Quantum Eraser Experiments

Quantum eraser experiments show that interference depends on the distinguishability of alternatives, not on whether a human observer has looked at a record. A system can lose visible interference when path alternatives become entangled with a marker system. If the marker is later measured in a basis that does not reveal the path, interference can reappear in appropriately conditioned subensembles.

The word “erase” is therefore potentially misleading. The experiment does not delete an event from the past, change an already recorded detection, or send a message backward in time. It changes which correlations are sorted together. The unconditional data remain compatible with ordinary no-signaling constraints.

For background, see Double-Slit Experiment, Complementarity, Delayed-Choice Experiments, and Local Measurement Statistics.

Which-Path Information

In a two-path interferometer, interference requires that the two alternatives contribute coherently to the same final event. If the path states are $\lvert u\rangle$ and $\lvert \ell\rangle$, a simple balanced preparation is

$$\lvert\psi\rangle = \frac{1}{\sqrt2} \left( \lvert u\rangle + e^{i\phi}\lvert \ell\rangle \right).$$

If the two paths are recombined without any path record, the relative phase $\phi$ can affect the detection probabilities. That is ordinary interference.

Now suppose the two alternatives become correlated with marker states $\lvert m_u\rangle$ and $\lvert m_\ell\rangle$. The joint state is

$$\lvert\Psi\rangle = \frac{1}{\sqrt2} \left( \lvert u\rangle\lvert m_u\rangle + e^{i\phi}\lvert \ell\rangle\lvert m_\ell\rangle \right).$$

The marker may be another photon, an internal atomic state, a polarization degree of freedom, a recoil state, or an environmental record. What matters is not consciousness or observation. What matters is whether the marker states are distinguishable in principle.

If the marker states are nearly identical, $\langle m_\ell\rvert m_u\rangle\approx 1$, the path alternatives remain coherent. If they are orthogonal,

$$\langle m_\ell\rvert m_u\rangle=0,$$

the marker carries complete path information and the local interference in the path system disappears.

Entanglement With Markers

The marker suppresses interference because the path system is no longer in a pure two-path superposition when the marker is ignored. Tracing out the marker gives

$$\rho_{\mathrm{path}} = \operatorname{Tr}_m \left( \lvert\Psi\rangle\langle\Psi\rvert \right).$$

For the state above,

$$\begin{aligned} \rho_{\mathrm{path}} = \frac12\Big( &\lvert u\rangle\langle u\rvert + \lvert \ell\rangle\langle \ell\rvert \\ &+ e^{-i\phi}\langle m_\ell\rvert m_u\rangle \lvert u\rangle\langle \ell\rvert \\ &+ e^{i\phi}\langle m_u\rvert m_\ell\rangle \lvert \ell\rangle\langle u\rvert \Big). \end{aligned}$$

The off-diagonal terms carry the phase coherence between the two paths. Their size is controlled by the marker overlap. This is the operational content of the phrase “which-path information destroys interference.”

For a position-sensitive screen with path amplitudes $\psi_u(x)$ and $\psi_\ell(x)$, the observed distribution has the schematic form

$$\begin{aligned} P(x) \propto &\lvert\psi_u(x)\rvert^2 + \lvert\psi_\ell(x)\rvert^2 \\ &+ 2\operatorname{Re} \left[ e^{i\phi} \langle m_\ell\rvert m_u\rangle \psi_\ell(x)\psi_u^*(x) \right]. \end{aligned}$$

When $\langle m_\ell\rvert m_u\rangle=0$, the interference term vanishes in the unsorted screen pattern. The global state may still be pure and coherent, but the coherence is stored in correlations with the marker rather than visible in the path subsystem alone.

This is why quantum eraser experiments naturally belong near entanglement and reduced-state ideas. For the formal operation, see Partial Trace and Reduced Density Operators.

Erasure of Distinguishability

An eraser measurement does not physically rewind the path system. It measures the marker in a basis that fails to reveal which path was taken.

