# This file is part of QuTiP: Quantum Toolbox in Python.
#
# Copyright (c) 2011 and later, Paul D. Nation and Robert J. Johansson.
# All rights reserved.
#
# Redistribution and use in source and binary forms, with or without
# modification, are permitted provided that the following conditions are
# met:
#
# 1. Redistributions of source code must retain the above copyright notice,
# this list of conditions and the following disclaimer.
#
# 2. Redistributions in binary form must reproduce the above copyright
# notice, this list of conditions and the following disclaimer in the
# documentation and/or other materials provided with the distribution.
#
# 3. Neither the name of the QuTiP: Quantum Toolbox in Python nor the names
# of its contributors may be used to endorse or promote products derived
# from this software without specific prior written permission.
#
# THIS SOFTWARE IS PROVIDED BY THE COPYRIGHT HOLDERS AND CONTRIBUTORS
# "AS IS" AND ANY EXPRESS OR IMPLIED WARRANTIES, INCLUDING, BUT NOT
# LIMITED TO, THE IMPLIED WARRANTIES OF MERCHANTABILITY AND FITNESS FOR A
# PARTICULAR PURPOSE ARE DISCLAIMED. IN NO EVENT SHALL THE COPYRIGHT
# HOLDER OR CONTRIBUTORS BE LIABLE FOR ANY DIRECT, INDIRECT, INCIDENTAL,
# SPECIAL, EXEMPLARY, OR CONSEQUENTIAL DAMAGES (INCLUDING, BUT NOT
# LIMITED TO, PROCUREMENT OF SUBSTITUTE GOODS OR SERVICES; LOSS OF USE,
# DATA, OR PROFITS; OR BUSINESS INTERRUPTION) HOWEVER CAUSED AND ON ANY
# THEORY OF LIABILITY, WHETHER IN CONTRACT, STRICT LIABILITY, OR TORT
# (INCLUDING NEGLIGENCE OR OTHERWISE) ARISING IN ANY WAY OUT OF THE USE
# OF THIS SOFTWARE, EVEN IF ADVISED OF THE POSSIBILITY OF SUCH DAMAGE.
###############################################################################
__all__ = ['Bloch']
import os
from numpy import (ndarray, array, linspace, pi, outer, cos, sin, ones, size,
sqrt, real, mod, append, ceil, arange)
from qutip.qobj import Qobj
from qutip.expect import expect
from qutip.operators import sigmax, sigmay, sigmaz
try:
import matplotlib.pyplot as plt
from mpl_toolkits.mplot3d import Axes3D
from matplotlib.patches import FancyArrowPatch
from mpl_toolkits.mplot3d import proj3d
class Arrow3D(FancyArrowPatch):
def __init__(self, xs, ys, zs, *args, **kwargs):
FancyArrowPatch.__init__(self, (0, 0), (0, 0), *args, **kwargs)
self._verts3d = xs, ys, zs
def draw(self, renderer):
xs3d, ys3d, zs3d = self._verts3d
xs, ys, zs = proj3d.proj_transform(xs3d, ys3d, zs3d, renderer.M)
self.set_positions((xs[0], ys[0]), (xs[1], ys[1]))
FancyArrowPatch.draw(self, renderer)
except:
pass
try:
from IPython.display import display
except:
pass
[docs]class Bloch():
"""Class for plotting data on the Bloch sphere. Valid data can be
either points, vectors, or qobj objects.
Attributes
----------
axes : instance {None}
User supplied Matplotlib axes for Bloch sphere animation.
fig : instance {None}
User supplied Matplotlib Figure instance for plotting Bloch sphere.
font_color : str {'black'}
Color of font used for Bloch sphere labels.
font_size : int {20}
Size of font used for Bloch sphere labels.
frame_alpha : float {0.1}
Sets transparency of Bloch sphere frame.
frame_color : str {'gray'}
Color of sphere wireframe.
frame_width : int {1}
Width of wireframe.
point_color : list {["b","r","g","#CC6600"]}
List of colors for Bloch sphere point markers to cycle through.
i.e. By default, points 0 and 4 will both be blue ('b').
point_marker : list {["o","s","d","^"]}
List of point marker shapes to cycle through.
point_size : list {[25,32,35,45]}
List of point marker sizes. Note, not all point markers look
the same size when plotted!
