A Very Common Fallacy in Quantum Mechanics

There is a very common fallacy, here called the separation fallacy, that is involved in the interpretation of quantum experiments involving a certain type of separation such as the:

double-slit experiments,
which-way interferometer experiments,
polarization analyzer experiments,
Stern-Gerlach experiments, and
quantum eraser experiments.

In each case, given an incoming quantum particle, the apparatus creates a labelled or tagged (a type of entanglement) superposition of certain eigenstates (the “separation”). Detectors can be placed in certain positions so that when the evolving superposition state is finally projected or collapsed by the detectors, then only one of the eigenstates can register at each detector (due to the labels or tags). The separation fallacy mistakes the creation of a tagged or entangled superposition for a measurement. Thus it treats the particle as if it had already been projected or collapsed to an eigenstate at the separation apparatus rather than at the later detectors. But if the detectors were suddenly removed while the particle was in the apparatus, then the superposition would continue to evolve and have distinct effects (e.g., interference patterns in the two-slit experiment).

Hence the separation fallacy makes it seem that by the delayed choice to insert or remove the appropriately positioned detectors, one can retro-cause either a projection to an eigenstate or not at the particle’s entrance into the separation apparatus.

The separation fallacy is remedied by:

taking superposition seriously, i.e., by seeing that the separation apparatus created an entangled superposition state of the alternatives (regardless of what happens later) which evolves until a measurement is taken, and
taking into account the role of detector placement (“contextuality”), i.e., by seeing that if a suitably positioned detector, as determined by the tags, can only detect one collapsed eigenstate, then it does not mean that the particle was already in that eigenstate prior to the measurement (e.g., it does not mean that the particle went through one slit, took one path in an interferometer, or was already in a polarization or spin eigenstate).

The separation fallacy will be first illustrated in a non-technical manner for the first four experiments. Then the lessons will be applied in a slightly more technical discussion of quantum eraser experiments where the labels or tags are erased after the separation apparatus and where, due to the separation fallacy, incorrect inferences about retrocausality have been rampant.

The double-slit experiment

In the well-known setup for the double-slit experiment, if a detector D₁ is placed a small distance after slit 1 so a particle “going through the other slit” cannot reach the detector, then a hit at the detector is usually interpreted as “the particle went through slit 1.”

But this is wrong. The particle is in a superposition state, which we might represent as $|Slit1\rangle + |Slit2\rangle$ , that evolves until it hits the detector which projects (or ‘collapses’) the superposition to one of (the evolved versions of) the slit-eigenstates. The particle’s state was not collapsed earlier so it was not previously in the $|Slit1\rangle$ eigenstate, i.e., it did not “go through slit 1.”

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 detector placement; it is not about which-slit.

By erroneously talking about the detector “showing the particle went through slit 1,” we imply a type of retro-causality. 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 retro-cause the particle to go through both slits or one slit only.

This sudden removal or insertion of detectors that can only detect one of the slit-eigenstates is a version of Wheeler’s delayed choice thought-experiment [Wheeler 1978].

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 … contribute to the record of a two-slit 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.” [Wheeler 1978, p. 13]

The separation fallacy is involved when Wheeler infers from the fact that one of the specially-placed detectors went off–that the photon had come from one of the slits as if there had been a projection to one of the slit eigenstates at the slits rather than later at the detectors.

Similar examples abound in the literature. For instance, concerning the quasar-galaxy 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 well-known 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. [Zeilinger 2008, 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) 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. [Zeilinger 2004, p. 207]

These descriptions using the separation fallacy are unfortunately common and have generated a spate of speculations about retrocausality.

Which-way interferometer experiments

Consider an interferometer with only one beam-splitter (e.g., half-silvered mirror) at the photon source which creates the superposition: $|LowerArm\rangle + |UpperArm\rangle$ .

When detector $D_{1}$ registers a hit, it is said that “the photon took the lower arm” of the interferometer and similarly for $D_{2}$ and the upper arm. This is the interferometer analogue of putting two up-close detectors after the 2 slits in the 2-slit experiment.

And this standard description is wrong 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 $|LowerArm\rangle$ eigenstate then only $D_{1}$ will get a hit, and similarly for $D_{2}$ and the upper arm. The point is that the placement of the detectors (like in the double-slit experiment) only captures one or the other of the projected eigenstates.

Now insert a second beam-splitter as in the following diagram.

It is said that the second beam-splitter “erases” the “which-way information” so that a hit at either detector could have come from either arm, and thus an interference pattern emerges.

