Why delayed choice experiments do Not imply retrocausality
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Abstract
Although retrocausality might be involved in quantum mechanics in a number of ways, the focus here is on the delaychoice arguments popularized by John Archibald Wheeler. There is a common fallacy that is often involved in the interpretation of quantum experiments involving a certain type of separation such as the: doubleslit experiments, whichway interferometer experiments, polarization analyzer experiments, SternGerlach experiments, and quantum eraser experiments. The fallacy leads not only to flawed textbook accounts of these experiments but to flawed inferences about retrocausality in the context of delayed choice versions of separation experiments.
Keywords
Retrocausality Delayed choice experiments Quantum eraser experiments1 Introduction: retrocausality in QM
There are a number of ways that the idea of retrocausality arises in quantum mechanics (QM). One way, which is analyzed here, is the argument largely due to Wheeler [19] that delayedchoice experiments reveal a type of retrocausality.
There is also the twovector approach to QM pioneered by Aharonov et al.:
in which a quantum system is described, at a given time, by two (instead of one) quantum states: the usual one evolving toward the future and the second evolving backwards in time from a future measurement [1, p. 1].
Cramer’s [4] transactional interpretation of QM also involves the idea of a second wave travelling backwards in time. The idea of QM as involving a wave travelling backwards in time goes back at least to Arthur Eddington’s Gifford Lectures in 1927:
The probability is often stated to be proportional to \(\psi ^{2}\), instead of \(\psi \), as assumed above. The whole interpretation is very obscure, but it seems to depend on whether you are considering the probability after you know what has happened or the probability for the purposes of prediction. The \(\psi ^{2}\) is obtained by introducing two symmetrical systems of \(\psi \)waves travelling in opposite directions in time; one of these must presumably correspond to probable inference from what is known (or is stated) to have been the condition at a later time [6, fn. pp. 216–217].
Finally the idea of retrocausality might arise when spacelike separated entangled systems are viewed from different inertial frames of reference. Abner Shimony popularized the idea of “peaceful coexistence” [17, p. 388] in spite of the “tension” between QM and special relativity.
In order to explore further the tension between quantum mechanics and relativity theory, let us consider an experimental arrangement in which [system 1] and [system 2] are tested by observers at rest in different inertial frames, and suppose that the tests are events of spacelike separation. If the reduction of \(\psi \) is to be interpreted causally, then which of the events is the cause and which is the effect? There is obviously no relativistically invariant way to answer this question. It could happen that in one frame of reference the testing of [system 1] is earlier than the testing of [system 2], and in the other frame the converse is the case [17, p. 387].
If a measurement of system 1 was taken as the “cause” and the reduction of system 2 the “effect” then in certain inertial frames the effect would precede the cause. This might be interpreted as a type of retrocausality or rather as a type of causal connection where the usual notions of “cause” and “effect” do not apply. As Shimony put it:
The wiser course is to say that quantum mechanics presents us with a kind of causal connection which is generically different from anything that could be characterised classically, since the causal connection cannot be unequivocally analysed into a cause and an effect [17, p. 387].
These other ways in which retrocausality might arise in QM are mentioned solely to emphasize that this paper is only concerned with Wheeler’s delayedchoice arguments.
2 Wheeler’s delayedchoice argument for retrocausality
Following Wheeler’s paper on delayed choice experiments, a number of quantum physicists, philosophers, and popular science writers seem to have just accepted the implication of retrocausality as part of quantum “weirdness.” In Wheeler’s own words:
There is an inescapable sense in which we, in the here and now, by a delayed setting of our analyzer of polarization to one or other angle, have an inescapable, an irretrievable, an unavoidable influence on what we have the right to say about what we call the past [21, p. 486].
Similar examples abound in the literature. For instance, concerning the quasargalaxy version of Wheeler’s delayed choice experiment, Anton Zeilinger remarks:
We decide, by choosing the measuring device, which phenomenon can become reality and which one cannot. Wheeler explicates this by example of the wellknown case of a quasar, of which we can see two pictures through the gravity lens action of a galaxy that lies between the quasar and ourselves. By choosing which instrument to use for observing the light coming from that quasar, we can decide here and now whether the quantum phenomenon in which the photons take part is interference of amplitudes passing on both side of the galaxy or whether we determine the path the photon took on one or the other side of the galaxy [24, pp. 191–192].