sphere_alpha : float {0.2}
Transparency of Bloch sphere itself.
sphere_color : str {'#FFDDDD'}
Color of Bloch sphere.
figsize : list {[7,7]}
Figure size of Bloch sphere plot. Best to have both numbers the same;
otherwise you will have a Bloch sphere that looks like a football.
vector_color : list {["g","#CC6600","b","r"]}
List of vector colors to cycle through.
vector_width : int {5}
Width of displayed vectors.
vector_style : str {'-|>', 'simple', 'fancy', ''}
Vector arrowhead style (from matplotlib's arrow style).
vector_mutation : int {20}
Width of vectors arrowhead.
view : list {[-60,30]}
Azimuthal and Elevation viewing angles.
xlabel : list {["$x$",""]}
List of strings corresponding to +x and -x axes labels, respectively.
xlpos : list {[1.1,-1.1]}
Positions of +x and -x labels respectively.
ylabel : list {["$y$",""]}
List of strings corresponding to +y and -y axes labels, respectively.
ylpos : list {[1.2,-1.2]}
Positions of +y and -y labels respectively.
zlabel : list {[r'$\\left|0\\right>$',r'$\\left|1\\right>$']}
List of strings corresponding to +z and -z axes labels, respectively.
zlpos : list {[1.2,-1.2]}
Positions of +z and -z labels respectively.
"""
def __init__(self, fig=None, axes=None, view=None, figsize=None,
background=False):
# Figure and axes
self.fig = fig
self.axes = axes
# Background axes, default = False
self.background = background
# The size of the figure in inches, default = [5,5].
self.figsize = figsize if figsize else [5, 5]
# Azimuthal and Elvation viewing angles, default = [-60,30].
self.view = view if view else [-60, 30]
# Color of Bloch sphere, default = #FFDDDD
self.sphere_color = '#FFDDDD'
# Transparency of Bloch sphere, default = 0.2
self.sphere_alpha = 0.2
# Color of wireframe, default = 'gray'
self.frame_color = 'gray'
# Width of wireframe, default = 1
self.frame_width = 1
# Transparency of wireframe, default = 0.2
self.frame_alpha = 0.2
# Labels for x-axis (in LaTex), default = ['$x$', '']
self.xlabel = ['$x$', '']
# Position of x-axis labels, default = [1.2, -1.2]
self.xlpos = [1.2, -1.2]
# Labels for y-axis (in LaTex), default = ['$y$', '']
self.ylabel = ['$y$', '']
# Position of y-axis labels, default = [1.1, -1.1]
self.ylpos = [1.2, -1.2]
# Labels for z-axis (in LaTex),
# default = [r'$\left|0\right>$', r'$\left|1\right>$']
self.zlabel = [r'$\left|0\right>$', r'$\left|1\right>$']
# Position of z-axis labels, default = [1.2, -1.2]
self.zlpos = [1.2, -1.2]
# ---font options---
# Color of fonts, default = 'black'
self.font_color = 'black'
# Size of fonts, default = 20
self.font_size = 20
# ---vector options---
# List of colors for Bloch vectors, default = ['b','g','r','y']
self.vector_color = ['g', '#CC6600', 'b', 'r']
#: Width of Bloch vectors, default = 5
self.vector_width = 3
#: Style of Bloch vectors, default = '-|>' (or 'simple')
self.vector_style = '-|>'
#: Sets the width of the vectors arrowhead
self.vector_mutation = 20
# ---point options---
# List of colors for Bloch point markers, default = ['b','g','r','y']
self.point_color = ['b', 'r', 'g', '#CC6600']
# Size of point markers, default = 25
self.point_size = [25, 32, 35, 45]
# Shape of point markers, default = ['o','^','d','s']
self.point_marker = ['o', 's', 'd', '^']
# ---data lists---
# Data for point markers
self.points = []
# Data for Bloch vectors
self.vectors = []
# Data for annotations
self.annotations = []
# Number of times sphere has been saved
self.savenum = 0
# Style of points, 'm' for multiple colors, 's' for single color
self.point_style = []
# status of rendering
self._rendered = False
# status of showing
if fig is None:
self._shown = False
else:
self._shown = True
[docs] def set_label_convention(self, convention):
"""Set x, y and z labels according to one of conventions.
