8. Delayed Choice and Quantum Eraser: How Later Choices Determine Present Statistics

One of the core principles of quantum mechanics, Complementarity, states that we cannot simultaneously know the particle-like properties (e.g., which path was taken?) and wave-like properties (e.g., interference pattern between paths) of a system perfectly. Delayed Choice and Quantum Eraser experiments are thought experiments and actual experiments that dramatically illustrate this principle. These experiments show that even if we perform a measurement to determine whether it is “wave-like” or “particle-like” much later than the point at which we would expect the object (e.g., a photon) to have already “completed” its choice, the results appear as if we had set that configuration from the beginning.

1. Fundamental Concepts

Wheeler’s Delayed Choice: An experiment where, after a photon has already entered an interferometer (e.g., Mach-Zehnder), we decide whether to observe an interference pattern (wave-like property) or determine the path (particle-like property).
Detailed Explanation: The Analogy of a Traveler at a Crossroads ✈️

Imagine a traveler (photon) departing from an airport (BS1) and boarding a flight to one of two routes (Path 1, Path 2).
1. Path Measurement (Particle): If we separate the arrival airports into two (D1, D2), the traveler arrives at either D1 or D2. We can 100% know whether they came via Path 1 or Path 2.
2. Interference Measurement (Wave): If we merge the two paths into a single “converging terminal” (BS2) just before arrival, travelers from both paths interfere with each other. Depending on the weather (phase difference), all travelers might end up going only to D1 or only to D2 (interference pattern).
Key Insight of Delayed Choice: Even if we decide whether to install this “converging terminal (BS2)” after the traveler is already in flight (after passing through BS1), the result remains the same. If we install the terminal (later choice), an interference pattern appears; if we remove it (later choice), path statistics emerge. This suggests that the photon does not pre-commit to being a “particle” or a “wave.”
Quantum Eraser: A technique where we intentionally erase interference patterns by attaching a “marker” that reveals the particle’s path, and later selectively restore the interference pattern by “erasing” that marker information through measurement.
Detailed Explanation: Double Slit and Polarizing Glasses 🕶️
1. Interference Erasure: In the double-slit experiment, place a vertical polarizer (\(|V\rangle\)) at slit 1 and a horizontal polarizer (\(|H\rangle\)) at slit 2. Now, by simply observing the polarization of a photon arriving at the screen, we can 100% determine which slit it passed through (\(|V\rangle \perp |H\rangle\)). In this way, when path information (particle-like property) is perfectly known, the interference pattern (wave-like property) completely disappears due to the principle of complementarity.
2. Information Erasure: Place a 45-degree polarizer (\(|D\rangle\)) in front of the screen. Photons passing through this polarizer become 45-degree polarized regardless of whether they came from \(|V\rangle\) or \(|H\rangle\). Thus, the original path information (\(|V\rangle\) or \(|H\rangle\)) is erased.
3. Interference Restoration: Remarkably, when we collect only the photons that passed through the 45-degree polarizer, their distribution on the screen again shows an interference pattern! Photons that did not pass through the 45-degree polarizer (i.e., those that passed through the 135-degree polarizer) show a different interference pattern (with the opposite phase). Complementarity and Information**: The clarity (visibility, \(V\)) of the interference pattern and our ability to distinguish the path (distinguishability, \(D\)) are related by the complementary relation \(V^2 + D^2 \le 1\). The quantum eraser demonstrates that in the \(D=1, V=0\) state, by performing an ‘erasure’ measurement, it selects a subset where \(D < 1\), thereby showing that \(V > 0\).

2. Symbols and Key Relations

Interference Amplitude: The amplitude of a system passing through two paths \(\psi_1, \psi_2\) is \(\mathcal{A} = \psi_1 + \psi_2\). The detection probability (intensity) is \(I \propto |\psi_1 + \psi_2|^2 = |\psi_1|^2 + |\psi_2|^2 + 2\text{Re}(\psi_1^* \psi_2)\).
Which-Path Marking: If the path information is stored in a marker state \(|m_p\rangle\), the total state becomes an entangled state: \(|\Psi\rangle = \psi_1(x) |m_1\rangle + \psi_2(x) |m_2\rangle\) In this case, the detection probability (intensity) ignoring the marker at the screen \(x\) is: \(I(x) \propto |\psi_1(x)|^2 + |\psi_2(x)|^2 + 2\text{Re}(\langle m_1|m_2\rangle \psi_1^*(x)\psi_2(x))\)
- \(\mu = \langle m_1|m_2\rangle\) is the inner product (overlap) of the marker states.
- If the marker is perfectly distinguishable (\(|m_1\rangle \perp |m_2\rangle\)), \(\mu = 0\), and the interference term (\(2\text{Re}(\dots)\)) disappears.
Quantum Erasure (Conditional Erasure): When the marker state is measured (projected) into the ‘erasure basis’ \(|e\rangle\), the conditional probability of finding the system at \(x\) is: \(P(x | e) = |\langle x| \langle e | \Psi \rangle|^2 = |\psi_1(x)\langle e|m_1\rangle + \psi_2(x)\langle e|m_2\rangle|^2\)
- If the erasure state \(|e\rangle\) is chosen such that \(\langle e|m_1\rangle = \langle e|m_2\rangle \neq 0\) (e.g., \(|e\rangle = \frac{1}{\sqrt{2}}(|m_1\rangle + |m_2\rangle)\)), the interference term is perfectly restored into the form \(\psi_1^*\psi_2\).
No Retrocausality:
💡 Does a later choice change the past? (No!)

