Another common textbook example of the fallacy is the treatment of polarization analyzers such as calcite crystals that are incorrectly said to create two orthogonally polarized beams in the upper and lower channels, say \(\left| v\right\rangle \) and \(\left| h\right\rangle \) from an arbitrary incident beam (Fig. 5).

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 vice-versa.

But here again, the story is about the spatial placement of the detector (or blocking mask); 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, a superposition state that correlates the compatible polarization and spatial modes for a particle. This could be schematically represented as:

$$\begin{aligned} \left| \text {vertical polarization, upper channel}\right\rangle +\left| \text {horizontal polarization, lower channel}\right\rangle . \end{aligned}$$

There is a certain ambiguity in the practice of representing two eigenstates or eigenvalues in a ket: \(\left| \text {state 1, state 2}\right\rangle \). This could be interpreted as shorthand for:

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 single-particle states.

We are using in this section the second single-particle version of the correlated superposition:

$$\begin{aligned} \left| \text {vertical, upper}\right\rangle +\left| \text {horizontal, lower}\right\rangle . \end{aligned}$$

This sort of a superposition state is thus formally similar to an entangled state but only involves a single particle.

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.

If the analyzer had in fact induced a collapse 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, \(\left| 45^{\circ }\right\rangle \) 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 [8] (Fig. 6).

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 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. Thus when the undetected particle rejoins the undivided “interstate,” there was no which-way information to be erased. 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 \(\left| \psi \right\rangle \) such as \(\left| 45^{\circ }\right\rangle \) in the above example. This polarization state can be represented or resolved in terms of the \(vh\)-basis as:

$$\begin{aligned} \left| \psi \right\rangle =\left\langle v|\psi \right\rangle \left| v\right\rangle +\left\langle h|\psi \right\rangle \left| h\right\rangle . \end{aligned}$$

The effect of the \(vh\)-analyzer \(P\) might be represented as correlating the vertical and horizontal polarization states with the upper (\(U\)) and lower (\(L\)) channels so the \(vh\)-analyzer puts an incident photon into the one-particle correlated superposition state:

$$\begin{aligned} \left\langle v|\psi \right\rangle \left| v,U\right\rangle +\left\langle h|\psi \right\rangle \left| h,L\right\rangle , \end{aligned}$$

not into a mixture of an eigenstate of \(\left| v\right\rangle \) in the upper channel (\(\left| v,U\right\rangle \)) or an eigenstate \(\left| h\right\rangle \) in the lower channel (\(\left| h,L\right\rangle \)).

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.

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 \(L\) spatial modes from the superposition state \(\left\langle v|\psi \right\rangle \left| v,U\right\rangle +\left\langle h|\psi \right\rangle \left| h,L\right\rangle \) so that a photon exits the loop in the superposition state:

$$\begin{aligned} \left\langle v|\psi \right\rangle \left| v\right\rangle +\left\langle h|\psi \right\rangle \left| h\right\rangle =\left| \psi \right\rangle . \end{aligned}$$

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 inverse \(vh\)-analyzer does remove the correlation with the two spatial modes so the original state \(\left\langle v|\psi \right\rangle \left| v\right\rangle +\left\langle h|\psi \right\rangle \left| h\right\rangle =\left| \psi \right\rangle \) is restored.