For ideal orthogonal markers, define the superposition marker basis

$$\lvert m_+\rangle = \frac{1}{\sqrt2} \left( \lvert m_u\rangle+\lvert m_\ell\rangle \right), \qquad \lvert m_-\rangle = \frac{1}{\sqrt2} \left( \lvert m_u\rangle-\lvert m_\ell\rangle \right).$$

Equivalently,

$$\lvert m_u\rangle = \frac{1}{\sqrt2} \left( \lvert m_+\rangle+\lvert m_-\rangle \right), \qquad \lvert m_\ell\rangle = \frac{1}{\sqrt2} \left( \lvert m_+\rangle-\lvert m_-\rangle \right).$$

Substituting into the joint state gives

$$\begin{aligned} \lvert\Psi\rangle = \frac12\Big[ &\left( \lvert u\rangle + e^{i\phi}\lvert \ell\rangle \right) \lvert m_+\rangle \\ &+ \left( \lvert u\rangle - e^{i\phi}\lvert \ell\rangle \right) \lvert m_-\rangle \Big]. \end{aligned}$$

Conditioned on the marker outcome $m_+$, the path state has one interference phase. Conditioned on $m_-$, the path state has the opposite phase. The two conditioned patterns are often called fringes and antifringes.

The essential point is conditionality. Interference is recovered only after sorting events by the appropriate marker outcome. If all marker outcomes are combined, the fringes and antifringes add to a noninterference distribution.

Conditional Interference

Suppose the screen distribution conditioned on the marker result $+$ is

$$P_+(x) \propto I_0(x) + I_1(x)\cos\theta(x),$$

while the distribution conditioned on $-$ is

$$P_-(x) \propto I_0(x) - I_1(x)\cos\theta(x).$$

If the two marker outcomes occur with equal probability, the unsorted screen pattern is

$$P_{\mathrm{unsorted}}(x) = \frac12P_+(x)+\frac12P_-(x) \propto I_0(x).$$

The interference term cancels. A person looking only at the screen, without access to the marker outcomes, sees no fringe pattern.

This is the reason coincidence sorting is central in optical quantum erasers. One detector registers the signal photon at a screen or output port. Another detector registers the marker or idler photon. Only after the records are compared can one form the conditional subensembles. The local signal data alone do not reveal the later marker basis or marker outcome.

Delayed-Choice Erasers

Delayed-choice quantum eraser experiments combine the eraser idea with Wheeler’s delayed-choice theme. The marker may be measured after the signal photon has already been detected. If the marker is measured in a which-path basis, the corresponding conditional data show path-tagged behavior. If it is measured in an erasing basis, the corresponding conditional data can show complementary interference patterns.

The timing sounds dramatic, but the operational statement is modest:

Later marker measurements determine how joint records are sorted.
They do not change the earlier local record.

No observer can decide later whether an already collected unsorted screen image contains visible interference. The unsorted image is fixed. The later choice affects which subset labels are attached when the two data streams are compared.

This is also why delayed-choice erasers do not allow faster-than-light communication. The marginal statistics of the signal system are independent of which measurement basis is chosen for the marker. The marker choice changes correlations, not the locally visible distribution.

For the formal no-signaling statement, see Local Measurement Statistics.

Common Misconceptions

• A quantum eraser does not prove that human knowledge collapses the wavefunction.
• “Erasing” means erasing distinguishability in a chosen measurement basis, not deleting a recorded physical event.
• Interference reappears in conditional subensembles, not in the unsorted local data.
• Delayed-choice erasers do not send messages to the past.
• The marker can be a physical degree of freedom that no human reads.
• The experiment is compatible with complementarity; it does not give full which-path information and full interference in one and the same sorted ensemble.
• Postselection is not a technical footnote. It is the reason the striking interference patterns can be displayed at all.

Historical Role

Scully and Druehl proposed quantum eraser arrangements to sharpen the relation between which-path information and interference. Later optical experiments implemented the idea with entangled photon pairs, polarization markers, interferometric markers, and delayed marker choices.

The experiments are important because they separate three ideas that are often conflated:

• physical availability of path information,
• actual measurement basis used for the marker,
• conditional sorting of joint data.

Once these are separated, the puzzle becomes a standard quantum-correlation phenomenon. The global state contains coherence. The local reduced state may not show it. A suitable joint measurement or conditional sorting can reveal phase-sensitive correlations without enabling signaling.

