‘Well! I’ve often seen a cat without a grin,’ thought Alice, ‘but a grin without a cat! It’s the most curious thing I ever saw in my life!’ – Alice in Wonderland

Pictured: The infamous Cheshire Cat. Its smile is in the top right but the rest of the cat is elsewhere! How does that work?

The Cheshire Cat may seem like the product of a mind descending into madness, but then so is Quantum Mechanics. It will therefore surprise absolutely no one that a quantum version of the Cheshire Cat story exists. The Quantum Cheshire Cat also continues the ongoing fascination that physicists have with cute fluffy cats (see Schrodinger’s cat for another example). I am a dog person myself, but I digress.

Let us break down the Cheshire cat into its parts. The cat has a large toothy grin, you see, but a cat grinning is not the weirdest part. The weirdness comes along when the body of the Cheshire disappears and then reappears somewhere else, but the grin stays behind, floating suspiciously without a body attached.

Inspired by this, Aharanov et al. presented a way to translate this previously fictional weirdness into Quantum Mechanics. Unable to perform experiments on actual cats, they proposed an experiment using photons instead. Photons have an intrinsic property called polarization, which basically tells you the direction that the electromagnetic waves are oscillating. In Aharanov’s experiment, the photons are set up such that they can move along on 2 possible paths – Left and Right, and the polarizations are along 2 possible directions – Horizontal and Vertical. The polarization is a part of the photon, since it doesn’t make much sense to speak of which direction a photon is oscillating in when the photon is not there.

In Aharanov’s experiment, however, it is possible to have the photon in the left path but the polarization on the right! Suppose we implement Aharanov’s experiment and we prepare the photons in exactly the same manner many times in a row. If we were to make a measurement to see if the photon travelled the left path, the detector will always click, 100% of the time. We will therefore conclude that the photon is travelling the left path, not a difficult conclusion to make. However, if we were to make a polarization measurement on the right path, that detector will start giving us clicks! This means there is photon polarization on the right path! We have previously ascertained, with 100% confidence that a photon prepared in the same manner will always travel the left path, so the photon must have performed a “Cheshire Cat”, by moving in the left path, but having its polarization appear on the right.

The above argument is called counterfactual reasoning. Counterfactual reasoning basically refers to arguments made on the following basis: we didn’t do this, but if we did, this would have happened instead. In the previous paragraph, we are measured the polarization on the right path, but suppose we didn’t and measured the path of the photon instead, then we will conclude that the photon is always on the left, but this somehow contradicts our polarization experiment. The apparent weirdness, or paradox, therefore arises because we employed counterfactual logic, since we didn’t (and couldn’t) measure the path of the photon on the same photon we are measuring the polarization, but made our conclusion supposing that we did.

Aharanov et al. of course realized this, and thinking that may be an avenue for criticism, also devised an alternative reasoning based off of weak measurements that does not rely on counterfactual reasoning. I personally don’t think it is a problem per se, but instead perfectly illustrates how our everyday ‘common sense logic’ simply does not apply to quantum mechanics. In any case, it is fun to ponder why counterfactual reasoning does not work in quantum mechanics. 

Link to Aharanov et al.’s paper : [1202.0631] Quantum Cheshire Cats – arXiv.org