## Quantum physics against intuition – Part II

In the first post we discussed the fact that classical first-order logic is distributive, that is, pizza and (lemonade or water) is the same as (pizza and lemonade) or (pizza and water); or symbolically,

$x \land (y \lor z) = (x \land y) \lor (x \land z)$ .

This time the aim will be to come up with an example demonstrating that this very intuitive identity does not always hold in quantum mechanics. To do that, we will need the uncertainty principle discussed in the previous post.

Quantum cyclist

We are going to use the uncertainty principle for position and momentum to construct a system which does not obey the distributive law. To make the numbers a bit simpler, we take $\hbar=1$, so the uncertainty relation looks like:

$\Delta X \Delta P\geq \frac{1}{2}$ .

Recall the example with a cyclist from the first post, we observed that the cyclist being in some interval and having some velocity is the same as the cyclist being in the first half of the interval with the same velocity or the cyclist being in the second half of the interval with the same velocity. Now consider a (tiny!) quantum cyclist; for concreteness, suppose the cyclist is in the interval $[0,1]$ and has the momentum in the interval $[0,\frac{1}{2}]$. For simplicity, we take the uncertainty to be the length of the interval1, so we are saying that the cyclist is equally likely to be anywhere between $0$ and $1$ and is equally likely to have any momentum between $0$ and $\frac{1}{2}$. Hence we have $\Delta X = 1$ and $\Delta P = \frac{1}{2}$. Now let $x$, $y$ and $z$ be the following statements about our system (i.e. about the cyclist):
$x$ = ‘cyclist has the momentum in $[0,\frac{1}{2}]$
$y$ = ‘cyclist is in $[0,\frac{1}{2}]$
$z$ = ‘cyclist is in $[\frac{1}{2},1]$‘.
The distributive law is:

$x \land (y \lor z) = (x \land y) \lor (x \land z)$ .

Note that the left-hand side of this identity is precisely what we have described above; the cyclist is in $[0,1]$ with momentum in $[0,\frac{1}{2}]$. We calculate $\Delta X\Delta P = 1\cdot \frac{1}{2} = \frac{1}{2}$, which satisfies the uncertainty condition, and so the system is physically possible. On the right-hand side, however, we have $(x \land y)$, that is, the cyclist is in $[0,\frac{1}{2}]$ with momentum in $[0,\frac{1}{2}]$, giving both $\Delta X$ and $\Delta P$ as $\frac{1}{2}$. But this violates the uncertainty bound, since $\Delta X\Delta P = \frac{1}{2}\cdot\frac{1}{2} = \frac{1}{4}$, which is certainly smaller than $\frac{1}{2}$! Since $(x \land z)$ gives the same uncertainties, we must conclude that both terms on the right-hand side are physically impossible, and thus false. This makes all of the right-hand side false; we must, therefore, conclude that this identity cannot hold in this case, as it equates a true statement about the physical system with a false one.

The example above raises many questions for classical logic. Must we conclude that its axioms and rules of inference don’t always hold? If yes, what would be the axioms, and how would they account for the fact that classical logic is distributive? If no, how do we account for the anomaly described above? It is not even clear if there should be one formal system of reasoning flawlessly applicable in all situations to all possible systems. No matter the answers to these questions, the example certainly opens up the space for development of a formal system correctly describing the logic of quantum mechanics.2

1This is actually not quite correct, e.g.  $\Delta X$ should really be $\frac{1}{\sqrt{12}}$. We can, however, get the uncertainties we want by scaling the intervals accordingly, but this doesn’t really contribute to the understanding, and so we drop the scaling for clarity.