But this is wrong. The evolving superposition state $|LowerArm\rangle + |UpperArm\rangle$ (which contains no which-way information) was always there until the detectors. The so-called “which-way information” was not there to be “erased” since the particle did not take one way or the other in the first place. The second beam-splitter only allows the two projected eigenstates from the superposition to be measured at each detector so, by shifting the phase $\phi$ , an interference pattern can be recorded at each detector.

By inserting or removing the second beam-splitter after the particle has traversed the first beam-splitter, the “bad talk” makes it seem that we can retro-cause the particle to go through both arms or only one arm.

Instead of inserting the second beam-splitter, we could rig up more mirrors, a lense, and a detector so that when the detector causes the collapse, then it is will register either arm-eigenstate.

This might also be (mis)interpreted as “erasing” the “which-way information” but in fact the photon did not go through just one arm so there was no such information to be erased. The point is the positioning of the detector so that when the evolved superposition $|LowerArm\rangle + |UpperArm\rangle$ is projected to one of the eigenstates, then both are detected. Any setup that would allow a detector to register both collapsed arm-eigenstates (and thus to register the interference effects of the evolving superposition) would be a setup that could be (mis)interpreted as “erasing” the “which-way information.”

Polarization analyzers and loops

Another common textbook example of the separation fallacy is the treatment of polarization analyzers such as calcite crystals that are said to create two orthogonally polarized beams in the upper and lower channels, say $|v\rangle$ and $|h\rangle$ from an arbitrary incident beam.

The output beams from the analyzer $P$ are routinely described as being “vertically polarized” and “horizontally polarized” as if the analyzer was 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 vice-versa.

But here again, the story is about detector (or blocking mask) placement; it is not about the analyzer supposedly projecting a photon into one or the other of the eigenstates. The analyzer puts the incident photons into a superposition state. If a detector is placed in, say, the upper beam, then that is the measurement that collapses the evolved superposition state. If the collapse is to the vertical polarization eigenstate then it will register only in the upper detector and similarly for a collapse to the horizontal polarization eigenstate for any detector placed in the lower position. 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.

If the analyzer had in fact performed the measurement collapsing to the eigenstates, then any prior polarization of the incident beam would be lost. Hence assume that the incident beam was prepared in a specific polarization of, say, $|45^{\circ}\rangle$ half-way between the states of vertical and horizontal polarization. Then follow the vh-analyzer $P$ with its inverse $P^{-1}$ to form an analyzer loop [French and Taylor 1978].

The characteristic feature of an analyzer loop is that it outputs the same polarization, in this case $|45^{\circ}\rangle$ , 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.

Very few textbooks realize there is even a problem with presenting a polarization analyzer such as a calcite crystal as creating two beams with 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 eigenstates.

One (partial) exception is Dicke and Wittke’s text [1960]. At first they present polarization analyzers as if they measured polarization and thus “destroyed completely any information that we had about the polarization” [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.” [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 beam unchanged.

They have some trouble squaring this with their prior statement about the $P$ analyzer destroying the polarization of the incident beam but they, unlike most texts, face up to the problem.

“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.” [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 (p1 or p2) which the photon followed in leaving the box P.” [p. 119]

There is some classical imagery (like Schrodinger’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 which-way 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 particle-as-car 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. Correct descriptions of quantum separation experiments require taking superposition seriously–so classical imagery should only be used cum grano salis.

This analysis might be rendered in a more technical but highly schematic way. The photons in the incident beam have a particular polarization $|\psi\rangle$ such as $|45^{\circ}\rangle$ in the above example. This polarization state can be represented or resolved in terms of the vh-basis as:

The effect of the vh-analyzer $P$ is then represented as tagging the vertical and horizontal polarization states with the upper and lower (or straight) channels so the vh-analyzer puts an incident photon into the superposition state:

$\langle v|\psi\rangle|v\rangle_{U} + \langle h|\psi\rangle|h\rangle_{L}$ .

If a blocker or detector were inserted in either channel, then this superposition state would project to one of the eigenstates, and then only vertically polarized photons would be found in the upper channel and horizontally polarized photons in the lower channel (as indicated by the tags).

The separation fallacy is to describe the vh-analyzer as if its effect by itself was to project an incident photon either into $|v\rangle$ in the upper channel or $|h\rangle$ in the lower channel–instead of only creating the above superposition state. The mistake of describing the unmeasured polarization analyzer as creating two beams of eigenstate polarized photons is analogous to the mistake of describing a particle as going through one slit or the other in the unmeasured-at-slits double-slit experiment–and similarly for the other separation experiments.

It is fallacious to reason that “we know the photons are in one polarization state in one channel and in the orthogonal polarization state in the other channel because that is what we find when we measure the channels,” just as it is fallacious to reason “the particle has to go through one slit or another (or one arm or another in the interferometer experiment) because that is what we find when we measure it.” The purely operational (or “Copenhagen”) description (“what we find when we measure”) does not take superposition seriously.

