The Quantum Eraser

A crystal emits two photons at once, entangled in polarisation: neither has a polarisation of its own, but whatever you find for one, the other matches. One photon — the signal — goes into a Mach-Zehnder interferometer. The other — the idler — flies off down a long fibre to a polariser you can rotate, and a detector.

A half-wave plate in arm A of the interferometer swaps that arm’s horizontal and vertical. Because the signal’s polarisation is tied to the idler’s, this makes the two arms carry two-photon states that are perfectly distinguishable. The signal detector D1 goes flat: no fringes, at any phase, no matter what.

Now run the scan and watch the plot fill in. The amber points — every D1 click, counted — sit on a flat line at 50%. The violet points, the subset whose idler passed the polariser, swing from 0 to 50% and back: a full-visibility fringe. The teal points, the rest, trace exactly the opposite fringe. Turn θ from 45° down to 0° and the two fringes collapse together onto a flat 25%. The amber line never moves.

Nothing you did to the idler changed what D1 recorded. That statement is exact, it is visible in the readout — the two coincidence rates always sum to the singles rate — and it is the single most important fact on this page.

What the analyser angle changes is how the shots get sorted. The idler on its own carries no which-arm information; the record lives in the joint signal–idler polarisation. At θ = 0° the polariser projects the idler onto horizontal, which leaves the signal horizontal in one arm and vertical in the other — the arms are still tagged, so each pile still looks flat. Reading the arm off would take one more step: measuring the signal’s own polarisation as well. At θ = 45° the projection onto diagonal leaves the signal with the same polarisation in both arms; the tag is gone, and sorting by that answer splits the same clicks into two piles that each show perfect interference, out of step with one another so their sum is still flat.

The fringes were never absent and never restored. Once the idler is measured every click belongs to one pile or the other — which pile is random shot by shot, and is only fixed when the idler is actually measured — but the statistics of every possible way of sorting were fixed by the state the moment the pair was created and the plates were set. The only thing the eraser changes is which partition of the data you are allowed to compute — and computing it requires the idler results, which arrive by ordinary classical means, at ordinary classical speed.

Make the idler’s fibre long enough and the signal photon hits D1 before the idler reaches the polariser. You can then set θ after the signal is already recorded. The coincidence curves come out identical. This is the delayed-choice configuration, and the timeline under the optical table is drawn to that order.

It is routinely described as a later choice reaching back to change an earlier event. It is not, and the model makes the reason concrete: the idler measurement and the signal’s trip through the interferometer act on different factors of the same state, so the two operations commute. The engine behind this page has a unit test that does them in both orders and gets the same numbers to twelve digits. Nothing needed to travel backwards, because the order was never load-bearing.

The reason the experiment feels retrocausal is a genuine subtlety about sorting. You cannot see a fringe in a single detection; a fringe is a property of an ensemble. The ensemble that shows fringes is defined by the idler results, and you do not hold those results until the idler has been measured and the record has been carried to wherever the signal record is. Until the two lists are brought together — by a wire, a courier, a person — there is nothing but a flat curve.

If it could work the other way, it would be a telephone. Suppose choosing θ = 45° really did put fringes into D1’s own record. Then whoever holds the idler could send a bit — fringes for 1, no fringes for 0 — instantly, at any distance. Quantum mechanics forbids that, and this page shows exactly how: the marginal distribution at D1 is fixed by the state alone and contains no θ at all.

Turn the marker off. With no half-wave plate the two arms are indistinguishable, and the fringes appear in the singles themselves — the amber curve starts oscillating. The coincidence curves become half of it, and stay in step with each other whatever θ you choose. There is nothing to erase because nothing was written.

Park θ at 22.5°. The coincidence visibility lands at |sin 2θ| = 0.707. Erasure is not a switch; it is a continuous trade between how much which-path information the idler still carries and how much fringe survives in the sorted data.

Watch the measured column. The visibility computed from the sampled counts wanders around the exact figure and settles as the shots pile up. That is shot noise, the same thing that makes real coincidence experiments take hours. The experiment this page is modelled on — Walborn, Terra Cunha, Pádua and Monken (Phys. Rev. A 65, 033818, 2002), with polarisation markers and a rotatable idler polariser, in the spirit of Kim, Yu, Kulik, Shih and Scully’s 2000 delayed-choice eraser — ran for exactly that reason.

Then go back to the Mach-Zehnder page and switch on its 45° detector polariser. That is the same erasure done with a single photon and no partner — same physics, no delay, and no mystery available to be misreported.