Quantum Eraser

The intuition — knowing the path is enough to kill the fringes

Send single photons through a double slit and they build up an interference pattern — alternating bright and dark fringes — even though each photon arrives as a single dot. The pattern says each photon went through both slits as a superposition and interfered with itself.

Now do one thing: arrange for some marker to record which slit each photon used. Put a polarizer behind each slit so that path A stamps the photon "horizontally polarized" and path B stamps it "vertically polarized" — two orthogonal tags. You don't have to read those stamps. You don't even have to build a detector. The instant the path becomes knowable in principle, the fringes vanish and the screen shows a plain, structureless blob — exactly what you'd get from two classical floodlights.

The quantum eraser is the second half of the trick: take a polarizer oriented at 45° and place it in front of the screen. That polarizer measures every photon along the diagonal, which gives the same answer for an H-tagged and a V-tagged photon. The which-path stamp is scrambled beyond recovery — erased — and the interference fringes snap back into existence. You destroyed information you never read, and recovered a wave pattern by deleting a label.

How it works — the cross term is everything

An interference pattern is the signature of a coherent superposition. Write the amplitude at the screen as the sum of the amplitude from each path:

P(x) = |ψ₁(x) + ψ₂(x)|²
     = |ψ₁|² + |ψ₂|² + 2 Re(ψ₁* ψ₂)

The first two terms are just the two single-slit blobs. The last term, 2 Re(ψ₁* ψ₂), is the cross term — it oscillates with position and produces the fringes. Kill the cross term and you are left with a featureless sum of two blobs.

Marking the path attaches a marker state to each amplitude — call them |A⟩ for path 1 and |B⟩ for path 2. The full state is now entangled:

|Ψ⟩ = ψ₁(x)|A⟩ + ψ₂(x)|B⟩

When you compute the probability at the screen you must trace over (sum out) the marker, and the cross term picks up the factor ⟨A|B⟩ — the overlap of the two marker states:

P(x) = |ψ₁|² + |ψ₂|² + 2 Re(ψ₁* ψ₂ ⟨A|B⟩)

If the markers are orthogonal (perfect which-path info, ⟨A|B⟩ = 0) the cross term is zero — no fringes. If the markers are identical (no which-path info, ⟨A|B⟩ = 1) the cross term is full strength — full fringes. Erasing means projecting both markers onto a common state |+⟩ = (|A⟩ + |B⟩)/√2, which restores a non-zero overlap within that subset and brings the fringes back.

The duality relation — distinguishability versus visibility

Bohr's complementarity is usually stated as an all-or-nothing rule, but Englert (1996) and Greenberger–Yasin (1988) made it quantitative. Define fringe visibility V and path distinguishability D:

V = (P_max − P_min) / (P_max + P_min)      fringe contrast, 0 to 1
D = path distinguishability                 0 = unknown, 1 = fully known

         D² + V² ≤ 1     (equality for pure states)

This is a tight trade-off, not a binary switch. Full path knowledge (D = 1) forces V = 0. Zero knowledge (D = 0) permits perfect fringes (V = 1). Everything in between is allowed: tag the two paths with polarizations 60° apart and the markers have overlap ⟨A|B⟩ = cos 60° = 0.5, giving partial distinguishability and a washed-out — but still visible — fringe pattern. The eraser slides you back along this curve from D = 1, V = 0 toward D = 0, V = 1.

The two canonical setups

Element	Polarization eraser (Walborn-style)	Delayed-choice eraser (Kim et al. 2000)
Marker	Quarter-wave plates at ±45° behind each slit — orthogonal circular tags (L vs R)	Entangled idler photon from down-conversion carries the path
Where the tag lives	On the same photon's polarization	On a separate, distant idler photon
Eraser	45° polarizer before the screen	Beam splitter routing idlers to "which-path" or "erasing" detectors
Timing	Erasure can be before or after the slit	Erasure happens after the signal photon already hit the screen
What you see directly	Fringes vanish, then return when polarizer inserted	Total screen pattern always smeared; fringes hide in subsets
Fringes recovered by	Inserting the diagonal polarizer	Coincidence-sorting signal hits against erasing-detector clicks
Headline puzzle	Deleting an unread label revives the wave pattern	The "choice" appears to act on an already-recorded photon

Numbers from real experiments

The delayed-choice quantum eraser of Kim, Yu, Kulik, Shih and Scully (Phys. Rev. Lett. 84, 1–5, 2000) used a 351.1 nm argon-ion pump laser through a beta-barium-borate (BBO) crystal to make entangled photon pairs by spontaneous parametric down-conversion.