In the analyzer loop, no measurement (detector or blocker) is made after the vh-analyzer. It is followed by the inverse vh-analyzer $P^{-1}$ which has the inverse effect of removing the $U$ and $L$ tags so that a photon exits the loop in the state:

The inverse vh-analyzer does not “erase” the which-polarization information since there was no measurement to reduce the superposition state to eigenstate polarizations in the channels of the analyzer loop–in the first place.

The Stern-Gerlach experiment

We have seen the separation fallacy in the standard treatments of the double-slit experiment, which-way interferometer experiments, and in polarization analyzers. In spite of the differences between those separation experiments, there was that common (mis)interpretative theme of premature projection. Since the “logic” of the polarization analyzers is followed in the Stern-Gerlach experiment (with spin playing the role of polarization), it is not surprising that the same fallacy occurs there.

And again, the fallacy is revealed by considering the Stern-Gerlach analogue of an analyzer loop. The idea of a Stern-Gerlach loop seems to have been first broached by David Bohm [1951, 22.11] and was later used by Eugene Wigner [1979]. One of the few texts to consider such a Stern-Gerlach analyzer loop is The Feynman Lectures on Physics: Quantum Mechanics (Vol. III) where it is called a “modified Stern-Gerlach apparatus” (p. 5-2).

Ordinarily texts represent the Stern-Gerlach apparatus as separating particles into spin eigenstates denoted by, say, $+S,0S,-S$ . But as in our other examples, the apparatus does not project the particles to eigenstates. Instead it creates a superposition state so that with a detector in a certain position, then as the detector causes the collapse to a spin eigenstate, the detector will only see particles of one spin state. Alternatively 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. In the context of the Stern-Gerlach 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….” [p. 5-9, italics in original]

The Separation Fallacy

We have seen the same fallacy of interpretation in two-slit experiments, which-way interferometer experiments, polarization analyzers, and Stern-Gerlach experiments. The common element in all the cases is that there is some ‘separation’ apparatus that puts a particle into a certain superposition of eigenstates in such a manner that when an appropriately positioned 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.’ The quantum erasers are elaborated versions of these simpler experiments, and a similar separation fallacy arises in that context.

A simple quantum eraser experiment

A simple quantum eraser can be devised using a single beam of photons as in Hilmer and Kwiat [2007]. We start with the standard two-slit setup.

After the two slits, a photon could be schematically represented as being in a superposition state $|s1\rangle+|s2\rangle$ (where $s1$ and $s2$ stand for the two slits) which evolves with interference to give the familiar pattern on the far wall.

A $-45^{\circ}$ polarizer might be placed in front of the two slits to control the incoming polarization to obtain a half-half split when a horizontal polarizer is placed behind of slit 1 and a vertical polarizer behind slit 2.

After the two slits and the polarizers, a photon is in a state that entangles the spatial slit states and the polarization states which might be represented as: $|s1\rangle \otimes |h\rangle +|s2\rangle \otimes |v\rangle$ . But as this superposition evolves, it cannot be separated into a superposition of the slit-states as before, so the interference disappears. The so-called “which slit” information is said to be marked with the polarization information.

Then a $+45^{\circ}$ polarizer is inserted between the two-slit screen and the wall. This transforms the evolving state to:

$|s1\rangle\otimes|45^{\circ}\rangle+|s2\rangle\otimes|45^{\circ}\rangle = [|s1\rangle+|s2\rangle]\otimes |45^{\circ}\rangle$

so that the $|s1\rangle+|s2\rangle$ term will show interference in a “fringe” pattern when the $+45^{\circ}$ polarized photons hit the wall. If we had inserted a $-45^{\circ}$ polarizer, then again interference in an “antifringe” pattern would appear as the $-45^{\circ}$ polarized photons hit the wall. The sum of the fringe and antifringe patterns gives the no-interference pattern of the previous figure.

A common description of this type of quantum eraser experiment is that the insertion of the h,v polarizers “marks” the photons with “which-slit information” (previous figure without $+45^{\circ}$ polarizer ) that destroys the interference–even if the horizontal or vertical polarization is not measured at the wall. If the horizontal or vertical polarization was measured at the wall, then the evolved superposition state $|s1\rangle \otimes |h\rangle +|s2\rangle \otimes |v\rangle$ would collapse to the evolved version of $|s1\rangle$ (if h was found) or $|s2\rangle$ (if v was found). This is said to reveal the so-called “which-slit information” that the photon went through slit 1 or slit 2, i.e., that at the slits, the photon was already in the state $|s1\rangle$ or $|s2\rangle$ instead of being in the superposition state. By incorrectly inferring that the photon was in one state or the other at the slits–while it would have to “go through both slits” to yield the interference pattern obtained by inserting the $+45^{\circ}$ polarizer–we seem to be able to retro-cause the particle to go through one slit or both slits by withdrawing or inserting the $+45^{\circ}$ polarizer after a photon has traversed the two slits.