Occasionally instead of stating that future actions can determine whether the particle passes “on both sides of the galaxy” (or through both slits of a twoslit experiment) or only “on one or the other side” (or through only one slit), the euphemism is used of saying the photon acts like a wave or particle depending on the future actions.
The important conclusion is that, while individual events just happen, their physical interpretation in terms of wave or particle might depend on the future; it might particularly depend on decisions we might make in the future concerning the measurement performed at some distant spacetime location in the future [23, p. 207].

doubleslit experiments,

whichway interferometer experiments,

polarization analyzer experiments, and

SternGerlach experiments, and

quantum eraser experiments.
And if after the particle had entered the apparatus, the delayedchoice is made to suddenly remove the detectors (prior to arrival of the particle), then the superposition would continue to evolve and have distinctive effects (e.g., interference patterns in the twoslit experiment). Hence the fallacy makes it seem that by the delayed choice to insert or remove the appropriately positioned detectors or measurement devices, one can retrocause either a collapse to an eigenstate or not at the particle’s entrance into the apparatus.
3 Doubleslit experiments
In the wellknown setup for the doubleslit experiment, if detectors \(D_{1}\) and \(D_{2}\) are placed a small finite distance after the slits so a particle “going through the other slit” cannot reach the detector, then this is seen as “measuring which slit the particle went through” and a hit at a detector is usually interpreted as “the particle went through that slit.”
The natural image for this behaviour is that of a particle that passes either through hole [slit 1] or through [slit 2], but not through both holes [2, p. 45].
Thus what is called “detecting which slit the particle went through” is a misinterpretation. It is only placing a detector in such a position so that when the superposition projects to an eigenstate, only one of the eigenstates can register in that detector. It is about the spatial detector placement; it is not about whichslit information.
By erroneously talking about the detector “showing the particle went through slit 1,” we imply a type of retrocausality. If the detector is suddenly removed after the particle has passed the slits, then the superposition state continues to evolve and shows interference on the far wall (not shown)—in which case people say “the particle went through both slits.” Thus the “bad talk” makes it seem that by removing or inserting the detector after the particle is beyond the slits, one can retrocause the particle to go through “both slits” or one slit only.
In Wheeler’s version of the experiment, there are two detectors which are positioned behind the removable screen so they can only detect one of the projected (evolved) slit eigenstates when the screen is removed. The choice to remove the screen or not is delayed until after a photon has traversed the two slits.
In the one case [screen in place] the quantum will transform a grain of silver bromide and contribute to the record of a twoslit interference fringe. In the other case [screen removed] one of the two counters will go off and signal in which beam—and therefore from which slit—the photon has arrived [19, p. 13].
The fallacy is involved when Wheeler infers from the fact that one of the speciallyplaced detectors went off that the photon had come from one of the slits—as if there had been a projection or collapse to one of the slit eigenstates at the slits rather than later at the detectors. Wheeler makes a similar mistake when he infers from a photon now registering a certain polarization upon measurement—that the photon always had that polarization. Hence by changing the angle of the polarization detector we could seem to change the polarization in the past.
4 Whichway interferometer experiments
When detector \(D_{1}\) registers a hit, it is said that “the photon was reflected and thus took the lower arm” of the interferometer and similarly for \(D_{2}\) and passing through into the upper arm.
Indeed, if we want to visualize what happens in this experiment, the only possible image is that “something” is either reflected, or transmitted, on the beamsplitter, but it is not split: this corresponds to the behavior of a classical particle [2, p. 58].
So we can say in this case, without fear of paradox, that each photon went through just one path through the beamsplitter. In fact, if the photon were to take both paths, it would be hard to understand why it should appear to have taken just one or the other paths, why, that is, it is detected at [D1] (say) rather than at both [D1 and D2] [9, p. 40].
This is the interferometer analogue of putting two upclose detectors after the two slits in the twoslit experiment.
