Hopf fibration

 To determine the value of the quantum state |ψ〉, set the values of the polar and azimuth angles either using the GUI in the upper right corner of the program, or using the keys located under the GUI. The intersection of a red-colored straight line with a horizontal plane sets the value of complex coordinates. This value is (ζ = ...) you can see it at the very bottom of the left part of the program window.

A short video that shows how to control this program.

This site shows an interactive visualization of the Hopf fibration or Hopf bundle. The corresponding program was created on the basis of the Riemann-Bloch sphere and the stereographic projection of points belonging to this sphere onto a Complex plane. When visualizing the Hopf fibration, each point of this sphere is displayed as a circle on the display screen. These circles are called either Hopf circles, or Clifford parallels, or Villarso circles. Both Hopf, Clifford and Villarso have left their mark on the study of this field of geometry. Quantum rotations were used to move points along the Riemann-Bloch sphere, and, consequently, circles on the torus. Quantum rotations are discussed in detail on this site and another site.

The work area of the program is divided into four parts. On the left side of the workspace is the Bloch-Riemann sphere. The red arrow on this sphere shows the Bloch vector. Each Bloch vector on the sphere corresponds to a quantum state vector . By specifying the Bloch vector (coordinates θ and φ on the sphere), we obtain (after appropriate calculations) the quantum state vector |ψ〉, the value of which is displayed under the Bloch sphere.

There is a control panel on the right side of the workspace. The coordinates θ and φ of the Bloch vector on the Bloch sphere can be set in the control panel using the polar_electron and azimuth_electron parameters, respectively. The most frequently used values of these parameters can also be set using the buttons, which are also located on the right side of the program workspace.

The program provides for the possibility of rotating the Bloch vector around a predetermined axis of rotation. The position of the rotation axis can be set using the polar_axis and azimuth_axis parameters in the control panel located on the right side of the program workspace. When you select any of the values of these parameters, the axis of rotation on the Bloch sphere appears automatically. The most frequently used values of the parameters used to set the position of the rotation axis can also be set using the buttons, which are also located on the right side

The middle part of the working area of the program is devoted to the output of images corresponding to Hopf bundles. The type of Hopf fibration is given by the position of the Bloch vector on the Riemann-Bloch sphere. In the right part of the work area of the program there is a button "Erase Hopf circles" which is designed to erase the circles that make up the Hopf fibration. Also on the right side of the program's workspace are three buttons "1", "2" and "3" ("Thick lines"), which set the thickness of the circles (each Hopf fibration is a circle called a Hopf circle). The "Axes" switch enables/disables the display of coordinate axes in the middle part of the working area of the program (in which Hopf circles are visualized).

At the very bottom of the program's workspace there is a row of buttons designed to rotate the Bloch vector on the Riemann-Bloch sphere:
 

The Riemann-Bloch sphere is the basis for visualizing the Hopf fibration

In order to understand how the Hopf fibration appears, it is first necessary to deal with the Riemann-Bloch sphere and the stereographic projection of points lying on this sphere onto a complex plane. When visualizing the Hopf fibration, each point on this sphere is displayed as a circle, which is called a "Hopf circle". Each Hopf circle is also called a "layer". Just in case, we note that the Riemann-Bloch sphere is a purely mathematical construction and should not be considered as a geometric object in the usual visual sense.

On the left side of the workspace, the program displays the Riemann-Bloch sphere. In general, the Hopf fibration is constructed based on a four-dimensional topology in which the Riemann-Bloch sphere is usually called the Riemann sphere. But since we will visualize the Hopf fibration in relation to the quantum mechanics of electron spin, the Riemann-Bloch sphere is usually called simply the Bloch sphere here, since Bernhard Riemann is mainly a mathematician, and Felix Bloch is a physicist. However, both in pure mathematics and in physics, the same methods are used to describe the Hopf fibration. Note also that when describing the state of polarization of light, the Riemann-Bloch sphere is usually called the Poincare sphere.