It is precisely the separation fallacy that leads to this inference of retrocausality. In the situation of prevous figure (before inserting the $+45^{\circ}$ polarizer), a photon stays in a superposition state $|s1\rangle \otimes |h\rangle +|s2\rangle \otimes |v\rangle$ until it hits the wall. The slit states are indeed marked, tagged, labelled, or entangled with polarization states but this is incorrectly called “which-slit information” as if it could “reveal” that the photon was in the state $|s1\rangle$ or $|s2\rangle$ at the slits, i.e., that it went through slit 1 or slit 2.

Also it might be noted that the insertion of a $+45^{\circ}$ or $-45^{\circ}$ polarizer does not “restore” the original interference pattern but picks out the fringe or antifringe interference patterns out of the previous “mush” of hits.

Marvin Chester has developed a user-drivable model of this quantum eraser experiment to illustrate how it works.

Two photon quantum eraser experiment

We now turn to one of the more elaborate quantum eraser experiments (the treatment in the Wikipedia link is muddled but it gives the link to the original paper by Walborn et al. [2002] for this experiment). As noted in the Walborn paper, their quantum eraser experiment is an optical realization of type of quantum eraser suggested by Scully et al. [1991] using maser cavities and atoms.

A photon hits a down-converter which emits a “signal” p-photon entangled with an “idler” s-photon with a superposition of orthogonal $|x\rangle$ and $|y\rangle$ polarizations:

$|\Psi\rangle = \frac{1}{\sqrt{2}} [|x\rangle_{s}\otimes |y\rangle_{p}+|y\rangle_{s}\otimes|x\rangle_{p}]$ .

The lower s-photon hits a double-slit screen, and will show an interference pattern on the $D_{s}$ detector as the detector is moved along the x-axis.

Next two quarter-wave plates are inserted before the 2-slit screen with the fast axis of the one over slit 1 oriented at $+45^{\circ}$ to the x-axis and the one over the slit 2 with its fast axis oriented at $-45^{\circ}$ to the x-axis.

Then Walborn et al. [2002] give the overall state of the system as (where the s1 and s2 tags refer to the two slits):

$|\Psi\rangle = \frac{1}{2}[(|L\rangle_{s1}\otimes|y\rangle_{p}+ i|R\rangle_{s1}\otimes|x\rangle_{p})+ (|R\rangle_{s2}\otimes|y\rangle_{p}-i|L\rangle_{s2}\otimes|x\rangle_{p})]$ .

Then by measuring the linear polarization of the p-photon at $D_{p}$ and the circular polarization at $D_{s}$ , “which-slit information” is said to be obtained and no interference pattern recorded at $D_{s}$ .

For instance measuring $|x\rangle$ at $D_{p}$ and $|L\rangle$ 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 $|x\rangle$ polarization at $D_{p}$ and $|L\rangle$ polarization at $D_{s}$ , so only when the superposition state collapses to $|x\rangle$ for the p-photon and $|L\rangle$ for the s-photon do we get a hit at both detectors.

This is the analogue of the one-beam-splitter 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 arm-eigenstate. The phrase “which-slit” or “which-arm information” is a misnomer in that it implies the system was already in a slit- or arm-eigenstate and the so-called measurement only revealed the information. Instead, it is only at the measurement that there is a collapse or projection to an evolved slit-eigenstate (not at the previous ‘separation’ due to the two slits).

Walborn et al. indulge in the separation fallacy when they discuss what the so-called “which-path 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 $|\Psi\rangle$ ], 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. [Walborn et al. 2007, p. 4]

Firstly, the measurement that p has polarization x after the s photon has traversed the $\lambda/4$ plates and two slits [see their Figure 1] does not retrocause the s photon to already have “polarization y before hitting the $\lambda/4$ plates.” After photon p is measured with polarization x, then the two particle system is in the superposition state:

$i|R\rangle_{s1}\otimes|x\rangle_{p}-i|L\rangle_{s2}\otimes|x\rangle_{p}=[i|R\rangle_{s1}-i|L\rangle_{s2}]\otimes|x\rangle_{p}$

which means that the s photon is still in the slit-superposition state: $i|R\rangle_{s1}-i|L\rangle_{s2}$ . Then only with the measurement of the circular polarization states L or R at $D_{s}$ do we have the collapse to (the evolved version of) one of the slit eigenstates s1 or s2 (in their notation). It is an instance of the separation fallacy to infer “the passage of s through slit 1″ or “slit 2″, i.e., s1 or s2, instead of the photon s being in the tagged superposition state $|\Psi\rangle$ after traversing the slits.