Quantity	Value / behaviour	Why it matters
Pump wavelength	351.1 nm (Ar-ion laser)	One UV photon splits into two ~702 nm IR photons
Down-converted pair	Signal + idler, momentum-entangled	Idler carries the which-path record for the signal
Path separation of idler	≈ 2.5 m extra travel before its detectors	Idler detected ~8 ns after the signal hit the screen
Detector D0 (signal)	Scanned across the focal plane	Records the position pattern on the "screen"
D1, D2 (erasing)	Idler path scrambled by a beam splitter	Coincidences with D1/D2 reveal fringes — opposite phase
D3, D4 (which-path)	Idler path preserved	Coincidences with D3/D4 show NO fringes — two blobs
Sum of all subsets	Featureless single-hump distribution	No FTL signalling — total pattern never shows fringes

The two complementary fringe subsets (from D1 and D2) are shifted by exactly half a fringe spacing, so they add up to the smooth no-fringe total. That cancellation is the mathematical guarantee that nothing observable changes at the screen when you flip the idler's fate — which is why no faster-than-light message can be sent.

Why it isn't disturbance, and isn't observation

The seductive wrong story is that "measuring kicks the photon and smears the fringes." Heisenberg's microscope makes that picture plausible, but the eraser refutes it. Marking with a quarter-wave plate or by entanglement transfers essentially no momentum — and you can compute, from the duality relation, that the fringe loss is exactly what orthogonal markers predict regardless of any kick. The killer is correlation, not collision.

The second wrong story is that "a conscious observer collapses the wavefunction." The fringes are already gone the moment the path is encoded in any degree of freedom that could in principle be read — long before a human looks. Decohere the photon into the lab and the interference is lost whether or not anyone ever measures. What the eraser turns on and off is the availability of distinguishing information, a property of the joint quantum state, not the presence of a mind.

Where it shows up

• Foundations of quantum mechanics. The cleanest demonstration that complementarity is about information, refining Bohr–Einstein debates with hard numbers via D² + V² ≤ 1.
• Quantum information. The marker-erasure logic is the same coherence/decoherence accounting used in qubit design; protecting interference means protecting against unwanted which-path entanglement with the environment.
• Interferometry & metrology. Mach–Zehnder and atom interferometers lose contrast exactly when stray entanglement leaks path information; understanding erasure tells you how to recover visibility.
• Quantum imaging & "ghost" imaging. Built on the same entangled signal/idler pairs from down-conversion, exploiting correlations the eraser made famous.
• Teaching tool. The canonical worked example for explaining decoherence, entanglement, and the measurement problem without invoking observers.
• Tests of nonlocal correlations. Delayed-choice versions probe how far the "no signalling" theorem holds when timing is pushed to extremes.

Common misconceptions and edge cases

• "It sends information back in time." No. Extracting fringes from the data always requires the erasing-detector record, which arrives later through ordinary forward-in-time channels. Without it you cannot even sort the photons into a fringe subset.
• "The total screen pattern changes when you erase." It never does. The raw, unsorted pattern is always a featureless smear. Only the coincidence-sorted subsets show fringes, and the complementary subsets are anti-phased so they sum back to the smear.
• "You can use it to signal faster than light." Impossible — the no-signalling theorem holds because the marginal distribution at the screen is independent of the distant choice.
• "Erasing un-measures the photon." Erasing chooses a measurement basis in which the path is unknowable, projecting both paths onto a shared state. It doesn't undo anything; it picks a question with no which-path answer.
• "It needs a conscious observer." The interference is already lost once the path is recoverable in any physical record; minds are irrelevant.
• "It only works with photons." Quantum-eraser logic has been demonstrated with photons, atoms, and even large molecules. Any system that can be put into a path superposition and entangled with a marker qualifies.