Parameters
----------
convention : string
One of the following:
- "original"
- "xyz"
- "sx sy sz"
- "01"
- "polarization jones"
- "polarization jones letters"
see also: http://en.wikipedia.org/wiki/Jones_calculus
- "polarization stokes"
see also: http://en.wikipedia.org/wiki/Stokes_parameters
"""
ketex = "$\\left.|%s\\right\\rangle$"
# \left.| is on purpose, so that every ket has the same size
if convention == "original":
self.xlabel = ['$x$', '']
self.ylabel = ['$y$', '']
self.zlabel = ['$\\left|0\\right>$', '$\\left|1\\right>$']
elif convention == "xyz":
self.xlabel = ['$x$', '']
self.ylabel = ['$y$', '']
self.zlabel = ['$z$', '']
elif convention == "sx sy sz":
self.xlabel = ['$s_x$', '']
self.ylabel = ['$s_y$', '']
self.zlabel = ['$s_z$', '']
elif convention == "01":
self.xlabel = ['', '']
self.ylabel = ['', '']
self.zlabel = ['$\\left|0\\right>$', '$\\left|1\\right>$']
elif convention == "polarization jones":
self.xlabel = [ketex % "\\nearrow\\hspace{-1.46}\\swarrow",
ketex % "\\nwarrow\\hspace{-1.46}\\searrow"]
self.ylabel = [ketex % "\\circlearrowleft", ketex %
"\\circlearrowright"]
self.zlabel = [ketex % "\\leftrightarrow", ketex % "\\updownarrow"]
elif convention == "polarization jones letters":
self.xlabel = [ketex % "D", ketex % "A"]
self.ylabel = [ketex % "L", ketex % "R"]
self.zlabel = [ketex % "H", ketex % "V"]
elif convention == "polarization stokes":
self.ylabel = ["$\\nearrow\\hspace{-1.46}\\swarrow$",
"$\\nwarrow\\hspace{-1.46}\\searrow$"]
self.zlabel = ["$\\circlearrowleft$", "$\\circlearrowright$"]
self.xlabel = ["$\\leftrightarrow$", "$\\updownarrow$"]
else:
raise Exception("No such convention.")
def __str__(self):
s = ""
s += "Bloch data:\n"
s += "-----------\n"
s += "Number of points: " + str(len(self.points)) + "\n"
s += "Number of vectors: " + str(len(self.vectors)) + "\n"
s += "\n"
s += "Bloch sphere properties:\n"
s += "------------------------\n"
s += "font_color: " + str(self.font_color) + "\n"
s += "font_size: " + str(self.font_size) + "\n"
s += "frame_alpha: " + str(self.frame_alpha) + "\n"
s += "frame_color: " + str(self.frame_color) + "\n"
s += "frame_width: " + str(self.frame_width) + "\n"
s += "point_color: " + str(self.point_color) + "\n"
s += "point_marker: " + str(self.point_marker) + "\n"
s += "point_size: " + str(self.point_size) + "\n"
s += "sphere_alpha: " + str(self.sphere_alpha) + "\n"
s += "sphere_color: " + str(self.sphere_color) + "\n"
s += "figsize: " + str(self.figsize) + "\n"
s += "vector_color: " + str(self.vector_color) + "\n"
s += "vector_width: " + str(self.vector_width) + "\n"
s += "vector_style: " + str(self.vector_style) + "\n"
s += "vector_mutation: " + str(self.vector_mutation) + "\n"
s += "view: " + str(self.view) + "\n"
s += "xlabel: " + str(self.xlabel) + "\n"
s += "xlpos: " + str(self.xlpos) + "\n"
s += "ylabel: " + str(self.ylabel) + "\n"
s += "ylpos: " + str(self.ylpos) + "\n"
s += "zlabel: " + str(self.zlabel) + "\n"
s += "zlpos: " + str(self.zlpos) + "\n"
return s
def _repr_png_(self):
from IPython.core.pylabtools import print_figure
self.render()
fig_data = print_figure(self.fig, 'png')
plt.close(self.fig)
return fig_data
def _repr_svg_(self):
from IPython.core.pylabtools import print_figure
self.render()
fig_data = print_figure(self.fig, 'svg').decode('utf-8')
plt.close(self.fig)
return fig_data
[docs] def clear(self):
"""Resets Bloch sphere data sets to empty.
