Plotting on the Bloch Sphere¶
Important
Updated in QuTiP version 3.0.
Introduction¶
When studying the dynamics of a twolevel system, it is often convent to visualize the state of the system by plotting the statevector or density matrix on the Bloch sphere. In QuTiP, we have created two different classes to allow for easy creation and manipulation of data sets, both vectors and data points, on the Bloch sphere. The qutip.Bloch
class, uses Matplotlib to render the Bloch sphere, where as qutip.Bloch3d
uses the Mayavi rendering engine to generate a more faithful 3D reconstruction of the Bloch sphere.
The Bloch and Bloch3d Classes¶
In QuTiP, creating a Bloch sphere is accomplished by calling either:
In [1]: b = Bloch()
which will load an instance of the qutip.Bloch
class, or using:
>>> b3d = Bloch3d()
that loads the qutip.Bloch3d
version. Before getting into the details of these objects, we can simply plot the blank Bloch sphere associated with these instances via:
In [2]: b.show()
or
In addition to the show() command, the Bloch class has the following functions:
As an example, we can add a single data point:
In [3]: pnt = [1/np.sqrt(3),1/np.sqrt(3),1/np.sqrt(3)]
In [4]: b.add_points(pnt)
In [5]: b.show()
<Figure size 500x500 with 1 Axes>
and then a single vector:
In [6]: vec = [0,1,0]
In [7]: b.add_vectors(vec)
In [8]: b.show()
<Figure size 500x500 with 1 Axes>
and then add another vector corresponding to the \(\left\rm up \right>\) state:
In [9]: up = basis(2,0)
In [10]: b.add_states(up)
In [11]: b.show()
<Figure size 500x500 with 1 Axes>
Notice that when we add more than a single vector (or data point), a different color will automatically be applied to the later data set (mod 4). In total, the code for constructing our Bloch sphere with one vector, one state, and a single data point is:
In [12]: b = Bloch()
In [13]: pnt = [1./np.sqrt(3), 1./np.sqrt(3), 1./np.sqrt(3)]
In [14]: b.add_points(pnt)
In [15]: vec = [0,1,0]
In [16]: b.add_vectors(vec)
In [17]: up = basis(2,0)
In [18]: b.add_states(up)
In [19]: b.show()
where we have removed the extra show()
commands. Replacing b=Bloch()
with b=Bloch3d()
in the above code generates the following 3D Bloch sphere.
We can also plot multiple points, vectors, and states at the same time by passing list or arrays instead of individual elements. Before giving an example, we can use the clear() command to remove the current data from our Bloch sphere instead of creating a new instance:
In [20]: b.clear()
In [21]: b.show()
<Figure size 500x500 with 1 Axes>
Now on the same Bloch sphere, we can plot the three states associated with the x, y, and z directions:
In [22]: x = (basis(2,0)+(1+0j)*basis(2,1)).unit()
In [23]: y = (basis(2,0)+(0+1j)*basis(2,1)).unit()
In [24]: z = (basis(2,0)+(0+0j)*basis(2,1)).unit()
In [25]: b.add_states([x,y,z])
In [26]: b.show()
<Figure size 500x500 with 1 Axes>
a similar method works for adding vectors:
In [27]: b.clear()
In [28]: vec = [[1,0,0],[0,1,0],[0,0,1]]
In [29]: b.add_vectors(vec)
In [30]: b.show()
<Figure size 500x500 with 1 Axes>
Adding multiple points to the Bloch sphere works slightly differently than adding multiple states or vectors. For example, lets add a set of 20 points around the equator (after calling clear()):
In [31]: xp = [np.cos(th) for th in np.linspace(0, 2*pi, 20)]
In [32]: yp = [np.sin(th) for th in np.linspace(0, 2*pi, 20)]
In [33]: zp = np.zeros(20)
In [34]: pnts = [xp, yp, zp]
In [35]: b.add_points(pnts)
In [36]: b.show()
<Figure size 500x500 with 1 Axes>
Notice that, in contrast to states or vectors, each point remains the same color as the initial point. This is because adding multiple data points using the add_points function is interpreted, by default, to correspond to a single data point (single qubit state) plotted at different times. This is very useful when visualizing the dynamics of a qubit. An example of this is given in the example . If we want to plot additional qubit states we can call additional add_points functions:
In [37]: xz = np.zeros(20)
In [38]: yz = [np.sin(th) for th in np.linspace(0, pi, 20)]
In [39]: zz = [np.cos(th) for th in np.linspace(0, pi, 20)]
In [40]: b.add_points([xz, yz, zz])
In [41]: b.show()
<Figure size 500x500 with 1 Axes>
The color and shape of the data points is varied automatically by the Bloch class. Notice how the color and point markers change for each set of data. Again, we have had to call add_points
twice because adding more than one set of multiple data points is not supported by the add_points
function.
What if we want to vary the color of our points. We can tell the qutip.Bloch
class to vary the color of each point according to the colors listed in the b.point_color
list (see Configuring the Bloch sphere below). Again after clear()
:
In [42]: xp = [np.cos(th) for th in np.linspace(0, 2*pi, 20)]
In [43]: yp = [sin(th) for th in np.linspace(0, 2*pi, 20)]
In [44]: zp = np.zeros(20)
In [45]: pnts = [xp, yp, zp]
In [46]: b.add_points(pnts,'m') # < add a 'm' string to signify 'multi' colored points
In [47]: b.show()
<Figure size 500x500 with 1 Axes>
Now, the data points cycle through a variety of predefined colors. Now lets add another set of points, but this time we want the set to be a single color, representing say a qubit going from the \(\left\rm up\right>\) state to the \(\left\rm down\right>\) state in the yz plane:
In [48]: xz = np.zeros(20)
In [49]: yz = [np.sin(th) for th in np.linspace(0, pi ,20)]
In [50]: zz = [np.cos(th) for th in np.linspace(0, pi, 20)]
In [51]: b.add_points([xz, yz, zz]) # no 'm'
In [52]: b.show()
<Figure size 500x500 with 1 Axes>
Again, the same plot can be generated using the qutip.Bloch3d
class by replacing Bloch
with Bloch3d
:
A more slick way of using this ‘multi’ color feature is also given in the example, where we set the color of the markers as a function of time.
Differences Between Bloch and Bloch3d¶
While in general the Bloch
and Bloch3d
classes are interchangeable, there are some important differences to consider when choosing between them.
The
Bloch
class uses Matplotlib to generate figures. As such, the data plotted on the sphere is in reality just a 2D object. In contrast theBloch3d
class uses the 3D rendering engine from VTK via mayavi to generate the sphere and the included data. In this sense theBloch3d
class is much more advanced, as objects are rendered in 3D leading to a higher quality figure.Only the
Bloch
class can be embedded in a Matplotlib figure window. Thus if you want to combine a Bloch sphere with another figure generated in QuTiP, you can not useBloch3d
. Of course you can always postprocess your figures using other software to get the desired result.Due to limitations in the rendering engine, the
Bloch3d
class does not support LaTex for text. Again, you can get around this by postprocessing.The user customizable attributes for the
Bloch
andBloch3d
classes are not identical. Therefore, if you change the properties of one of the classes, these changes will cause an exception if the class is switched.
Configuring the Bloch sphere¶
Bloch Class Options¶
At the end of the last section we saw that the colors and marker shapes of the data plotted on the Bloch sphere are automatically varied according to the number of points and vectors added. But what if you want a different choice of color, or you want your sphere to be purple with different axes labels? Well then you are in luck as the Bloch class has 22 attributes which one can control. Assuming b=Bloch()
:
Attribute 
Function 
Default Setting 

