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

  1. Once you identify all elementary outcomes, or events, you may then assign probabilities to them, where…
  2. … a probability is a number between 0 and 1, and an event which is certain has probability 1.
  3. 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 emitted from a source S can reach a detector D by taking two different paths, e.g. through an upper or a lower slit in a barrier between the source and the detector. After sufficiently many repetitions of this experiment we can evaluate the frequency of clicks in the detector D and show that it is inconsistent with the predictions based on probability theory. Let us use the quantum approach to show how the discrepancy arises.

The particle emitted from S can reach detector D by taking two different paths, which are assigned probability amplitudes z_1 and z_2, respectively. We may then say that the upper slit is taken with probability p_1=|z_1|^2 and the lower slit with probability p_2=|z_2|^2. These are two mutually exclusive20 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 p_1+p_2=|z_1|^2+|z_2|^2. But this is 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 \begin{aligned} p &= |z|^2 \\& = |z_1 + z_2|^2 \\& = |z_1|^2 + |z_2|^2 + z_1^\star z_2 + z_1 z_2^\star \\& = p_1 + p_2 + |z_1||z_2|\left( e^{i(\varphi_2-\varphi_1)} + e^{-i(\varphi_2-\varphi_1)} \right) \\& = p_1 + p_2 + \underbrace{2 \sqrt{p_1 p_2} \cos(\varphi_2-\varphi_1)}_{\text{interference terms}} \end{aligned} \tag{$\ddagger$} where we have expressed the amplitudes in their polar forms: \begin{aligned} z_1 &= |z_1|e^{i\varphi_1} \\z_2 &= |z_2|e^{i\varphi_2}. \end{aligned} The appearance of the interference terms marks the departure from the classical theory of probability. The probability of any two seemingly mutually exclusive events is the sum of the probabilities of the individual events p_1 + p_2 modified by the interference term 2\sqrt{p_1p_2}\cos(\varphi_2-\varphi_1). Depending on the relative phase \varphi_2-\varphi_1, the interference term can be either negative (giving what we call destructive interference) or positive (constructive interference), leading to either suppression or enhancement (respectively) of the total probability p.

The algebra is simple; our focus is on the physical interpretation. Firstly, note that the important quantity here is the relative phase \varphi_2-\varphi_1 rather than the individual phases \varphi_1 and \varphi_2. This observation is not trivial at all: if a particle reacts only to the difference of the two phases, each pertaining to a separate path, then it must have, somehow, experienced the two paths, right? That is, we cannot say that the particle has travelled 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.


  1. 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).↩︎

  2. 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.↩︎

  3. 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.↩︎