"""
self.points = []
self.vectors = []
self.point_style = []
self.annotations = []
[docs] def add_points(self, points, meth='s'):
"""Add a list of data points to bloch sphere.
Parameters
----------
points : array/list
Collection of data points.
meth : str {'s', 'm', 'l'}
Type of points to plot, use 'm' for multicolored, 'l' for points
connected with a line.
"""
if not isinstance(points[0], (list, ndarray)):
points = [[points[0]], [points[1]], [points[2]]]
points = array(points)
if meth == 's':
if len(points[0]) == 1:
pnts = array([[points[0][0]], [points[1][0]], [points[2][0]]])
pnts = append(pnts, points, axis=1)
else:
pnts = points
self.points.append(pnts)
self.point_style.append('s')
elif meth == 'l':
self.points.append(points)
self.point_style.append('l')
else:
self.points.append(points)
self.point_style.append('m')
[docs] def add_states(self, state, kind='vector'):
"""Add a state vector Qobj to Bloch sphere.
Parameters
----------
state : qobj
Input state vector.
kind : str {'vector','point'}
Type of object to plot.
"""
if isinstance(state, Qobj):
state = [state]
for st in state:
vec = [expect(sigmax(), st),
expect(sigmay(), st),
expect(sigmaz(), st)]
if kind == 'vector':
self.add_vectors(vec)
elif kind == 'point':
self.add_points(vec)
[docs] def add_vectors(self, vectors):
"""Add a list of vectors to Bloch sphere.
Parameters
----------
vectors : array_like
Array with vectors of unit length or smaller.
"""
if isinstance(vectors[0], (list, ndarray)):
for vec in vectors:
self.vectors.append(vec)
else:
self.vectors.append(vectors)
[docs] def add_annotation(self, state_or_vector, text, **kwargs):
"""Add a text or LaTeX annotation to Bloch sphere,
parametrized by a qubit state or a vector.
Parameters
----------
state_or_vector : Qobj/array/list/tuple
Position for the annotaion.
Qobj of a qubit or a vector of 3 elements.
text : str/unicode
Annotation text.
You can use LaTeX, but remember to use raw string
e.g. r"$\\langle x \\rangle$"
or escape backslashes
e.g. "$\\\\langle x \\\\rangle$".
**kwargs :
Options as for mplot3d.axes3d.text, including:
fontsize, color, horizontalalignment, verticalalignment.
"""
if isinstance(state_or_vector, Qobj):
vec = [expect(sigmax(), state_or_vector),
expect(sigmay(), state_or_vector),
expect(sigmaz(), state_or_vector)]
elif isinstance(state_or_vector, (list, ndarray, tuple)) \
and len(state_or_vector) == 3:
vec = state_or_vector
else:
raise Exception("Position needs to be specified by a qubit " +
"state or a 3D vector.")
self.annotations.append({'position': vec,
'text': text,
'opts': kwargs})
[docs] def make_sphere(self):
"""
Plots Bloch sphere and data sets.
"""
self.render(self.fig, self.axes)
def run_from_ipython(self):
try:
__IPYTHON__
return True
except NameError:
return False
[docs] def render(self, fig=None, axes=None):
"""
Render the Bloch sphere and its data sets in on given figure and axes.