And this standard description is incorrect for the same reasons. The photon stays in the superposition state until the detectors force a projection to one of the (evolved) eigenstates. If the projection is to the evolved \(\left R1\right\rangle \) eigenstate then only \(D_{1}\) will get a hit, and similarly for \(D_{2}\) and the evolved version of \(\left T1\right\rangle \). The point is that the placement of the detectors (like in the doubleslit experiment) only captures one or the other of the projected eigenstates—but that does not mean the photon was in that eigenstate prior to the detection/measurement.
It is said that the second beamsplitter “erases” the “whichway information” so that a hit at either detector could have come from either arm, and thus an interference pattern emerges by varying the phase \(\phi \).
By inserting or removing the second beamsplitter after the particle has traversed the first beamsplitter (as in [19] or [20]), the fallacy makes it seem that we can retrocause the particle to go through both arms or only one arm.
5 Polarization analyzers and loops
The output from the analyzer \(P\) is routinely described as a “vertically polarized” beam and “horizontally polarized” beam as if the analyzer was itself a measurement that collapsed or projected the incident beam to either of those polarization eigenstates. This seems to follow because if one positions a detector in the upper beam then only vertically polarized photons are observed and similarly for the lower beam and horizontally polarized photons. A blocking mask in one of the beams has the same effect as a detector to project the photons to eigenstates. If a blocking mask in inserted in the lower beam, then only vertically polarized photons will be found in the upper beam, and viceversa.
 1.
the tensor product of two states \(\left \text {state 1}\right\rangle \otimes \left \text {state 2}\right\rangle \) of two particles so a superposition would be an entangled state–or
 2.
it could be interpreted as giving two eigenstates of one particle for two compatible observables (e.g., as in Dirac’s complete set of commuting observables) and then we could consider a superposition that correlates those singleparticle states.
If a polarization detector is spatially placed in, say, the upper channel and it registers a hit, then that is the measurement that collapses the evolved superposition state to \(\left \text {vertical, upper}\right\rangle \) so only a vertically polarized photon will register in the upper detector, and similarly for the lower channel. Thus it is misleadingly said that the “upper beam” was already vertically polarized and the “lower beam” was already horizontally polarized as if the analyzer had already done the projection to one of the eigenstates.
The characteristic feature of an analyzer loop is that it outputs the same polarization, in this case \(\left 45^{\circ }\right\rangle \), as the incident beam. This would be impossible if the \(P\) analyzer had in fact rendered all the photons into a vertical or horizontal eigenstate thereby destroying the information about the polarization of the incident beam. But since no collapsing measurement was in fact made in \(P\) or its inverse, the original beam can be the output of an analyzer loop.
Some texts do not realize there is a problem with presenting a polarization analyzer such as a calcite crystal as creating two beams with orthogonal eigenstate polarizations—rather than creating a superposition state so that appropriately positioned detectors can detect only one eigenstate when the detectors cause the projections to the eigenstates.
One (partial) exception is Dicke and Wittke’s text [5]. At first they present polarization analyzers as if they measured polarization and thus “destroyed completely any information that we had about the polarization” [5, p. 118] of the incident beam. But then they note a problem:
The equipment [polarization analyzers] has been described in terms of devices which measure the polarization of a photon. Strictly speaking, this is not quite accurate [5, p. 118].
They then go on to consider the inverse analyzer \(P^{1}\) which combined with \(P\) will form an analyzer loop that just transmits the incident photon unchanged.
They have some trouble squaring this with their prior statement about the \(P\) analyzer destroying the polarization of the incident beam but they struggle with getting it right.
Stating it another way, although [when considered by itself] the polarization \(P\) completely destroyed the previous polarization \(Q\) [of the incident beam], making it impossible to predict the result of the outcome of a subsequent measurement of \(Q\), in [the analyzer loop] the disturbance of the polarization which was effected by the box \(P\) is seen to be revocable: if the box \(P\) is combined with another box of the right type, the combination can be such as to leave the polarization \(Q\) unaffected [5, p. 119].
They then go on to correctly note that the polarization analyzer \(P\) did not in fact project the incident photons into polarization eigenstates.