It is easy to misunderstand delayed-choice or quantum erasure as implying that a future measurement changes a past event (whether the photon was a particle or a wave). This is not true.
- Overall statistics are invariant: Whether or not we use an ‘erasure’ measurement, the total distribution of photons accumulated on the screen is always the one without interference fringes (the state with path information).
- Conditional selection: The ‘erasure’ measurement is a post-selection process that selects only the data among the photons that have already reached the screen, satisfying a specific condition (e.g., passing through a 45° polarizer).
In other words, a later choice does not change the past physical reality, but rather determines which statistical subset we observe. It is like an encrypted message that remains unchanged, but only a specific key (the erasure measurement) allows us to decode it.

3. Examples with Deeper Insight

Example A1: Mach-Zehnder Interferometer
- BS2 Removed (Particle Path): The photons that pass through the first beam splitter (BS1) definitely follow path 1 or 2. Path 1 goes to detector D1, and path 2 goes to detector D2. Even when changing \(\Delta\phi\), the probabilities for D1 and D2 remain constant at 50:50. \(\to\) Path Determination (Which-Path).
- BS2 Inserted (Wave Interference): The second beam splitter (BS2) recombines the two paths. Depending on the phase difference \(\Delta\phi\) between the two paths, interference fringes appear at D1 as \(I \propto \cos^2(\Delta\phi/2)\) (constructive/destructive interference), and at D2 as \(I \propto \sin^2(\Delta\phi/2)\). \(\to\) Interference (Wave-like).
Example A2: Double Slit and Polarization Tagging
- Tagging (Interference Suppression): Slit 1 \(\to |H\rangle\) polarization, Slit 2 \(\to |V\rangle\) polarization. Since \(\langle H|V\rangle = 0\), interference disappears. Two single-slit fringes overlap on the screen.
- Erasing (Interference Restoration): Measurement is performed with a 45-degree (\(|D\rangle\)) polarizer in front of the screen.
  - Conditional probability \(P(x|D) \propto |\psi_1(x)\langle D|H\rangle + \psi_2(x)\langle D|V\rangle|^2 = |\frac{1}{\sqrt{2}}\psi_1(x) + \frac{1}{\sqrt{2}}\psi_2(x)|^2\). The interference fringes of \(\psi_1\) and \(\psi_2\) are restored.
  - If measured with a 135-degree (\(|A\rangle\)) polarizer, \(P(x|A) \propto |\frac{1}{\sqrt{2}}\psi_1(x) - \frac{1}{\sqrt{2}}\psi_2(x)|^2\). Interference fringes with opposite phase are restored.
Example B1: Delayed-Choice Quantum Eraser Using Entanglement
- Setup: An entangled photon pair (Signal, Idler) is generated.
- Signal: Passes through a double slit and is recorded on a screen (D0). (Attempt to create interference fringes)
- Tagging (Idler): The Idler photon’s path is set to depend on which slit the Signal photon passed through (i.e., the Idler acquires “path information”).
- Delayed Choice: After the Signal photon has reached D0, we decide the measurement method for the Idler photon.
  1. Path Reading: Measure which path the Idler came from (D1 or D2).
  2. Path Erasing: Combine the two paths of the Idler using a beam splitter (BS) and measure (D3 or D4).
- Result: The entire data recorded on D0 shows no interference fringes. However, when selecting data under the condition “Idler detected at D3”, interference fringes appear, and when selecting under the condition “Idler detected at D1”, no interference fringes appear.