b.axes 
Matplotlib axes instance for animations. Set by 

b.fig 
User supplied Matplotlib Figure instance. Set by 

b.font_color 
Color of fonts 
‘black’ 
b.font_size 
Size of fonts 
20 
b.frame_alpha 
Transparency of wireframe 
0.1 
b.frame_color 
Color of wireframe 
‘gray’ 
b.frame_width 
Width of wireframe 
1 
b.point_color 
List of colors for Bloch point markers to cycle through 
[‘b’,’r’,’g’,’#CC6600’] 
b.point_marker 
List of point marker shapes to cycle through 
[‘o’,’s’,’d’,’^’] 
b.point_size 
List of point marker sizes (not all markers look the same size when plotted) 
[55,62,65,75] 
b.sphere_alpha 
Transparency of Bloch sphere 
0.2 
b.sphere_color 
Color of Bloch sphere 
‘#FFDDDD’ 
b.size 
Sets size of figure window 
[7,7] (700x700 pixels) 
b.vector_color 
List of colors for Bloch vectors to cycle through 
[‘g’,’#CC6600’,’b’,’r’] 
b.vector_width 
Width of Bloch vectors 
4 
b.view 
Azimuthal and Elevation viewing angles 
[60,30] 
b.xlabel 
Labels for xaxis 
[‘$x$’,’’] +x and x (labels use LaTeX) 
b.xlpos 
Position of xaxis labels 
[1.1,1.1] 
b.ylabel 
Labels for yaxis 
[‘$y$’,’’] +y and y (labels use LaTeX) 
b.ylpos 
Position of yaxis labels 
[1.2,1.2] 
b.zlabel 
Labels for zaxis 
[‘$left0\right>$’,’$left1\right>$’] +z and z (labels use LaTeX) 
b.zlpos 
Position of zaxis labels 
[1.2,1.2] 
Bloch3d Class Options¶
The Bloch3d sphere is also customizable. Note however that the attributes for the Bloch3d
class are not in onetoone
correspondence to those of the Bloch
class due to the different underlying rendering engines. Assuming b=Bloch3d()
:
Attribute 
Function 
Default Setting 

b.fig 
User supplied Mayavi Figure instance. Set by 

b.font_color 
Color of fonts 
‘black’ 
b.font_scale 
Scale of fonts 
0.08 
b.frame 
Draw wireframe for sphere? 
True 
b.frame_alpha 
Transparency of wireframe 
0.05 
b.frame_color 
Color of wireframe 
‘gray’ 
b.frame_num 
Number of wireframe elements to draw 
8 
b.frame_radius 
Radius of wireframe lines 
0.005 
b.point_color 
List of colors for Bloch point markers to cycle through 
[‘r’, ‘g’, ‘b’, ‘y’] 
b.point_mode 
Type of point markers to draw 
sphere 
b.point_size 
Size of points 
0.075 
b.sphere_alpha 
Transparency of Bloch sphere 
0.1 
b.sphere_color 
Color of Bloch sphere 
‘#808080’ 
b.size 
Sets size of figure window 
[500,500] (500x500 pixels) 
b.vector_color 
List of colors for Bloch vectors to cycle through 
[‘r’, ‘g’, ‘b’, ‘y’] 
b.vector_width 
Width of Bloch vectors 
3 
b.view 
Azimuthal and Elevation viewing angles 
[45,65 ] 
b.xlabel 
Labels for xaxis 
[‘x>’, ‘’] +x and x 
b.xlpos 
Position of xaxis labels 
[1.07,1.07] 
b.ylabel 
Labels for yaxis 
[‘$y$’,’’] +y and y 
b.ylpos 
Position of yaxis labels 
[1.07,1.07] 
b.zlabel 
Labels for zaxis 
[‘0>’, ‘1>’] +z and z 
b.zlpos 
Position of zaxis labels 
[1.07,1.07] 
These properties can also be accessed via the print command:
In [53]: b = Bloch()
In [54]: print(b)
Bloch data:

Number of points: 0
Number of vectors: 0
Bloch sphere properties:

font_color: black
font_size: 20
frame_alpha: 0.2
frame_color: gray
frame_width: 1
point_color: ['b', 'r', 'g', '#CC6600']
point_marker: ['o', 's', 'd', '^']
point_size: [25, 32, 35, 45]
sphere_alpha: 0.2
sphere_color: #FFDDDD
figsize: [5, 5]
vector_color: ['g', '#CC6600', 'b', 'r']
vector_width: 3
vector_style: >
vector_mutation: 20
view: [60, 30]
xlabel: ['$x$', '']
xlpos: [1.2, 1.2]
ylabel: ['$y$', '']
ylpos: [1.2, 1.2]
zlabel: ['$\\left0\\right>$', '$\\left1\\right>$']
zlpos: [1.2, 1.2]
Animating with the Bloch sphere¶
The Bloch class was designed from the outset to generate animations. To animate a set of vectors or data points the basic idea is: plot the data at time t1, save the sphere, clear the sphere, plot data at t2,… The Bloch sphere will automatically number the output file based on how many times the object has been saved (this is stored in b.savenum). The easiest way to animate data on the Bloch sphere is to use the save()
method and generate a series of images to convert into an animation. However, as of Matplotlib version 1.1, creating animations is builtin. We will demonstrate both methods by looking at the decay of a qubit on the bloch sphere.
Example: Qubit Decay¶
The code for calculating the expectation values for the Pauli spin operators of a qubit decay is given below. This code is common to both animation examples.
from qutip import *
from scipy import *
def qubit_integrate(w, theta, gamma1, gamma2, psi0, tlist):
# operators and the hamiltonian
sx = sigmax(); sy = sigmay(); sz = sigmaz(); sm = sigmam()
H = w * (cos(theta) * sz + sin(theta) * sx)
# collapse operators
c_op_list = []
n_th = 0.5 # temperature
rate = gamma1 * (n_th + 1)
if rate > 0.0: c_op_list.append(sqrt(rate) * sm)
rate = gamma1 * n_th
if rate > 0.0: c_op_list.append(sqrt(rate) * sm.dag())
rate = gamma2
if rate > 0.0: c_op_list.append(sqrt(rate) * sz)
# evolve and calculate expectation values
output = mesolve(H, psi0, tlist, c_op_list, [sx, sy, sz])
return output.expect[0], output.expect[1], output.expect[2]
## calculate the dynamics
w = 1.0 * 2 * pi # qubit angular frequency
theta = 0.2 * pi # qubit angle from sigma_z axis (toward sigma_x axis)
gamma1 = 0.5 # qubit relaxation rate
gamma2 = 0.2 # qubit dephasing rate
# initial state
a = 1.0
psi0 = (a* basis(2,0) + (1a)*basis(2,1))/(sqrt(a**2 + (1a)**2))
tlist = linspace(0,4,250)
#expectation values for ploting
sx, sy, sz = qubit_integrate(w, theta, gamma1, gamma2, psi0, tlist)
Generating Images for Animation¶
An example of generating images for generating an animation outside of Python is given below:
import numpy as np
b = Bloch()
b.vector_color = ['r']
b.view = [40,30]
for i in range(len(sx)):
b.clear()
b.add_vectors([np.sin(theta),0,np.cos(theta)])
b.add_points([sx[:i+1],sy[:i+1],sz[:i+1]])
b.save(dirc='temp') #saving images to temp directory in current working directory
Generating an animation using ffmpeg (for example) is fairly simple:
ffmpeg r 20 b 1800 i bloch_%01d.png bloch.mp4
Directly Generating an Animation¶
Important
Generating animations directly from Matplotlib requires installing either mencoder or ffmpeg. While either choice works on linux, it is best to choose ffmpeg when running on the Mac. If using macports just do: sudo port install ffmpeg
.
The code to directly generate an mp4 movie of the Qubit decay is as follows:
from pylab import *
import matplotlib.animation as animation
from mpl_toolkits.mplot3d import Axes3D
fig = figure()
ax = Axes3D(fig,azim=40,elev=30)
sphere = Bloch(axes=ax)
def animate(i):
sphere.clear()
sphere.add_vectors([np.sin(theta),0,np.cos(theta)])
sphere.add_points([sx[:i+1],sy[:i+1],sz[:i+1]])
sphere.make_sphere()
return ax
def init():
sphere.vector_color = ['r']
return ax
ani = animation.FuncAnimation(fig, animate, np.arange(len(sx)),
init_func=init, blit=True, repeat=False)
ani.save('bloch_sphere.mp4', fps=20)
The resulting movie may be viewed here: bloch_decay.mp4