Let us take a new polarization space basis of $|+\rangle = +45 ^{\circ}$ to the x-axis and $|-\rangle = -45 ^{\circ}$ to the x-axis. Then the overall state can be rewritten in terms of this basis as (see original paper for the details):

$|\Psi\rangle = \frac{1}{2}[(|+\rangle_{s1}-i|+\rangle_{s2})\otimes |+\rangle_{p}+ i(|-\rangle_{s1}+i|-\rangle_{s2})\otimes |-\rangle_{p}]$ .

Then a $|+\rangle$ polarizer or a $|-\rangle$ polarizer is inserted in front of $D_{p}$ to select $|+\rangle_{p}$ or $|-\rangle_{p}$ respectively. In the first case, this reduces the overall state $|\Psi\rangle$ to $(|+\rangle_{s1}-i|+\rangle_{s2})$ which exhibits an interference pattern, and similarly for the $|-\rangle_{p}$ selection. This is misleadingly said to “erase” the so-called “which-slit information” so 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 no-interference pattern obtained before inserting the polarizer. The polarizer simply selects one of the interference patterns out of the mush of their merged non-interference pattern. Thus instead of “erasing which-slit information,” it selects one of two interference patterns out of the both-patterns mush.

Even though the polarizer may be inserted after the s-photon has traversed the 2 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 non-delayed setup (except to increase the woo-woo factor). Instead of inserting the $|+\rangle$ or $|-\rangle$ polarizer in front of $D_{p}$ , insert it in front of $D_{s}$ and it would have the same effect of selecting $(|+\rangle_{s1}-i|+\rangle_{s2})$ or $(|-\rangle_{s1}+i|-\rangle_{s2})$ each of which exhibits interference. Then it is much like the simple quantum eraser of the previous section.

The delayed quantum eraser

If the upper arm is extended so the $D_{p}$ detector is triggered last (“delayed erasure”), the same results are obtained. The entangled state is collapsed at $D_{s}$ . A coincidence counter (not pictured) is used to correlate the hits at $D_{s}$ with the hits at $D_{p}$ for each fixed polarizer setting, and the same interference pattern is obtained.

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 $|+\rangle_{p}$ , then out of the mush of hits at $D_{s}$ obtained years before, the coincidence counter will pick out the ones from $|+\rangle_{s1}-i|+\rangle_{s2}$ which will show interference.

Again, the years-later $D_{p}$ detections do not retro-cause anything at $D_{s}$ , e.g., do not “erase which-way 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 years-earlier 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 post-selection of subensembles with maximal fringe visibility.” [Kwiat et al. 1999, p. 79]

The same sort of analysis could be made of the delayed choice quantum eraser experiment described in the paper by Kim et al. & Scully [Kim 2000]. A good analysis of this experiment which avoids the separation fallacy (and avoids any implication of retro-causality) is given by Brian Greene of PBS fame [Greene 2004, pp. 194-199].

References

Bohm, David 1951. Quantum Theory. Englewood Cliffs NJ: Prentice-Hall.

Dicke, Robert H. and James P. Wittke 1960. Introduction to Quantum Mechanics. Reading MA: Addison-Wesley.

Feynman, Richard P., Robert B. Leighton and Matthew Sands 1965. The Feynman Lectures on Physics: Quantum Mechanics (Vol. III). Reading MA: Addison-Wesley.

French, A.P. and Edwin F. Taylor 1978. An Introduction to Quantum Physics. New York: Norton.

Greene, Brian 2004. The Fabric of the Cosmos. New York: Alfred A. Knopf.

Hilmer, Rachel and Paul G. Kwiat 2007. A Do-It-Yourself Quantum Eraser. Scientific American. 296 (5 May): 90-95.

Kim, Yoon-Ho, R. Yu, S. P. Kulik, Y. H. Shih and Marlan O. Scully 2000. Delayed choice quantum eraser. Physical Review Letters. 84 (1 ) [quant-ph/9903047].

Kwiat, P. G. , P. D. D. Schwindt and B.-G. Englert 1999. What Does a Quantum Eraser Really Erase? In Mysteries, Puzzles, and Paradoxes in Quantum Mechanics. Rodolfo Bonifacio ed., Woodbury NY: American Institute of Physics: 69-80.

Scully, Marlan O., Berthold-Georg Englert and Herbert Walther 1991. Quantum optical tests of complementarity. Nature. 351 (May 9, 1991): 111-116.