4. Exercises

Visibility and Marker Inner Product: Show that when two paths (\(\psi_1, \psi_2\)) are entangled with marker states (\(|m_1\rangle, |m_2\rangle\)), the total intensity \(I(x)\) at screen \(x\) takes the form \(I(x) \propto |\psi_1(x)|^2 + |\psi_2(x)|^2 + 2\text{Re}(\mu \psi_1^*(x)\psi_2(x))\), where \(\mu = \langle m_1|m_2\rangle\). Why does interference vanish when \(\mu=0\)?
Conditional Interference Restoration: Explain what conditions \(\langle e|m_1\rangle\) and \(\langle e|m_2\rangle\) must satisfy for the interference term to be restored in the conditional intensity \(I(x|e)\) when conditioned on the marker analyzer state \(|e\rangle\) in a quantum eraser.
Mach-Zehnder Transition Amplitude: Using the transition matrices of a 50:50 beam splitter and mirrors, calculate the probabilities of detection at each of the two detectors both when and when the final beam splitter (BS2) is present, as functions of the phase difference \(\Delta\phi\).
Delayed Choice and Causality: Logically explain why a “later decision” does not alter the past, from the perspective of the invariance of conditional statistics and the overall joint distribution.
Complementarity Relation: Summarize how Bohr’s complementarity principle (the arrangement of measurement devices exclusively reveals either wave or particle behavior) is consistent with the results of delayed-choice and quantum eraser experiments.

