## 1.2 Quantum interference: the failure of probability theory

Modern mathematical probability theory is based on three axioms, proposed by Andrey Nikolaevich Kolmogorov (1903–1987) in his monograph with the impressive German title *Grundbegriffe der Wahrscheinlichkeitsrechnung* (“Foundations of Probability Theory”).
The **Kolmogorov axioms** are simple and intuitive:^{19}

- Once you identify all elementary outcomes, or events, you may then assign
*probabilities*to them, where… - … a
*probability*is a number between0 and1 , and an event which is certain has probability1 . - Finally, the probability of any event can be calculated using a deceptively simple rule — the
**additivity axiom**:*whenever an event can occur in several mutually exclusive ways, the probability for the event is the sum of the probabilities for each way considered separately.*

Obvious, isn’t it?
So obvious, in fact, that probability theory was accepted as a mathematical framework, a language that can be used to describe actual physical phenomena.
Physics should be able to identify elementary events and assign numerical probabilities to them.
Once this is done we may revert to mathematical formalism of probability theory.
The Kolmogorov axioms will take care of the mathematical consistency and will guide us whenever there is a need to calculate probabilities of more complex events.
This is a very sensible approach, apart from the important fact that *it does not always work*!
Today, we know that probability theory, as ubiquitous as it is, fails to describe many common quantum phenomena.
In order to see the need for quantum theory let us consider a simple experiment in which probability theory fails to give the right predictions.

### 1.2.1 The double-slit experiment

In a double-slit experiment, a particle (such as a photon) emitted from a source

The particle emitted from ^{20} events.
With the two slits open, allowing the particle to take either path, probability theory declares (by the Kolmogorov additivity axiom) that the particle should reach the detector with probability *not* what happens experimentally!

Let us see what happens if we instead follow the two “quantum rules”: first we add the amplitudes, then we square the absolute value of the sum to get the probability.
Thus, the particle will reach the detector with probability
*modified* by the **interference term** **relative phase** **destructive** interference) or positive (**constructive** interference), leading to either suppression or enhancement (respectively) of the total probability

The algebra is simple; our focus is on the physical interpretation.
Firstly, note that the important quantity here is the *relative* phase *either* through the upper or the lower slit, because it has travelled through *both*.
In the same way, quantum computers follow, in some tangible way, *all* computational paths simultaneously, producing answers that depend on *all* these alternative calculations.
Weird, but this is how it is!

Secondly, what has happened to the additivity axiom in probability theory?
What was wrong with it?
One problem is the assumption that the processes of taking the upper or the lower slit are mutually exclusive; in reality, as we have just mentioned, the two transitions *both occur, simultaneously*.
However, we cannot learn this from probability theory, nor from any other *a priori* mathematical construct — we can only observe this by repeated scientific experiments in our physical world.^{21}

There is no fundamental reason why Nature should conform to the additivity axiom.

We find out how nature works by making “intelligent” guesses, running experiments, checking what happens and formulating physical theories.
If our guess disagrees with experiments then it is wrong, so we try another intelligent guess, and another, etc.
Right now, quantum theory is the best guess we have: it offers good explanations and predictions that have not been falsified by any of the existing experiments.
This said, rest assured that one day quantum theory *will* be falsified, and then we will have to start guessing all over again.

It’s an interesting coincidence that the two basic ingredients of modern quantum theory — probability and complex numbers — were discovered by the same person, an extraordinary man of many talents: a gambling scholar by the name of Girolamo Cardano (1501–1576).↩︎

That is, if one happens then the other one cannot. For example, “heads” and “tails” are mutually exclusive outcomes of flipping a coin, but “heads” and “6” are

*not*mutually exclusive outcomes of simultaneously flipping a coin and rolling a dice.↩︎According to the philosopher Karl Popper (1902–1994) a theory is genuinely scientific only if it is possible, in principle, to establish that it is false. Genuinely scientific theories are never finally confirmed because, no matter how many confirming observations have been made, observations that are inconsistent with the empirical predictions of the theory are always possible.↩︎