As you know, any complex number can be written in exponential form. The numbers a and b in exponential form look like this:
  a = r_a⋅e^iφ_a
  b = r_b⋅e^iφ_b
The state vector |ψ〉 in this case will take the following form:
  |ψ〉 = r_a⋅e^iφ_a|0〉 + r_b⋅e^iφ_b|1〉
Let's introduce the following notation
  φ = φ_b-φ_a  
and therefore,
  φ_b = φ + φ_a
Then the previous expression for the state vector |ψ〉 can be rewritten in this form
  |ψ〉 = r_a⋅e^iφ_a|0〉 + r_b⋅e^iφ_b|1〉 = r_a⋅e^iφ_a|0〉 + r_b⋅e^iφ⋅e^iφ_a|1〉 = e^iφ_a(r_a|0〉 + r_b⋅e^iφ|1〉)

For the state vector, only the ratio in which the probability amplitudes are located between each other matters. Therefore, it is possible to painlessly multiply both composing vectors |ψ〉 by the same number equal to e^-iφ_a. After this multiplication, we get the following expression for |ψ〉
  |ψ〉 = r_a⋅|0〉 + r_b⋅e^iφ⋅|1〉
Note that the state vector |ψ〉 can be adjusted to any number - real, complex or purely imaginary - the physical meaning will not change from this. The angle φ_a is called the "global phase". Since the global phase appears only as a result of mathematical transformations, it has no physical meaning and therefore can be ignored (but only in quantum mechanical calculations). And when visualizing the Hopf fibration, the global phase plays a fundamental role.
Unlike the global phase, the angle φ has a direct physical meaning, which determines the phase difference between the components of the state vector. The angle φ is called the "local phase". The local phase plays an important role in the interference of electrons.
The probability amplitudes a and b are usually normalized to 1. Therefore, the sum of r_a² + r_b² should also be equal to 1:
 r_a² + r_b² = 1
Based on this, it is convenient to enter the following normalization (because always cos²+sin²=1)
  r_a = cos(θ/2)
  r_b = sin(θ/2)
Then the expression for the state vector |ψ〉 will take the form
  r_a = cos(θ/2)
  r_b = sin(θ/2)

Sphere S³ in C²

A sphere S³ of a unit radius in four-dimensional space R⁴ is a set of points located at a unit distance from the origin. If in this space we consider the real coordinates x₁, y₁ and x₂, y₂, then this sphere will be given by the equation   x₁² + y₁² + x₂² + y₂² = 1

We can assume that in the expression   |ψ〉 = a⋅|0〉 + b⋅|1〉   the values of the complex quantities a and b are defined as follows:
   a = x₁ + i⋅y₁,   b = x₂ + i⋅y₂.
Then the sphere S³ will be represented as a set of complex pairs (a, b). Note that |a|² + |b|² = 1. Thus, we will get an image of our sphere S³ in a two-dimensional complex space C².

The entire three-dimensional sphere S³ is filled with circles, and each such circle is associated with a point on a two-dimensional sphere S² - the Riemann-Bloch sphere. No two circles intersect. This partition of a three-dimensional sphere S³ on a circle is called a Hopf fibration.

Let's turn to the qubit again. Suppose we rotate our qubit |ψ〉 by some angle γ (global phase). Just looking at this qubit by itself, this phase is physically unobservable, since the probabilities do not change, i.e. |ψ〉 ∼λ⋅|ψ〉 = e^iγ⋅|ψ〉   (it is obvious that the module |λ| = |e^iγ| = 1). However, to visualize the Hopf circles, we will be interested in exactly the entire value e^iγ⋅|ψ〉, in which the angle γ will vary from 0° to 360° thereby defining a circle (this circle is often called a Hopf circle).

Despite the fact that the global phase has no physical meaning (as they say in all textbooks on quantum mechanics), in mathematics, using it you can get very beautiful results. But it turns out not only in mathematics, but also in some sections of modern physics.

Despite the fact that the global phase has no physical meaning (as they say in all textbooks on quantum mechanics), in mathematics, using it you can get very beautiful results. But it turns out not only in mathematics, but also in some sections of modern physics. Why the Hopf fibration based on the global phase (which has no physical meaning in quantum mechanics) is used in physics, I do not know. But by the end of the 1970s, it became clear to some well-known scientists that the Hopf bundle plays a fundamentally important role in gauge approaches to quantum field theory. In addition, in fact, as the core of the entire model, the Hopf bundle appeared in Roger Penrose's theory of twistors, and later in a number of other approaches to the theory of quantum gravity.

