Today’s video is short, but it introduces a topic dear to me: **How do we know that quantum superpositions are “real” superpositions?** ?

The theme of this video was actually born from the setting of a futuristic video about quantum decoherence. When we talk about quantum superposition, we generally say that the signature of superposition is the ability to perform interference. This is well illustrated already in the case of the interferences that can result with electrons and Young’s slits. We have the phenomenon of interference because what happens when the two slits are open is not the sum of what happens when both slits are open, which leads us to say that d in a certain way the electron passed through both slits at the same time.

Mathematically, it comes from the fact that the probability amplitude associated with the sum of the two configurations is not the sum of the amplitudes of each

\(|\Psi_1 + \Psi_2 | ^2 \neq | \Psi_1 | ^2 + | \Psi_2 | ^2\)

But if intrusions are really a general way to highlight overlays, then **It should be possible to interfere with any quantum system** ? And especially in the simple case of a two-state system (aka one qbit). So I asked myself what is the lowest experiment I can imagine, the results of which highlight the overlaps in the two-state system, and this in the most extreme and amazing way possible (in a similar way to what I said about the quantum cake of Bell inequalities).

I wanted something very simple, which if we present the results to someone, make them realize that yes, in fact, the superposition phenomenon is very “real”, and by that I mean it is not a question and it is not out of simple ignorance the true state of the system. This is how I came to imagine my thought experiment with the machine and the quantum bits.

This way of phrasing matches my knowledge of the simplest possible way of explaining interventions: **One observable, two classic cases, and the counterintuitive outcome to the extreme**.

As I mentioned in the video, my “machine” actually does a fairly simple operation: rotate \(\pi / 4\) in a space of Hilbert states. Now cycles of this type are easy to get by evolution operators. Let’s break it down a bit for those who want to.

The rotation in a two-state Hilbert space is simply represented by the matrix \(SU(2)\)

It is therefore a monadic shift (as an evolution operator should be), and thus one might wonder how it was created as an evolution operator. For this we need a Hamiltonian since we have it

\(U = \exp (\frac{i}{\hbar }H t)\)

Fortunately, there is a theorem that tells us that any unitary matrix can be put into exponential form. Here we will take the Hamiltonian of the form

It is, in a way, one of the simplest Hamiltonians one can imagine that couples the two fundamental states of a system as closely as possible. Once exponential, this Hamiltonian will generate an evolution factor which is a simple rotation in the Hilbert space of states, where the angle of rotation is proportional to the time of motion t.

If we make this verb Hamiltonian for a period like \(\theta = \pi / 4\), we get a twist which is what I wanted in the video. IThe base states \(|P \rangle \) and \(|F \rangle \) are transmitted via overlays, while the superposition state \(|P \rangle + | F \rangle \) will be transmitted over \(|P \rangle \ ) (or \(|F \rangle \) depending on which direction we’re headed).

Taking the other values of \(\theta \), we pass through all the intermediate values in the measured ratios, which correspond to all the “gradient” between black and white that we have when we overlap the margins.