Walborn, S. P., M. O. Terra Cunha, S. Padua and C. H. Monken 2002. Double-slit quantum eraser. Physical Review A. 65 (3).

Wheeler, John A. 1978. The “Past” and the “Delayed-Choice” Double-Slit Experiment. In Mathematical Foundations of Quantum Theory. A. R. Marlow ed., New York: Academic Press: 9-48.

Wigner, Eugene P. 1979. The Problem of Measurement. In Symmetries and Reflections, Woodbridge CT: Ox Bow Press: 153-170.

Zeilinger, Anton 2004. Why the Quantum? “It” from “bit”? A participatory universe? Three far-reaching challenges from John Archibald Wheeler and their relation to experiment. In Science and Ultimate Reality: Quantum Theory, Cosmology, and Complexity. John Barrow, Paul Davies and Charles Harper eds., Cambridge: Cambridge University Press: 201-220. Available at: http://www.quantum.at/zeilinger

Zeilinger, Anton 2008. On the Interpretation and Philosophical Foundation of Quantum Mechanics. In Grenzen menschlicher Existenz. Hans Daub ed., Petersberg: Michael Imhof Verlag. Available at: http://www.quantum.at/zeilinger

Published November 30, 2011

Filed in Applications,Philosophy,Quantum Mechanics and tagged delayed choice, double slit, measurement, quantum eraser, retrocausality, separation fallacy, Stern-Gerlach, superposition

22 replies

Write comment | Trackback

Comments

Luboš Motl wrote:

Good article. I quoted it at http://motls.blogspot.com/2011/12/contextuality-of-our-quantum-world-is.html

December 1, 2011 @ 1:37 am
James Gallagher wrote:

Hi,

yes the results of experiments are pretty trivial to predict via machinary of QM, sometimes people think you can carry out the analysis using classical Maxwell waves, so you might have emphasised why that doesn’t work.

But for the “years-later” case you would not have coincident detections at Dp and Ds, I assume you mean after applying an offset (of several years) to the Dp detections.

One question: in the case of the “years later” case with detections at Dp, does it matter if someone mucks around with the polarizer or maybe even swaps the polarizer for a different one in the intervening years, let’s say they replace the polarizer just the day before detections at Dp – would you still retreive an interference pattern in the (offset) coincidence matches?

December 1, 2011 @ 5:01 am
David Ellerman wrote:

Yes, in the years later case, you have to take into account the time difference to determine a coincidence–which I hope is not as hard as determining the spatial difference between CERN and Rome.
Concerning the question, you are gathering coincidences in the years-later D_p measurements so you need to keep the same polarizer in place at D_p during the years-later sampling period.

December 1, 2011 @ 6:49 am
James Gallagher wrote:

haha, yes the practical execution of such an experiment would be tricky, but let’s assume it is doable for argument’s sake. Now, while I agree with the majority of your post and the general point about the ‘separation fallacy’ I’m not sure it is such a clear case in the ‘year’s later’ delayed choice eraser (although many people would not agree with me here). I realise you only briefly mention this particular case at the end, but I think it is the one scenario where the analysis is not so easy.

eg Does the polariser at Dp have to be in place when the Ds detections are made or can it be put it in place (by a preprogrammed computer) at any time in the intervening years?

December 1, 2011 @ 8:13 am
David Ellerman wrote:

Any time in the intervening years. It will only determine whether the coincidence counter picks out the fringe or the antifringe interference pattern out of all the years-earlier hits recorded by D_s.

December 1, 2011 @ 8:35 am
James Gallagher wrote:

hmm, but doesn’t that imply the ‘separation fallacy’ your blog article is refuting?

The Ds detections are measured and fixed macroscopically (eg on computer disk), but the idler photons on their way to the Dp detector are still in a superposition of polarisation states until measured at the Dp detector (years later) – but to extract an interference pattern out of the ‘mush’ the detections at Dp must match the embedded pattern on the computer disk record of the Ds detections.

So, if you can place the polariser in front of Dp at any intervening time then the idler photons on their way to Dp are not in a superposition of polarisation states, because it must already be predetermined which ones will pass through the polariser and hit the detector Dp.

December 1, 2011 @ 9:52 am
David Ellerman wrote:

I should have numbered for formulas. Anyway, if we take that long superposition formula where everything is expressed in the polarization basis of |+> and |->, then it is collapsed years earlier at Ds. Each photon that hits Ds will come from a |+>-i|+> hit or a |->+i|-> hit, and Ds doesn’t know which (since it registered a hit but did not determine polarization). But that collapses the signal photon winging toward Dp into a fixed polarization state, either a |+>_p or |->_p respectively, due to the entanglement. If you had, say, a |+> filter on Dp, then it would pick out all the signal photons whose years-earlier idler partner gave a |+>-i|+> hit and thus would show the fringe interference pattern, and similarly with the |-> filter at Dp and the other antifringe interference pattern.

