D2 branch: the setup destroys path information and replaces it with a complementary phase tag → path access is effectively low, so interference can reappear in the conditioned subensemble.

And the delayed choice?

• In QTT, everything lives on the absolute time axis \(T\) with a fixed ordering of events.
• The “delay” is about when a particular lab clock records its data, not about the fundamental ordering in \(T\).
• The Access structure just requires that the joint outcomes are consistent across the time plane—it never allows you to send information backwards in \(T\), and it never changes recorded outcomes at D0.

So QTT agrees with standard QM on the actual predictions (✅), but gives a more mechanical language:

“The idler choice doesn’t change the past; it only changes which kind of information the reality-ledger allows to be extracted about that pair, and that in turn controls whether the signal subensemble is allowed to show interference.”

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2. QTT-style math for the quantum eraser

Now let’s do the usual math for the DCQE, then overlay the QTT Access Law.

2.1 Standard state structure (paths + markers) ✅

Label:

• Signal photon path states: \(|A\rangle_s\) and \(|B\rangle_s\) (slit A vs B).
• Idler photon marker states: \(|1\rangle_i\) and \(|2\rangle_i\) (correlated with A vs B).

A typical entangled state just after the crystal and path tagging is

[
|\Psi\rangle
= \frac{1}{\sqrt{2}}
\Bigl(
|A\rangle_s \otimes |1\rangle_i
+
|B\rangle_s \otimes |2\rangle_i
\Bigr).
\tag{1}
]

Case 1: no eraser, which-path read out.

If we measure the idler directly in the \(\{|1\rangle_i,|2\rangle_i\}\) basis, we get:

• Outcome \(|1\rangle_i\) ⇒ signal collapses to \(|A\rangle_s\).
• Outcome \(|2\rangle_i\) ⇒ signal collapses to \(|B\rangle_s\).

If you don’t condition on the idler outcome, the signal’s reduced density matrix is

[
\rho_s
= \mathrm{Tr}_i\bigl(|\Psi\rangle\langle\Psi|\bigr)
= \frac{1}{2}\Bigl(|A\rangle\langle A| + |B\rangle\langle B|\Bigr).
\tag{2}
]

There are no off-diagonal terms like \(|A\rangle\langle B|\), so no interference when you compute the intensity on the screen: you just get the sum of two single-slit patterns. ✅

Case 2: eraser basis on the idler.

Define the eraser basis:

[
|+\rangle_i = \frac{1}{\sqrt{2}}\bigl(|1\rangle_i + |2\rangle_i\bigr),
\qquad
|-\rangle_i = \frac{1}{\sqrt{2}}\bigl(|1\rangle_i – |2\rangle_i\bigr).
\tag{3}
]

Rewrite \(|\Psi\rangle\) in this basis:

[
|\Psi\rangle
= \frac{1}{2}
\Bigl[
(|A\rangle_s + |B\rangle_s)\otimes |+\rangle_i
+
(|A\rangle_s – |B\rangle_s)\otimes |-\rangle_i
\Bigr].
\tag{4}
]

So:

• Conditional on idler outcome \(|+\rangle_i\), the signal is in \[ |\psi_+\rangle_s = \frac{1}{\sqrt{2}}\bigl(|A\rangle_s + |B\rangle_s\bigr), \tag{5} \] a coherent superposition that gives interference fringes.
• Conditional on idler outcome \(|-\rangle_i\), the signal is in \[ |\psi_-\rangle_s = \frac{1}{\sqrt{2}}\bigl(|A\rangle_s – |B\rangle_s\bigr), \tag{6} \] which produces complementary “anti-fringes”.

If you keep both \(|+\rangle_i\) and \(|-\rangle_i\) together (i.e. ignore the idler), the fringes and anti-fringes wash out, reproducing the mixture \(\rho_s\) in (2). Everything up to here is standard quantum mechanics. ✅

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2.2 Adding the screen: wavefunctions at position x ✅

Let \(\psi_A(x)\) be the wavefunction at screen position \(x\) for a photon that went through slit A, and \(\psi_B(x)\) for slit B.

• For the mixed state \(\rho_s\) in (2), the intensity is \[ I_{\rm mix}(x) = \frac{1}{2}\Bigl(|\psi_A(x)|^2 + |\psi_B(x)|^2\Bigr). \tag{7} \] No interference.
• For the coherent state \(|\psi_+\rangle\) in (5), \[ \psi_+(x) = \frac{1}{\sqrt{2}}\bigl(\psi_A(x)+\psi_B(x)\bigr), \] so the intensity is \[ I_+(x) = |\psi_+(x)|^2 = \frac{1}{2}\Bigl(|\psi_A|^2 + |\psi_B|^2 + \psi_A^\star\psi_B + \psi_B^\star\psi_A\Bigr). \tag{8} \] The cross terms give the interference fringes.
• For \(|\psi_-\rangle\) in (6), \[ \psi_-(x) = \frac{1}{\sqrt{2}}\bigl(\psi_A(x)-\psi_B(x)\bigr), \] so \[ I_-(x) = |\psi_-(x)|^2 = \frac{1}{2}\Bigl(|\psi_A|^2 + |\psi_B|^2 – \psi_A^\star\psi_B – \psi_B^\star\psi_A\Bigr). \tag{9} \] Same fringes but with opposite phase → anti-fringes.

Summing \(I_+(x)+I_-(x)\) kills the interference terms, bringing us back to (7).