Cross-Links

• Foundations Experiments and Quantum Reality
• What These Experiments Do and Do Not Prove
• Delayed-Choice Experiments
• Double-Slit Experiment
• Complementarity
• Probability Amplitudes
• Born Rule
• Measurement in a Chosen Basis
• Projective Measurement
• State Update Rule
• Partial Trace
• Reduced Density Operators
• Local Measurement Statistics
• Entanglement in Foundations
• Common Historical Misconceptions

References

• M. O. Scully and K. Druehl, “Quantum eraser: A proposed photon correlation experiment concerning observation and delayed choice in quantum mechanics,” Physical Review A 25, 2208-2213, 1982, DOI: 10.1103/PhysRevA.25.2208.
• P. G. Kwiat, A. M. Steinberg, and R. Y. Chiao, “Observation of a quantum eraser: A revival of coherence in a two-photon interference experiment,” Physical Review A 45, 7729-7739, 1992, DOI: 10.1103/PhysRevA.45.7729.
• T. J. Herzog, P. G. Kwiat, H. Weinfurter, and A. Zeilinger, “Complementarity and the Quantum Eraser,” Physical Review Letters 75, 3034-3037, 1995, DOI: 10.1103/PhysRevLett.75.3034.
• Y.-H. Kim, R. Yu, S. P. Kulik, Y. Shih, and M. O. Scully, “Delayed Choice Quantum Eraser,” Physical Review Letters 84, 1-5, 2000, DOI: 10.1103/PhysRevLett.84.1.
• X.-S. Ma, J. Kofler, A. Qarry, N. Tetik, T. Scheidl, et al., “Quantum erasure with causally disconnected choice,” Proceedings of the National Academy of Sciences 110, 1221-1226, 2013, DOI: 10.1073/pnas.1213201110.
• X.-S. Ma, J. Kofler, and A. Zeilinger, “Delayed-choice gedanken experiments and their realizations,” Reviews of Modern Physics 88, 015005, 2016, DOI: 10.1103/RevModPhys.88.015005.

Exercises

1. In the marker-entangled state

$$\lvert\Psi\rangle = \frac{1}{\sqrt2} \left( \lvert u\rangle\lvert m_u\rangle + e^{i\phi}\lvert \ell\rangle\lvert m_\ell\rangle \right),$$

why does the unsorted path interference vanish when $\langle m_\ell\rvert m_u\rangle=0$?

Solution

The reduced path state contains off-diagonal terms proportional to the marker overlap. When $\langle m_\ell\rvert m_u\rangle=0$, those terms vanish after the marker is ignored:

$$\rho_{\mathrm{path}} = \frac12 \left( \lvert u\rangle\langle u\rvert + \lvert \ell\rangle\langle \ell\rvert \right).$$

Without off-diagonal coherence between the path alternatives, the local path statistics contain no interference term.

2. Why do the $m_+$ and $m_-$ eraser outcomes produce complementary fringe and antifringe patterns?

Solution

Conditioning on $m_+$ leaves a path state proportional to

$$\lvert u\rangle+e^{i\phi}\lvert \ell\rangle,$$

while conditioning on $m_-$ leaves a path state proportional to

$$\lvert u\rangle-e^{i\phi}\lvert \ell\rangle.$$

The relative minus sign shifts the interference phase by $\pi$. Maxima in one conditional pattern correspond to minima in the other, so adding the two sorted patterns cancels the interference term.

3. Why can a delayed-choice quantum eraser not be used to send a message to the earlier signal detector?

Solution

The signal detector sees the marginal distribution of the signal system. That marginal distribution is obtained by ignoring, or tracing over, the marker outcome. Different marker measurement bases can change the joint correlations and the conditional subensembles, but they do not change the unsorted local signal distribution. Since the earlier detector cannot see which marker basis will later be used, no controllable message is transmitted.

4. Give one physical example of a marker system that can carry which-path information without being read by a person.

Solution

One example is photon polarization. If a two-path interferometer marks the upper path with horizontal polarization and the lower path with vertical polarization, the polarization degree of freedom carries path information. Even if no person reads a polarization detector, the orthogonal marker states can suppress local interference until the data are conditioned in an erasing polarization basis.