Quantum Erasure and Conditional Fringes

10.1 Entangled path markers and coincidence sorting

10.1.1 Tagging interferometer arms with orthogonal marker states

Suppose we entangle the two paths \(\lvert L\rangle\) and \(\lvert R\rangle\) of an interferometer with two orthogonal states \(\lvert H\rangle\) and \(\lvert V\rangle\) of an ancillary marker system (such as the polarization of the photon itself or of a twin herald photon). Because \(\langle V\vert H\rangle=0\), the marker carries complete which-path information, and the unconditional signal counts at the interferometer output become completely flat (\(P(0)=1/2\) for all \(\phi\)), showing zero interference visibility.

10.1.2 Sorting signal arrivals by diagonal marker outcomes

Now, instead of measuring the marker in the path-distinguishing basis \(\{\lvert H\rangle,\lvert V\rangle\}\), rotate the marker analyzer to the diagonal superposition basis \(\lvert\pm\rangle=(\lvert H\rangle\pm\lvert V\rangle)/\sqrt{2}\) that projects equally onto both path labels. When each signal detection is paired in coincidence with the marker outcome \(+\) or \(-\), the two sorted sub-ensembles recover full-contrast interference fringes—shifted in phase by \(\pi\) relative to one another!

10.1.3 Complementary sub-ensembles versus the flat marginal sum

In the interactive plot below, three curves appear simultaneously: the blue fringe \(P(0\mid +)\) conditioned on marker outcome \(+\), the dashed gold antiphase fringe \(P(0\mid -)\) conditioned on marker outcome \(-\), and the flat dotted violet line \(P(0)=1/2\) representing all unsorted signal detections. Quantum erasure never alters the flat unsorted signal distribution; it extracts complementary conditional correlations by cross-tabulating signal and marker records.

10.2 Basis rotation of the marker and antiphase fringes

10.2.1 Rewriting the path–marker state in the diagonal basis

Before the recombining beam splitter, the maximally entangled path–marker state is

\[\lvert\Psi(\phi)\rangle = \frac{1}{\sqrt{2}}\bigl(\lvert L\rangle\lvert H\rangle + e^{i\phi}\lvert R\rangle\lvert V\rangle\bigr).\]

Substituting \(\lvert H\rangle=(\lvert +\rangle+\lvert -\rangle)/\sqrt{2}\) and \(\lvert V\rangle=(\lvert +\rangle-\lvert -\rangle)/\sqrt{2}\) rewrites the exact same state vector in the diagonal marker basis without any physical disturbance to the signal arm:

\[\lvert\Psi(\phi)\rangle = \frac{1}{2}\bigl(\lvert L\rangle + e^{i\phi}\lvert R\rangle\bigr)\lvert +\rangle + \frac{1}{2}\bigl(\lvert L\rangle - e^{i\phi}\lvert R\rangle\bigr)\lvert -\rangle.\]

10.2.2 Opposite-phase conditional fringes and exact marginal cancellation

After the signal modes \(\lvert L\rangle\) and \(\lvert R\rangle\) recombine at the output beam splitter \(H\), the conditional probabilities of a signal click at port \(D_0\) given marker outcomes \(+\) and \(-\) are

\[P(0\mid +) = \frac{1+\cos\phi}{2}, \qquad P(0\mid -) = \frac{1-\cos\phi}{2} = \frac{1+\cos(\phi+\pi)}{2}.\]

Because each marker outcome \(m\in\{+,-\}\) occurs with marginal probability \(P(m)=1/2\), the total unconditional probability at port \(D_0\) is

\[P(0) = P(+)\,P(0\mid +) + P(-)\,P(0\mid -) = \frac{1}{2}\left(\frac{1+\cos\phi}{2}\right) + \frac{1}{2}\left(\frac{1-\cos\phi}{2}\right) = \frac{1}{2},\]

independent of \(\phi\). The crest of the \(+\) fringe sits directly over the trough of the \(-\) fringe, so their sum is identically flat.

10.2.3 Commuting local measurements and post-hoc coincidence sorting

When the signal and marker live on spatially separated photons, their local measurement operators \(M_{\mathrm{sig}}\otimes I\) and \(I\otimes M_{\mathrm{mark}}\) commute: \((M_{\mathrm{sig}}\otimes I)(I\otimes M_{\mathrm{mark}})=M_{\mathrm{sig}}\otimes M_{\mathrm{mark}}\). Consequently, the joint probability \(P(0,\pm)\) is completely invariant under whether the marker is measured before or years after the signal photon hits its detector. Delayed-choice quantum erasure simply sorts already-recorded signal timestamps using the classical log of later marker outcomes.

10.3 No-signalling and coincidence-window discipline

10.3.1 Why quantum erasure cannot transmit a faster-than-light signal

Can an experimenter at the marker station send an instantaneous message to the signal station by choosing whether to measure the marker in the \(\{\lvert H\rangle,\lvert V\rangle\}\) basis or the \(\{\lvert +\rangle,\lvert -\rangle\}\) basis? No: in both cases, the signal observer who lacks access to the classical marker outcomes observes the exact same flat marginal probability \(P(0)=1/2\). Extracting either interference fringe requires receiving the classical bit stream of marker outcomes \(\pm\) over an ordinary communication channel.

10.3.2 Reporting both conditional subsets and accidental coincidences

A rigorous quantum-eraser report displays both conditional curves \(P(0\mid +)\) and \(P(0\mid -)\), their relative weights \(P(+)\) and \(P(-)\), the flat unsorted sum \(P(0)\), and the coincidence time window used to pair signal and marker clicks.

10.3.3 Distinguishing coherent basis rotation from deleting classical files

Quantum erasure requires projecting the marker onto the coherent superposition states \(\lvert\pm\rangle\) before the marker decoheres into a macroscopic environment. Merely erasing a computer file after measuring the marker in the \(\{\lvert H\rangle,\lvert V\rangle\}\) basis leaves the signal in an incoherent mixture and never restores fringes.

Explore the experiment

Switch among the sorted \(+\) subset, the sorted \(-\) subset, and the unsorted marginal, and sweep \(\phi\) across \([0^\circ,360^\circ]\). Verify that the blue (\(+\)) and dashed gold (\(-\)) curves oscillate \(180^\circ\) out of phase and average to the flat line \(1/2\).

A worked example

At phase \(\phi=0^\circ\), calculate the joint probabilities \(P(0,+)\) and \(P(0,-)\) as well as the unconditional probability \(P(0)\) at detector \(D_0\).

Try it yourself

What to carry forward

When path alternatives are entangled with a marker, unconditional fringes vanish while complementary conditional fringes \(P(0\mid\pm)=(1\pm\cos\phi)/2\) remain encoded in joint correlations that can be extracted only by coincidence sorting with a diagonal-basis marker record.

Sources and further exploration

These readings provide historical context, experimental implementations, or deeper derivations. The interactive display above is an educational model, not a reproduction of their measured data.

• Kim et al.: A delayed-choice quantum eraser