Thus, the three-dimensional sphere S³ is embedded in a four-dimensional Euclidean space R⁴ identified with a two-dimensional complex space C²(a, b), where a and b are complex numbers. Next, let's consider the map of a p three-dimensional sphere S³on a two-dimensional sphere S² - the Bloch sphere.
   p:  S³ ⟶ S².
For this mapping, we take a point from a three-dimensional sphere S³ with coordinates (a, b) and using the p mapping, we will map this pair of complex numbers into one complex number λ = a/b:
   p:  (a, b)  ⟶ λ = a / b
We will consider the point λ = a/b to be a point on the complex line С¹. But the complex line С¹ is simultaneously a two-dimensional plane R². Since in general the complex number b can turn to 0, then it is necessary to add to С¹ an infinitely distant point. That is, to replenish the two-dimensional plane R² with one infinitely distant point. As a result of these actions, a two-dimensional sphere S² is obtained. On this sphere, all points λ = a/b are finite, but one point is located at infinity. Let this infinitely distant point be located at the south pole of a two-dimensional sphere S². As a result of all these manipulations, a three-dimensional sphere S³ was mapped onto a two-dimensional sphere S².

Here is a drawing showing the stereographic projection of a unit sphere from the south pole S to the plane z = 0. The stereographic projection maps the northern hemisphere to the area lying inside the unit circle. The southern hemisphere is mapped to an area outside the unit circle. The equator coincides with the unit circle.
 
Let O(0, 0, 0) be the center of the sphere, N(0, 0, 1) be the north pole, and S(0, 0, -1) the south pole. Let P'(x', y', 0) be the intersection of a straight line SP with the equatorial plane z = 0, andQ(0, 0, z) – projection P(x, y, z) on the z axis. The point P' is called the stereographic projection of the point P. From similar triangles SOP' and SQP we find:
 
Now let's introduce a complex variable in the plane z = 0ζ
 
In the visualization program, the complex variable ζ is denoted in the standard way accepted for complex numbers ζ = z = x + iy. In this expression, x and y are coefficients before the real and imaginary parts of the variable z, and no longer spatial coordinates (x, y, z). In the visualization program, the complex plane is located on the spatial plane z = 0. The complex number ζ = z = x + iy is located purely geometrically as a point P' intersection a projecting line PS with a plane z = 0.

The representation of the state vector on the Bloch sphere is made in such a way that it is possible to display the true size of the angles θ, rather than their half size θ/2. But it should be remembered that in fact the basis vectors |0〉 and |1〉 are orthogonal vectors despite the fact that they are shown lying on the same straight line. This is nothing more than just a convention adopted specifically for the representation of state vectors on the Bloch sphere.

Displaying Hopf circles on the screen

Let's emphasize that each point on the sphere the Riemann-Bloch is displayed as a separate Hopf circle. However, we recall once again that the Riemann-Bloch sphere is a purely mathematical construction and therefore it is impossible to draw a circle at any point of this sphere. Everything looks a little more complicated.
To visualize a Hopf circle, first of all you need to find the position of the center of this circle and its radius. To do this, draw a straight line, as shown in the following figure, and find the intersection point of this straight line with the OXY plane , which is designated in the program as "Complex plane". We also need to set some scale in which we will display the circles. It is easy to understand that there will be only one circle above the North pole - the Bloch vector directed from point O to point N (North pole) at any value of the parameter value azimuth_electron (geographic longitude) will always be directed vertically upwards. Therefore, we can take the size ON = OA as the unit radius of the Hopf circle. All other radii of the circles will be measured relative to this radius (the main thing is to maintain the proportionality of the radii of all Hopf circles).
 

