Does the Cheshire Cat exist in the world of quantum mechanics?

The Cheshire Cat is a strange cat that appears in Lewis Carroll's 'Alice's Adventures in Wonderland'. It is a cat that grins, and even more strangely, it begins to disappear from its tail, eventually vanishing completely, leaving only its grin behind.

People including Aharonov, one of the proponents of weak values, have written a paper [1] stating that the same phenomenon as this Cheshire Cat can be realized in quantum mechanics.

[1] Yakir Aharonov, Daniel Rohrlich, Sandu Popescu, Paul Skrzypczyk, http://arxiv.org/abs/1202.0631 .

Instead of a cat, they considered a single photon. This photon has a spin degree of freedom called polarization, which can be right-handed or left-handed. They argued that if the spatially localized photon body is the Cheshire Cat's body, then the polarization can be considered its grin. By using a beam splitter and a method called 'weak measurement', the photon's body (spatial degree of freedom) and its polarization degree of freedom can be spatially separated. Aharonov and his colleagues argue that this property is an interesting phenomenon of weak values that could have many applications, as it suggests the possibility that the mass and charge of elementary particles could also be separated from the body like the cat's grin.

I noticed the following article for the general public that mentions their research.

So, this time, let's think in a little more detail about whether the 'quantum Cheshire Cat' really exists.

First, the weak value of a physical quantity corresponding to a Hermitian matrix A is determined for two states, a pre-selected state |Ψ> and a post-selected state |Φ>,

and is defined by the following formula. In general, it takes a complex value, and if the inner product of the two states in the denominator is zero, it is a quantity that can diverge if the numerator is not zero. The physical interpretation of such weak values has not yet been established, and it is a situation that continues to be debated among researchers. Aharonov and his colleagues are considering these weak values for the following optical two-path experiment of a single photon.

The path of a single photon incident from the lower-left region is stochastically divided into path 1 and path 2 by the first beam splitter. Then, a quantum measurement is performed in the upper-right region where the two paths merge again.

Here, let the state where the photon is in path 1 be |1〉, and the state where it is in path 2 be |2〉. Also, as basis vectors for the polarization state of this photon, we choose the eigenvectors of the z-component of the Pauli matrix.

Whether a single photon is in path 1 or path 2 can be determined by measuring the physical quantity corresponding to the projection matrix in Equation (3) below.

In this case, since we are only interested in which path the photon took, the part corresponding to the polarization degree of freedom of the photon is the identity matrix I. Aharonov and his colleagues considered the pre-selection state of equation (4) for this photon in the lower-left region.

Then, we perform measurements in the upper-right region and collect only the data for photons that resulted in the following post-selection state.

Next, we discuss the weak value of the physical quantity. First, the inner product of the two states is given by

Next, in the case where the photon passes through path 1, we have

and therefore

is obtained. This means that the weak value for the projection matrix of the photon being in path 1 is zero.

Similarly, let us analyze the case for path 2.

From

it is calculated that the weak value of the projection matrix for the photon being in path 2 is

With this result, Aharonov and his colleagues argue that in this experiment, the photon passes only through path 2 and does not pass through path 1. Next, with the same setup, we calculate the weak value of the z-component of the photon's polarization in each path.

First, since

is calculated, we obtain

Therefore, the weak value of the z-component of the polarization of the photon in path 1 is 1.

For the polarization z-component of the photon in path 2

from

it follows that the weak value of the z-component of the polarization of the photon in path 2 vanishes.

From this result, they argue that although the photon itself is in path 2, only the property of the photon's polarization exists in path 1 and does not exist in path 2. They explained in their paper that this means the polarization degree of freedom, as the "grin" of the Cheshire Cat, has separated from the photon itself.

However, should we take their claim at face value?

In the first place, there is no strong reason to claim that the photon really passed through path 2 using the weak value results of equations (7) and (8). Similarly, there is no reason to claim from the result of equation (11) that polarization does not exist in path 2. Weak values are strange quantities that can become complex numbers rather than real numbers depending on other pre- and post-selections. For example, while the expectation value of a projection matrix P can be interpreted as a probability taking a value between 0 and 1, the weak value of that P can take values from negative infinity to positive infinity even when limited to real numbers, and can further become a complex number. Just because such a quantity happens to be 0 or 1, there is no rationality in judging that the phenomenon absolutely will not occur or will definitely occur. In other words, as an interpretation, it is not justified at all at present.

Furthermore, the polarization degree of freedom as the "grin" still remains in the photon in path 2 even from the perspective of this weak value. For example, when a photon is in path 2, consider the following three components of polarization.

Specifically, for the x-component,

since the calculation

can be performed, we obtain

Equation (12): Weak value of the polarization x-component of the photon in path 2

Equation (13): Square of the absolute value of the polarization degree of freedom of the photon in path 2

Equation (14): Weak value for the square of the absolute value of the polarization degree of freedom of the photon in path 2

This calculation shows that it has not vanished in path 2 either. Ultimately, the proposal by Aharonov and his colleagues is neither a mysterious "quantum Cheshire Cat" nor a mystical "quantum out-of-body experience." It is something like a "hallucination" created by forcibly interpreting weak values as probabilities. Also, while there are attempts to treat weak values as reality, it is essential that they are "statistical quantities" that can only be evaluated by repeating the same experiment many times, unlike ordinary physical quantities such as spin or energy whose values can be clearly observed in a single experimental trial. If weak values were real, there should be a method to measure a weak value in a single experimental trial, but no such method has been proposed to date. From this as well, I think it is a reasonable view that "weak values are real" is an untenable argument.