However, it should be noted that in this particular case [sic!], the first box \(P\) in [the first half of the analyzer loop] did not really measure the polarization of the photon: no determination was made of the channel \(\ldots \) which the photon followed in leaving the box \(P\) [5, p. 119].
Their phrase “in this particular case” makes it seem that the delayed choice to not add or add the second half \(P^{1}\) of the analyzer loop will retrocause a measurement to (respectively) be made or not made in the first box \(P\).
There is some classical imagery (like Schrödinger’s cat running around one side or the other side of a tree) that is sometimes used to illustrate quantum separation experiments when in fact it only illustrates how classical imagery can be misleading. Suppose an interstate highway separates at a city into both northern and southern bypass routes—like the two channels in a polarization analyzer loop. One can observe the bypass routes while a car is in transit and find that it is in one bypass route or another. But after the car transits whichever bypass it took without being observed and rejoins the undivided interstate, then it is said that the whichway information is erased so an observation cannot elicit that information.
This is not a correct description of the corresponding quantum separation experiment since the classical imagery does not contemplate superposition states. The particleascar is in a superposition of the two routes until an observation (e.g., a detector or “road block”) collapses the superposition to one eigenstate or the other. Thus when the undetected particle rejoins the undivided “interstate,” there was no whichway information to be erased. Correct descriptions of quantum separation experiments require taking superposition seriously—so classical imagery should only be used cum grano salis.
If a blocker or detector were inserted in either channel, then this superposition state would project to one of the eigenstates, and then, as indicated by the spatial modes that bring detector placement into play, only vertically polarized photons would be found in the upper channel and horizontally polarized photons in the lower channel.
The separation fallacy is to describe the \(vh\)analyzer as if the analyzer’s effect by itself was to project an incident photon either into \(\left v\right\rangle \) in the upper channel or \(\left h\right\rangle \) in the lower channel (a mixed state)—instead of only creating the above correlated superposition state.
6 SternGerlach experiments
But a careful “expert” analysis of the experiment (e.g., [10, p. 171]) shows the apparatus does not project the particles to eigenstates. Instead it creates a single particle superposition state that correlates spin with spatial location so that with a detector in a certain position, it will only see particles of one spin state (analogous to the previous polarization example). If the collapse is caused by placing blocking masks over two of the beams, then the particles in the third beam will all be those that have collapsed to the same eigenstate. It is the detectors or blockers that cause the collapse or projection to eigenstates, not the prior separation apparatus.
We previously saw how a polarization analyzer, contrary to the statement in many texts, does not lose the polarization information of the incident beam when it “separates” the beam (into a positionallycorrelated superposition state). In the context of the SternGerlach apparatus, Feynman similarly remarks:
“Some people would say that in the filtering by \(T\) we have ‘lost the information’ about the previous state (\(+S\)) because we have ‘disturbed’ the atoms when we separated them into three beams in the apparatus \(T\). But that is not true. The past information is not lost by the separation into three beams, but by the blocking masks that are put in...” [7, p. 5–19 (italics in original)].
7 The separation fallacy
We have seen the same fallacy of interpretation in twoslit experiments, whichway interferometer experiments, polarization analyzers, and SternGerlach experiments. The common element in all the cases is that there is some separation apparatus that puts a particle into a certain superposition of spatially “entangled” or correlated eigenstates in such a manner that when an appropriately spatiallypositioned detector induces a collapse to an eigenstate, then the detector will only register one of the eigenstates. The separation fallacy is that this is misinterpreted as showing that the particle was already in that eigenstate in that position as a result of the previous “separation.” In fact the superposition evolves until some distinction is made that constitutes a measurement, and only then is the state reduced to an eigenstate. The quantum erasers are more elaborate versions of these simpler experiments, and a similar separation fallacy arises in that context.
8 One photon quantum eraser experiment
The key point is that in spite of the bad terminology of “whichway” or “whichslit” information, the polarization markings do NOT create a halfhalf mixture of horizontally polarized photons going through slit 1 and vertically polarized photons going through slit 2. It creates the superposition (pure) state \(\frac{1}{\sqrt{2}}\left[ \left S1\right\rangle \otimes \left H\right\rangle +\left S2\right\rangle \otimes \left V\right\rangle \right] \) which evolves until measured at the wall.