Thus, only one circle will be located above the north pole of the Riemann-Bloch sphere. If we replace the geograph (parameter polar_electron), then in fact new Riemann-Bloch circles will appear lying on the corresponding parallel of this sphere. In the following figure, each point on the Riemann-Bloch sphere corresponds to one circle. To designate the parameter polar_electron equal to 120°, this entire parallel will be filled with Hopf circles - "layers". The totality of all the circles on this parallel forms a torus. To get this drawing - after the launch programs set polar_electron) = 120° when connected to the inside of the program. I think that press lightly, turning Z+ over and over again, and you will see a circle in front of you.
If you want to get the value polar_electron) = 180° - all the Hopf circles turn into one vertical straight line.
 

And here's what images formed by Hopf circles look like at parameter values polar_electron) respectively equal to 30°, 60° and 90°:
 
It can be seen from the screenshot that with an increase in the polar_electron parameter, the tori "swell". And with the value of the parameter polar_electron = 180°, instead of a torus, we will see a vertical straight line.

There are many tori formed at once when rotating around the Z axis:
       

The thickness of the tube of such a torus varies depending on the location of the latitude between the horizontal plane and the point from which the projection takes place (the south pole of the sphere). As the latitude shifts from the projection point, the torus passes through all intermediate states between the two limits. In one limit, becoming thinner and thinner, it degenerates into a circle (polar_electron = 0°).
In the opposite case, the torus thickens to a state where its "hole" degenerates into a straight line (on the Bloch sphere, in this case, the parameter polar_electron = 180°). In other words, Hopf circles fill the entire space with nested tori. But the most important thing here is this. Each point of the Bloch sphere located on the line of latitude on the surface of the torus corresponds to a circle line capturing the "doughnut hole" and obliquely encircling the pipe. Just as a multitude of points fills the entire circumference of latitude, similarly, a set of such rings, hooked to each other, completely covers the surface of the corresponding torus.

If you rotate the Bloch vector around the axis Z in a spiral mode (you need to install check-box), then you can see the following picture:
       

The Hopf circles that make up the torus are also called Villarso circles after the French mathematician and astronomer Yvon Villarso, who noticed that on the torus, in addition to the two standard families of circles, there are two other families of circles. This is a pair of circles obtained by cutting the torus of rotation with a diagonal tangent plane passing through the center of the torus:
   

All Villarso circles on the torus turn out to be interlocked with each other, and in the topological context of knot theory, this configuration is called the "Hopf bundle" and represents the simplest non-trivial (that is, unconnectable) knot. Here's what two hooked Hopf circles look like in close-up:
  

However, the principle of duality is sometimes used in geometry. For example, between points and straight lines. In our case, let's try to match each circle with a separate straight line. Let's create a visualization program so that the Villarso circles (Clifford parallels) turn into straight lines. This will result in a hyperboloid of rotation:
   
    

You can try to take some other geometric objects instead of circles and straight lines and match them with points on the Riemann-Bloch sphere. The resulting visualizations will not make any deep sense at all. But the resulting images themselves may be of interest - you will get bundles of geometric objects that are quite difficult to obtain in another way. Any two of the circles on the torus are interconnected and to create a program depicting this coupling, it is difficult to come up with another option than the one proposed for constructing the Hopf bundle.

The following screenshot shows, so to speak, the "bundles" obtained when, instead of Hopf circles, a square, a pentagon and a heptagon are taken:
&   

Several buttons have been added to the program interface to select geometric objects:
         

However, we are not only interested in rotation around the Z axis. This is what individual Hopf circles look like (from a set of such circles consist of tori) with the value of the parameter azimuth_electron) = 0° and various values polar_electron parameter. This screenshot shows rotation no longer around the Z axis, and around the Y axis:
  

  When rotating around the Y axis, an asymmetric torus was obtained:   

  When rotating around the X-axis, an asymmetric torus is also obtained:

  

Rotation can be performed around an arbitrarily directed N axis in space. This is how two tori obtained by rotating around two different axes look like. In the screenshot, we see not only the hooked Hopf circles, but also two hooked asymmetric tori:
  

And this is what four tori look like - rotation implemented around four different axes:
  

In this visualization, the Riemann-Bloch sphere and vector rotations play a main role in the Hopf fibration. As mentioned above, each point on the sphere corresponds to a single Hopf circle. With each rotation of the Bloch vector, we set a new point on the sphere. Rotations on the Riemann-Bloch sphere are quantum rotations.