Delayed Choice and Causal Interpretation

A delayed-choice experiment selects whether to measure path or interference after a quantum system has entered an apparatus. In standard quantum mechanics the calculation uses the actual later interaction; it does not require that this interaction rewrite an earlier event.

Open and closed interferometers

Use orthogonal path states $\lvert0\rangle,\lvert1\rangle$. After the first balanced beam splitter and a phase shift, the path state is

$$\lvert\psi_\phi\rangle= \frac{\lvert0\rangle+e^{i\phi}\lvert1\rangle}{\sqrt2}.$$

An open interferometer measures the paths directly: $p(0)=p(1)=1/2$, independent of $\phi$. A closed interferometer applies a second balanced beam splitter, represented here by a Hadamard matrix $H$, before the same detectors. Our phase convention gives

$$p(0\mid{\rm closed})=\cos^2(\phi/2),\qquad p(1\mid{\rm closed})=\sin^2(\phi/2).$$

The second apparatus therefore measures the $X$ basis of the incoming path qubit, whereas the open apparatus measures $Z$. Different measurements of one state need not have the same statistics. A delayed setting choice is meaningful when the setting is implemented after entry and unwanted advance communication is controlled.

What “choice” changes

The setting changes the transformation or observable to which the detectors respond. It does not establish that the system possessed both an unmeasured definite path and an interference phase as simultaneous classical properties. Nor does the absence of a second beam splitter prove that the particle had followed a classical trajectory throughout the apparatus.

For an incoming density matrix $\rho$ and setting $s$, write the probability as $\operatorname{tr}(\rho E_{a|s})$. This formula refers to one preparation and a family of alternative measurement effects. It contains no requirement to decide in advance which ontological story the system should enact.

A quantum-controlled beam splitter extends the calculation: a control qubit coherently selects applying $I$ or $H$. The resulting control–path state can become entangled. Calling this a simultaneous wave and particle is shorthand for correlations in a controlled circuit, not a new classical category.

Distinguishing delayed choice from an eraser

In a quantum eraser, interference can emerge in conditional subensembles when a remote marker is measured. The unsorted screen distribution remains unchanged. In the open/closed interferometer, the final optical element physically acts on the same system before detection. Neither case permits a later setting to send a message into an earlier freely accessible record.

The review by Ma, Kofler, and Zeilinger surveys these different designs. A claim about one realization should identify its spacetime arrangement, setting implementation, detection assumptions, and measured quantity. A successful delayed-choice experiment alone does not decide between all interpretations that reproduce quantum predictions.

Exercise

Let a random setting close the interferometer with probability $q$. At $\phi=0$, what is the probability of detector $0$ if the setting record is ignored? Does this indicate a superposition of measurement settings?

Solution

It is $q+(1-q)/2=(1+q)/2$. This is an ordinary probabilistic mixture of two procedures. It differs from coherent quantum control, for which the joint state and measurements on the control can reveal off-diagonal coherence. Ignoring a classical record cannot establish that coherence.

References

• Ma, X.-S., Kofler, J., and Zeilinger, A., “Delayed-choice gedanken experiments and their realizations,” Reviews of Modern Physics 88, 015005 (2016), arXiv:1407.2930 — historical proposals, implementations, and distinctions among protocols.