This can be seen by inserting a \(+45^{\circ }\) polarizer between the twoslit screen and the far wall.
 1.
\(\left H\right\rangle =\left( \left +45^{\circ }\right\rangle \left\langle +45^{\circ }\right +\left 45^{\circ }\right\rangle \left\langle 45^{\circ }\right \right) \left H\right\rangle =\left\langle +45^{\circ }H\right\rangle \left +45^{\circ }\right\rangle +\left\langle 45^{\circ }H\right\rangle \left 45^{\circ }\right\rangle \) and since a horizontal vector at \(0^{\circ }\) is the sum of the \(+45^{\circ } \) vector and the \(45^{\circ }\) vector, \(\left\langle +45^{\circ }H\right\rangle =\left\langle 45^{\circ }H\right\rangle =\frac{1}{\sqrt{2}}\) so that: \(\left H\right\rangle =\frac{1}{\sqrt{2}}\left[ \left +45^{\circ }\right\rangle +\left 45^{\circ }\right\rangle \right] \).
 2.
\(\left V\right\rangle =\left( \left +45^{\circ }\right\rangle \left\langle +45^{\circ }\right +\left 45^{\circ }\right\rangle \left\langle 45^{\circ }\right \right) \left V\right\rangle =\left\langle +45^{\circ }V\right\rangle \left +45^{\circ }\right\rangle +\left\langle 45^{\circ }V\right\rangle \left 45^{\circ }\right\rangle \) and since a vertical vector at \(90^{\circ }\) is the sum of the \(+45^{\circ }\) vector and the negative of the \(45^{\circ }\) vector, \(\left\langle +45^{\circ }V\right\rangle =\frac{1}{\sqrt{2}}\) and \(\left\langle 45^{\circ }V\right\rangle =\frac{1}{\sqrt{2}}\) so that: \(\left V\right\rangle =\frac{1}{\sqrt{2}}\left[ \left +45^{\circ }\right\rangle \left 45^{\circ }\right\rangle \right] \).
The allthephotons sum of the fringe and antifringe patterns reproduces the “mush” noninterference pattern of Fig. 10.
 1.
The insertion of the horizontal and vertical polarizers marks the photons with “whichslit” information that eliminates the interference pattern.
 2.
The insertion of a \(+45^{\circ }\) or \(45^{\circ }\) polarizer “erases” the whichslit information so an interference pattern reappears.
 1.
The markings by insertion of the horizontal and vertical polarizers creates the halfhalf mixture where each photon is reduced to either a horizontally polarized photon that went through slit 1 or a vertically polarized photon that went through slit 2. Hence the photon “goes through one slit or the other.”
 2.
The insertion of the \(+45^{\circ }\) polarizer erases that whichslit information so interference reappears which means that the photon had to “go through both slits.”

go through both slits, or

to only go through one slit or the other.
 1.
If a photon had been, say, in the state \(\left S1\right\rangle \otimes \left H\right\rangle \) then, with \(50{\%}\) probability, the photon would have passed through the filter in the state \(\left S1\right\rangle \otimes \left +45^{\circ }\right\rangle \), but that would not yield any interference pattern at the wall since their was no contribution from slit 2.
 2.
And similarly if a photon in the state \(\left S2\right\rangle \otimes \left V\right\rangle \) hits the \(+45^{\circ }\) polarizer.
Thus a correct interpretation of the quantum eraser experiment removes any inference of retrocausality and fully accounts for the experimentally verified facts given in the figures.
9 Two photon quantum eraser experiment
For instance measuring \(\left x\right\rangle \) at \(D_{p}\) and \(\left L\right\rangle \) at \(D_{s}\) imply \(S2\), i.e., slit \(2\). But as previously explained, this does not mean that the \(s\)photon went through slit \(2\). It means we have positioned the two detectors in polarization space, say to measure \(\left x\right\rangle \) polarization at \(D_{p}\) and \(\left L\right\rangle \) polarization at \(D_{s}\), so only when the superposition state collapses to \(\left x\right\rangle \) for the \(p\)photon and \(\left L\right\rangle \) for the \(s\)photon do we get a hit at both detectors.