✅ Conclusion:

• Which-path info available → mixture (2) → no interference.
• Which-path info erased in a complementary basis → coherent superpositions (5–6) → conditional interference.

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2.3 QTT’s Access Law for measurement ⭐⭐

QTT modifies the usual canonical commutator to encode information access:

[
[X,P] = J\,\hbar\,(I – M),
\tag{10}
]

where:

• \(X\), \(P\) are position and momentum operators,
• \(J\) is the “reality-dimension” unit with \(J^2 = -1\) (plays the role of \(i\), but tied to the hidden \(w\)-direction),
• \(M\) is an operator that represents which degrees of freedom are accessible to measurement (the “Access operator”).

From this you get a modified uncertainty relation:

[
\Delta X\,\Delta P \;\ge\; \frac{\hbar}{2}\,(1-\eta),
\qquad
\eta := \mathrm{Tr}(\rho M),
\tag{11}
]

where \(\eta \in [0,1]\) is the accessible fraction in the state \(\rho\).

• If \(\eta=0\) (no access to that observable), you recover the usual \(\Delta X\,\Delta P \ge \hbar/2\).
• If \(\eta=1\) (full access), the lower bound formally goes to 0 in that degree of freedom: the state behaves more classically in that channel.

In the DCQE, the relevant observable is not literally “position on the screen” but which path vs relative phase between paths. In QTT language:

• Define the path projectors \[ P_A = |A\rangle\langle A|, \qquad P_B = |B\rangle\langle B|. \]
• The which-path Access operator can be modeled as \[ M_{\rm path} = P_A \otimes |1\rangle\langle 1| + P_B \otimes |2\rangle\langle 2|. \tag{12} \]

Then:

• Which-path setup (idler in \(|1\rangle,|2\rangle\) basis):
  • The measurement basis commutes with \(M_{\rm path}\).
  • For the post-selected pairs, \(\eta_{\rm path} = \mathrm{Tr}(\rho M_{\rm path})\) is close to 1.
  • QTT says: in that subensemble, the path is high-access → the conjugate “phase between paths” is squeezed out in the sense of (11); the cross-terms in (8–9) cannot manifest on the screen → you get the mixed pattern (7).
• Eraser setup (idler in \(|\pm\rangle\) basis):
  • The measurement projects onto states that are superpositions of \(|1\rangle\) and \(|2\rangle\).
  • Those are not eigenstates of \(M_{\rm path}\); in the conditioned subensemble, the expectation \(\eta_{\rm path}\) is effectively driven down toward 0.
  • Path info is genuinely inaccessible (you can’t reconstruct slit A vs B from a \(|+\rangle\) or \(|-\rangle\) click).
  • QTT says: low path-access → the conjugate phase degree of freedom is free to fluctuate → you get the full interference terms in (8–9).

So the Access Law is a compact way to say:

Interference is not “mystically destroyed” or “restored”. It is permitted or forbidden depending on whether the physical setup makes the path information high-access or low-access.

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2.4 Where the “delayed choice” sits in the time plane ⭐⭐

QTT has a time plane with:

• Absolute time \(T\),
• Reality dimension \(w\),
• Lab time \(t\) as a tilted axis in that plane.

The total time mapping (ignoring cosmological drift for lab experiments) is

[
dt_{\rm lab} = I_{\rm clk}\,N(x,v)\,dT,
\tag{13}
]

with \(I_{\rm clk} = \cos(\pi/8)\) the universal tilt factor and \(N(x,v)\) the usual GR/SR dilation factor.

For the DCQE on a tabletop:

• \(N(x,v)\approx 1\) (no big gravitational/velocity effects).
• The distances are so small that any differences between the absolute ordering in \(T\) and the lab ordering in \(t\) are negligible.

The “delayed” part is just:

• In lab time \(t\), the screen detection at D0 happens at \(t_0\).
• The idler detection at D1–D4 happens later at \(t_1 > t_0\).
• On the absolute clock \(T\), they both lie along the same fixed order of events; there is no “going back” in \(T\).

The Access Law operates on the joint state across the time plane. It enforces that:

• The joint correlations between S and I are consistent with the Access structure defined by the actual optical elements.
• You can’t use the DCQE to send a signal to your past along \(T\), because:
  • The D0 pattern alone carries no interference signature.
  • The interference only appears after post-selecting on the idler outcomes, which you can’t know before the idler is measured.

So QTT’s verdict on the “paradox” is:

• ✅ It fully agrees with standard QM that there’s no retrocausality.
• ⭐⭐ It adds that the apparent weirdness is a symptom of ignoring the Access operator and the time plane: once you track which observable is accessible for each subensemble, the story becomes: “Future choices at D1–D4 don’t change the past. They only decide which part of the reality-ledger you’re allowed to read out—and that in turn decides whether you see interference in the conditioned subensemble.”

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2.5 Summary in one sentence

• Layman version:
  The delayed-choice quantum eraser looks spooky because we sort data after the fact, but nothing ever changes the past; it’s just correlations plus clever bookkeeping.
• QTT version:
  The experiment is a clean demonstration of the Access Law: whenever the setup lets you access which-path information, interference is suppressed; whenever that information is erased in a complementary basis, interference is allowed to show up in the conditioned subset—no retrocausality, just a reality-covariant ledger in the time plane telling you which quantum fuzz is allowed where.