What is wrong with the image of the Bloch sphere on wikipedia, on many sites and articles
I recently published an article on HABR about the visualization of the quantum spin and the Bloch sphere. After some thought, I decided to publish the following text in addition to this article.
The Bloch sphere, in my opinion, is depicted incorrectly on Wikipedia and in many articles and books, to put it mildly. According to Wikipedia, the Bloch sphere looks like this:
In the figure, |ψ› is the state vector. But in reality it is not so. There should be a Bloch vector here, but not a state vector. The Bloch vector is an ordinary vector specifying the position of a point in spherical coordinates.
However, each Bloch vector corresponds its state vector. But these two vectors are completely different.
This is how the Bloch vector is depicted in Arthur Eckert’s book:
And here is what Arthur Eckert himself says about this vector in his lecture
IQIS Lecture 2.5 – The Bloch sphere.
Here is a translation of his words from about 55 seconds:
“I can think of another vector which is called the Bloch vector and which I somehow I bind mentally with a state vector, and this Bloch vector, which you can see here in this diagram in terms of spherical coordinates, will be a unit vector that is characterized or defined by two angles.
Polar angle θ and azimuth angle φ, so I have this one-to-one correspondence.
Instead of thinking that the qubit is in this particular state,
i can think of a universal vector in 3d euclidean space,
pointing in some direction, which is determined by θ and φ.”
I refer to Arthur Eckert because he is very famous scientist.
It is easy to find documentation on the quantum computing library on the Internet qiskit and see a lot of screenshots with the Bloch vector, and not with the state vector. The state vector can only be obtained using the Bloch vector. But these two vectors are absolutely not the same. The value of the state vector is usually displayed not on the Bloch sphere, but somewhere nearby and in a separate line.
Therefore, I am perplexed by the large number of sites, articles and books in which the state vector is depicted on the Bloch sphere instead of the Bloch vector.
Personally, this approach to the state vector confused me very much and greatly delayed my understanding of the mathematics of quantum spin.