This is the analogue of the onebeamsplitter interferometer where the positioning of the detectors would only record one collapsed state which did not imply the system was all along in that particular armeigenstate. The phrase “whichslit” or “whicharm information” is a misnomer in that it implies the system was already in a slit or wayeigenstate and the socalled measurement only revealed the information. Instead, it is only at the measurement that there is a collapse or projection to an evolved sliteigenstate (not at the previous separation due to the two slits).
Walborn et al. indulge in the separation fallacy when they discuss what the socalled “whichpath information” reveals.
Let us consider the first possibility [detecting \(p\) before \(s\)]. If photon \(p\) is detected with polarization \(x\) (say), then we know that photon \(s\) has polarization \(y\) before hitting the \(\lambda /4\) plates and the double slit. By looking at [the above formula for \(\left \Psi \right\rangle \)], it is clear that detection of photon \(s\) (after the double slit) with polarization \(R\) is compatible only with the passage of \(s\) through slit \(1\) and polarization \(L\) is compatible only with the passage of \(s\) through slit \(2\). This can be verified experimentally. In the usual quantum mechanics language, detection of photon \(p\) before photon \(s\) has prepared photon \(s\) in a certain state [18, p. 4].
Then a \(\left +\right\rangle \) polarizer or a \(\left \right\rangle \) polarizer is inserted in front of \(D_{p}\) to select \(\left +\right\rangle _{p}\) or \(\left \right\rangle _{p}\) respectively. In the first case, this reduces the overall state \(\left \Psi \right\rangle \) to \(\left +,S1\right\rangle i\left +,S2\right\rangle \) which exhibits an interference pattern, and similarly for the \(\left \right\rangle _{p}\) selection. This is misleadingly said to “erase” the socalled “whichslit information” so that the interference pattern is restored.
The first thing to notice is that two complementary interferences patterns, called “fringes” and “antifringes,” are being selected. Their sum is the nointerference pattern obtained before inserting the polarizer. The polarizer simply selects one of the interference patterns out of the mush of their merged noninterference pattern. Thus instead of “erasing whichslit information,” it selects one of two interference patterns out of the bothpatterns mush.
Even though the polarizer may be inserted after the \(s\)photon has traversed the two slits, there is no retrocausation of the photon going though both slits or only one slit as previously explained.
One might also notice that the entangled \(p\)photon plays little real role in this setup (as opposed to the “delayed erasure” setup considered next). Instead of inserting the \(\left +\right\rangle \) or \(\left \right\rangle \) polarizer in front of \(D_{p}\), insert it in front of \(D_{s}\) and it would have the same effect of selecting \(\left +,S1\right\rangle i\left +,S2\right\rangle \) or \(\left ,S1\right\rangle +i\left ,S2\right\rangle \) each of which exhibits interference. Then it is very close to the onephoton eraser experiment of the last section.
10 Delayed quantum eraser
The interesting point is that the \(D_{p}\) detections could be years after the \(D_{s}\) hits in this delayed erasure setup. If the \(D_{p}\) polarizer is set at \(\left +\right\rangle _{p}\), then out of the mush of hits at \(D_{s}\) obtained years before, the coincidence counter will pick out the ones from \(\left +,S1\right\rangle i\left +,S2\right\rangle \) which will show interference.
Again, the yearslater \(D_{p}\) detections do not retrocause anything at \(D_{s}\), e.g., do not “erase whichway information” years after the \(D_{s}\) hits are recorded (in spite of the “delayed erasure” talk). They only pick (via the coincidence counter) one or the other interference pattern out of the yearsearlier mush of hits at \(D_{s}\).
We must conclude, therefore, that the loss of distinguishability is but a side effect, and that the essential feature of quantum erasure is the postselection of subensembles with maximal fringe visibility [15, p. 79].
The same sort of analysis could be made of the delayed choice quantum eraser experiment described in the papers by Scully et al. [16] or Kim et al. [13]. Brian Greene [11, pp. 194–199] gives a good popular analysis of the Kim et al. experiment which avoids the separation fallacy and thus avoids any implication of retrocausality.
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