December 1, 2011 @ 10:17 am
James Gallagher wrote:

But if you are saying the Dp photons have a fixed polarisation before they are measured then you are essentially proposing hidden variables, which is equivalent to the ‘separation fallacy’ you discussed in the blog post.

The idler photons in this experimental scenario are not particularly special, in nature photons are regularly entangled with other photons which are ‘measured’ (by being absorbed in a collision for example) so you would have fixed polarisations all over the place.

December 1, 2011 @ 11:41 am
David Ellerman wrote:

No hidden variables, just entanglement. Look at the math, i.e., look at the long superposition formula (formula 14 in Walborn et al.) which gives the superposition state in the |+> and |-> basis of both photons before any collapse in the delayed case. When the idler photon state first collapses to, say, |+>-i|+> then its entangled signal photon has to collapse at that time to |+>, and similarly for the other option.

December 1, 2011 @ 3:25 pm
James Gallagher wrote:

I think I mixed up idler/signal designation, you’re calling the Ds photons the idler photons, ok let’s stick with that.

I still protest that the entangled signal photons can not collapse to a polarisation state BEFORE they are MEASURED, unless you are proposing hidden variables. In nature, in general, photons and other particles are in multi-particle entanglements, and if the “collapse” of a single particle in the multi-particle entanglement causes “collapse” of all the other particles (before they are measured) then you don’t have quantum superpositions you have definite predetermined values for various quantum states.

I don’t think this is how nature works, I think the Dp polarizer must be in place at the time the Ds photons are measured, and must remain in place until the Dp photons are detected in order that the CORRELATIONS between the Ds detections and Dp detections are not destroyed.

This would be a doable experiment over earth based satellite distances, I would like to see it performed, but I’m sure it has to work like this.

The Walborn experiment makes this explicit, if we have a ‘years-later’ setup and the location where the Dp detector and polariser is positioned gets destroyed before the Dp photons reach it, then we have a stream of Dp bound photons in a fixed polarisation state even though they have never been and maybe never will be measured.

December 2, 2011 @ 5:04 am
James Gallagher wrote:

I meant “… then we *would* have a stream of Dp bound photons in a fixed polarisation state…” (which can’t be the case without hidden variables)

December 2, 2011 @ 5:53 am
David Ellerman wrote:

I think your difficulty is not with this specific experiment but with understanding the effects of entanglement (which Einstein’s famously referred to as “spooky action at a distance”) as requiring hidden variables. In the delayed choice version, the signal p-photon collapses to a fixed plus or minus polarization state, as per formula 14 in the Walborn et al. paper, when its entangled idler s-photon hits the Ds detector (which does not measure polarization so the experimenter does not know the already determined plus or minus polarization of the signal p photon until the years later measurement at Dp).
Far from requiring hidden variables, Bell’s inequality and the later experiments showing it was violated put some of the last nails in the coffin of hidden variable theories. Surely you know I can’t go into that long story in these comments.

December 2, 2011 @ 7:59 am
James Gallagher wrote:

Yes I know that hidden variables are not possible (except in some crazy explicitly non-local constructions) that’s why I’m trying to say the analysis of the delayed choice erasure is not so easy, the entangled photon arm doesn’t just add a ‘woo-hoo’ factor it very clearly brings to the forefront the confrontation of QM locality and relativistic locality.

To avoid hiding behind the math, which is a bit of a cop out (yes we know it makes the correct predictions for MEASUREMENTS), I asked a simple practical question regarding the placement of the Dp polarizer, and I have to say, I think your answer is incorrect. Although until the experiment is actually carried out I can not prove this.

The point is, that with the “years-later” DCQE we really need a consistent theory of the time evolution of macroscopic wavefunctions (even universe-wide) to determine what will happen. All the other scenarios you mentioned are static setups which can just be analysed with the standard QM machinery.

December 2, 2011 @ 10:37 am
maalpu wrote:

How does superposition reasoning explain the apparent erasure in a simple polarised tagging/erasing case, where the standard double slit interference pattern is as expected destroyed by adding different circular polarisers ahead of each slit, but then reinstated by erasing path information with a linear polariser ahead of the detector (as detailed in Section V. THE “QUANTUM ERASER” starting on page 5 of http://arxiv.org/abs/quant-ph/9501016, with the “removed” and “missing” images actually at the end of the PDF) ?