5. Explanation

The total state is \(|\Psi\rangle = \psi_1(x)|m_1\rangle + \psi_2(x)|m_2\rangle\). The probability density at the screen \(x\) is the diagonal component of the density matrix \(\rho_S(x) = \text{Tr}_m(|\Psi\rangle\langle\Psi|)\), which is obtained by taking the partial trace over the marker state. \(\rho_S(x) = |\psi_1(x)|^2 |m_1\rangle\langle m_1| + |\psi_2(x)|^2 |m_2\rangle\langle m_2| + \psi_1(x)\psi_2^*(x) |m_1\rangle\langle m_2| + \dots\) \(I(x) = \langle x|\text{Tr}_m(\rho_S(x))|x\rangle\) … (this derivation is complex). Easier method: \(I(x) = \text{Tr}(\hat{P}_x |\Psi\rangle\langle\Psi|) = \langle\Psi|\hat{P}_x|\Psi\rangle = ( \psi_1^*\langle m_1| + \psi_2^*\langle m_2| ) (\psi_1|m_1\rangle + \psi_2|m_2\rangle) = |\psi_1|^2\langle m_1|m_1\rangle + |\psi_2|^2\langle m_2|m_2\rangle + \psi_1^*\psi_2 \langle m_1|m_2\rangle + \psi_2^*\psi_1 \langle m_2|m_1\rangle\). Assuming \(|m_p\rangle\) is normalized (\(\langle m_p|m_p\rangle=1\)), \(I(x) \propto |\psi_1|^2 + |\psi_2|^2 + 2\text{Re}(\mu \psi_1^*\psi_2)\) where \(\mu=0\) (orthogonal markers), the \(\psi_1^*\psi_2\) term becomes zero and interference disappears.
\(I(x|e) \propto |\psi_1(x)\langle e|m_1\rangle + \psi_2(x)\langle e|m_2\rangle|^2\). The interference term is \(2\text{Re}(\psi_1^*\psi_2 \langle m_1|e\rangle \langle e|m_2\rangle)\). For this term to survive, both \(\langle e|m_1\rangle\) and \(\langle e|m_2\rangle\) must be non-zero. For maximum interference, \(|\langle e|m_1\rangle| = |\langle e|m_2\rangle|\) must hold.
BS matrix \(U_{BS} = \frac{1}{\sqrt{2}}\begin{pmatrix} 1 & i \\ i & 1 \end{pmatrix}\), mirror \(U_M = \begin{pmatrix} 0 & 1 \\ 1 & 0 \end{pmatrix}\), phase \(U_\phi = \begin{pmatrix} e^{i\phi_1} & 0 \\ 0 & e^{i\phi_2} \end{pmatrix}\).
- Removing BS2: D1 probability is only path 1, D2 probability is only path 2. Each is 50% (independent of phase difference).
- Inserting BS2: Calculating \(U_{BS} U_\phi U_{BS} |0\rangle\), D1 probability is \(\cos^2(\Delta\phi/2)\), D2 probability is \(\sin^2(\Delta\phi/2)\) (\(\Delta\phi = \phi_2-\phi_1+\pi/2\) etc.) and interference appears.
In delayed-choice, ‘BS2 insertion’ and ‘BS2 removal’ represent different experimental setups. The total data distribution obtained in each setup is different. In quantum erasure, the total data (including all idler measurement results) always shows the same (non-interfering) distribution, regardless of the ‘erasure’ setting. The ‘erasure’ measurement merely provides a later criterion for classifying a specific subset of the total data, without altering already recorded individual photon data. Therefore, there is no retrocausality.
Bohr’s complementarity seems contradictory when clinging to the classical picture of ‘wave’ and ‘particle’. However, delayed-choice/erasure experiments show that the “overall setup (context)” of the measurement device determines the “statistical phenomenon” we ultimately obtain. The setup inserting BS2 is designed to produce ‘interference statistics’, while the setup removing BS2 is designed to produce ‘path statistics’. What the photon was when passing through BS1 is not important; the entire experimental setup (light source, BS1, path, BS2/erasure, detector) as an integrated system determines the final statistics, fully consistent with complementarity.
The total state is \(|\Psi\rangle = \psi_1(x)|m_1\rangle + \psi_2(x)|m_2\rangle\). The probability density at screen \(x\) is the diagonal components of the density matrix \(\rho_S(x) = \text{Tr}_m(|\Psi\rangle\langle\Psi|)\), which is obtained by tracing out the marker state. \(\rho_S(x) = |\psi_1(x)|^2 |m_1\rangle\langle m_1| + |\psi_2(x)|^2 |m_2\rangle\langle m_2| + \psi_1(x)\psi_2^*(x) |m_1\rangle\langle m_2| + \dots\) \(I(x) = \langle x|\text{Tr}_m(\rho_S(x))|x\rangle\) … (this derivation is complex). Easier method: \(I(x) = \text{Tr}(\hat{P}_x |\Psi\rangle\langle\Psi|) = \langle\Psi|\hat{P}_x|\Psi\rangle = ( \psi_1^*\langle m_1| + \psi_2^*\langle m_2| ) (\psi_1|m_1\rangle + \psi_2|m_2\rangle) = |\psi_1|^2\langle m_1|m_1\rangle + |\psi_2|^2\langle m_2|m_2\rangle + \psi_1^*\psi_2 \langle m_1|m_2\rangle + \psi_2^*\psi_1 \langle m_2|m_1\rangle\). Assuming \(|m_p\rangle\) is normalized (\(\langle m_p|m_p\rangle=1\)), \(I(x) \propto |\psi_1|^2 + |\psi_2|^2 + 2\text{Re}(\mu \psi_1^*\psi_2)\). If \(\mu=0\) (orthogonal markers), the \(\psi_1^*\psi_2\) term becomes zero and interference disappears. 2. \(I(x|e) \propto |\psi_1(x)\langle e|m_1\rangle + \psi_2(x)\langle e|m_2\rangle|^2\). The interference term is \(2\text{Re}(\psi_1^*\psi_2 \langle m_1|e\rangle \langle e|m_2\rangle)\). For this term to survive, both \(\langle e|m_1\rangle\) and \(\langle e|m_2\rangle\) must be non-zero. For maximum interference, \(|\langle e|m_1\rangle| = |\langle e|m_2\rangle|\) must hold. 3. BS matrix \(U_{BS} = \frac{1}{\sqrt{2}}\begin{pmatrix} 1 & i \\ i & 1 \end{pmatrix}\), mirror \(U_M = \begin{pmatrix} 0 & 1 \\ 1 & 0 \end{pmatrix}\), phase \(U_\phi = \begin{pmatrix} e^{i\phi_1} & 0 \\ 0 & e^{i\phi_2} \end{pmatrix}\). - Removing BS2: D1 probability is only path 1, D2 probability is only path 2. Each is 50% (independent of phase difference). - Inserting BS2: Calculating \(U_{BS} U_\phi U_{BS} |0\rangle\), D1 probability is \(\cos^2(\Delta\phi/2)\), D2 probability is \(\sin^2(\Delta\phi/2)\) (\(\Delta\phi = \phi_2-\phi_1+\pi/2\) etc.), showing interference. 4. In delayed choice, ‘BS2 insertion’ and ‘BS2 removal’ represent different experimental setups. The total data distributions obtained in each setup differ. In quantum erasure, the total data (including all idler measurement results) always show the same (non-interfering) distribution regardless of the ‘erasure’ setup. The ‘erasure’ measurement merely provides a criterion to classify a subset of this total data at a later stage, without altering already recorded individual photon data. Thus, there is no retrocausality. 5. Bohr’s complementarity appears contradictory when clinging to the classical picture of ‘wave’ and ‘particle’. However, the delayed choice/eraser experiment shows that the “overall setup of the measurement device (Context)” determines the “statistical phenomenon” we ultimately obtain. The setup with BS2 inserted is designed to produce “interference statistics,” while the setup with BS2 removed is designed to produce “path statistics.” What the photon was when passing through BS1 is not important, and the fact that the entire experimental setup (light source, BS1, path, BS2/eraser, detector) acts as an integrated system determining the final statistics is fully compatible with complementarity.