December 5, 2011 @ 11:42 pm
David Ellerman wrote:

@maalpu The separation fallacy is illustrated first in the simpler cases of the double-slit experiment and the which-way experiment to drive home the point that a certain detector placement in physical space (and in polarization space as well in the q-eraser case) will only register one collapsed slit or arm-eigenstate. But that does not mean the particle was in that eigenstate all along before the measurement. The separation apparatus always puts it in superposition state that evolves (with a certain entanglement with the apparatus) until a measurement is made.
In the case you reference, the photon is in a superposition state until it hits the linear polarizer in front of the detector (and thus it didn’t go through one slit or the other so there as no such information to erase). That polarizer selects either the fringe or antifringe interference pattern out of what would otherwise be a mush of hits at the detector showing no overall interference pattern. Incidently, this is exactly the point I make in the last paragraph of the section entitled “Quantum eraser experiment” about the entanglement between the two particles not being essential in the non-delayed case.
Look at the long superposition formula (formula 14 in Walborn et al.) which gives the superposition state in the |+> and |-> basis of both photons before any collapse. The inserted linear polarizer for |+> collapses the idler photon state |+>_s1-i|+>_s2 which shows the fringe interference pattern, and similarly for the other option.

December 6, 2011 @ 7:38 am
maalpu wrote:

Which can be summarised as: the basic double-slit shows interference because both paths can contribute to every detection; adding circular polarizers before (or after?) the slits shows no interference because the polarizations are mutually exclusive and cannot be combined so only one path can contribute to each detection; adding the linear polarizer to that gives interference because now again both paths can contribute to each detection ?

December 7, 2011 @ 12:22 am
David Ellerman wrote:

@maalpu Yes, I have added a section to the paper on A Simple Quantum Eraser that treats this case (using linear rather than circular polarizers) and I added a user-driven demo supplied by Marvin Chester who wrote the text, Primer of Quantum Mechanics (Dover).

December 7, 2011 @ 5:21 pm
maalpu wrote:

Key to all this is that polarisers, splitters, mirrors etc, do not collapse the evolving state, while “detectors” do.

But why – all interact with the particle/wave state, so what makes “detectors” different ?

For example, does not a mirror recoil when hit, which is in principle something measurable, and so should qualify as detection (or does Quantum Mechanics not honor Newton’s action-reaction principle) ?

December 11, 2011 @ 1:48 pm
David Ellerman wrote:

@maalpu Roughly put, a separation apparatus creates a type of superposition state while the detectors go from a superposition state to an eigenstate. But a more detailed story would see a certain type of entanglement between the incident particle and the apparatus which does not destroy the information in the incident particle’s state since it can be reversed in the loop apparatus. Paul Kwiat’s treatment of quantum erasers has emphasized this role of entanglement between the particle and apparatus.
In the different cases, there should be some detailed solid state physics story about what happens. That sort of detailed scrutiny might be necessary to answer your mirror question. But off-hand, I would think that a mirror plus recoil detector would be a measurement apparatus but that the half-silvered mirror by itself would not be.
I am only trying to isolate a class of examples where there seems to be a common fallacy of interpretation that when combined with the delayed choice variations will lead to inferences of retrocausality. If we reject the retrocausality implications, then we need to go back to where the original fallacy of interpretation got us into that trouble, and that is what the paper tried to do.

December 12, 2011 @ 7:43 am
Dan Tannenbaum wrote:

You do realize that you have not only kicked the sacred cow, you have trammeled it to bits!!! Wonderful!!!!

December 21, 2011 @ 3:57 pm
maalpu wrote:

I can see how that works in most cases but Kim et al (http://arxiv.org/abs/quant-ph/9903047) has me puzzled. They certainly are just selecting points from the mush, and their twelve step high-level equation sequence gets the right numbers, but what is the in-principle relationship between signal detection and D1 versus D2 detection?

The final splitter should have randomly sent idlers to D1 or D2, but somehow it manages to selectively send to D1 idlers that pick out an interference pattern at D0 (ignoring any pattern at D1, which is used as a simple yes/no hit detector), and similarly for D2 (whose corresponding D0 pattern is complementary so the two add to a roughly smooth curve).

And why can idler paths A and B both contribute to those hits for each photon, when only one idler path ever contributes to the D3 and D4 hits ?

January 13, 2012 @ 9:47 pm
David Ellerman wrote:

In the Kim et al. paper, the in-principle relationship between the signal photon and the idler detection at D1 or D2 is their entanglement given by the down-converter. Since there is no phase-shifter in the arms (which would allow one to find at pattern at D1 or at D2), the only interference detected is at D0 as x is varied.
For your last question, the point is the presence of the beam-splitter BS which is in both arms, as opposed to BSA or BSB, which are only in one arm (so those hits correlated with D0 have the which-arm information and thus no interference pattern).

January 14, 2012 @ 1:59 pm