"""
if self._rendered:
self.axes.clear()
self._rendered = True
# Figure instance for Bloch sphere plot
if not fig:
self.fig = plt.figure(figsize=self.figsize)
if not axes:
self.axes = Axes3D(self.fig, azim=self.view[0], elev=self.view[1])
if self.background:
self.axes.clear()
self.axes.set_xlim3d(-1.3, 1.3)
self.axes.set_ylim3d(-1.3, 1.3)
self.axes.set_zlim3d(-1.3, 1.3)
else:
self.plot_axes()
self.axes.set_axis_off()
self.axes.set_xlim3d(-0.7, 0.7)
self.axes.set_ylim3d(-0.7, 0.7)
self.axes.set_zlim3d(-0.7, 0.7)
self.axes.grid(False)
self.plot_back()
self.plot_points()
self.plot_vectors()
self.plot_front()
self.plot_axes_labels()
self.plot_annotations()
def plot_back(self):
# back half of sphere
u = linspace(0, pi, 25)
v = linspace(0, pi, 25)
x = outer(cos(u), sin(v))
y = outer(sin(u), sin(v))
z = outer(ones(size(u)), cos(v))
self.axes.plot_surface(x, y, z, rstride=2, cstride=2,
color=self.sphere_color, linewidth=0,
alpha=self.sphere_alpha)
# wireframe
self.axes.plot_wireframe(x, y, z, rstride=5, cstride=5,
color=self.frame_color,
alpha=self.frame_alpha)
# equator
self.axes.plot(1.0 * cos(u), 1.0 * sin(u), zs=0, zdir='z',
lw=self.frame_width, color=self.frame_color)
self.axes.plot(1.0 * cos(u), 1.0 * sin(u), zs=0, zdir='x',
lw=self.frame_width, color=self.frame_color)
def plot_front(self):
# front half of sphere
u = linspace(-pi, 0, 25)
v = linspace(0, pi, 25)
x = outer(cos(u), sin(v))
y = outer(sin(u), sin(v))
z = outer(ones(size(u)), cos(v))
self.axes.plot_surface(x, y, z, rstride=2, cstride=2,
color=self.sphere_color, linewidth=0,
alpha=self.sphere_alpha)
# wireframe
self.axes.plot_wireframe(x, y, z, rstride=5, cstride=5,
color=self.frame_color,
alpha=self.frame_alpha)
# equator
self.axes.plot(1.0 * cos(u), 1.0 * sin(u),
zs=0, zdir='z', lw=self.frame_width,
color=self.frame_color)
self.axes.plot(1.0 * cos(u), 1.0 * sin(u),
zs=0, zdir='x', lw=self.frame_width,
color=self.frame_color)
def plot_axes(self):
# axes
span = linspace(-1.0, 1.0, 2)
self.axes.plot(span, 0 * span, zs=0, zdir='z', label='X',
lw=self.frame_width, color=self.frame_color)
self.axes.plot(0 * span, span, zs=0, zdir='z', label='Y',
lw=self.frame_width, color=self.frame_color)
self.axes.plot(0 * span, span, zs=0, zdir='y', label='Z',
lw=self.frame_width, color=self.frame_color)
def plot_axes_labels(self):
# axes labels
opts = {'fontsize': self.font_size,
'color': self.font_color,
'horizontalalignment': 'center',
'verticalalignment': 'center'}
self.axes.text(0, -self.xlpos[0], 0, self.xlabel[0], **opts)
self.axes.text(0, -self.xlpos[1], 0, self.xlabel[1], **opts)
self.axes.text(self.ylpos[0], 0, 0, self.ylabel[0], **opts)
self.axes.text(self.ylpos[1], 0, 0, self.ylabel[1], **opts)
self.axes.text(0, 0, self.zlpos[0], self.zlabel[0], **opts)
self.axes.text(0, 0, self.zlpos[1], self.zlabel[1], **opts)
for a in (self.axes.w_xaxis.get_ticklines() +
self.axes.w_xaxis.get_ticklabels()):
a.set_visible(False)
for a in (self.axes.w_yaxis.get_ticklines() +
self.axes.w_yaxis.get_ticklabels()):
a.set_visible(False)
for a in (self.axes.w_zaxis.get_ticklines() +
self.axes.w_zaxis.get_ticklabels()):
a.set_visible(False)
def plot_vectors(self):
# -X and Y data are switched for plotting purposes
for k in range(len(self.vectors)):
xs3d = self.vectors[k][1] * array([0, 1])
ys3d = -self.vectors[k][0] * array([0, 1])
zs3d = self.vectors[k][2] * array([0, 1])
color = self.vector_color[mod(k, len(self.vector_color))]
if self.vector_style == '':
# simple line style
self.axes.plot(xs3d, ys3d, zs3d,
zs=0, zdir='z', label='Z',
lw=self.vector_width, color=color)
else:
# decorated style, with arrow heads
a = Arrow3D(xs3d, ys3d, zs3d,
mutation_scale=self.vector_mutation,
lw=self.vector_width,
arrowstyle=self.vector_style,
color=color)
self.axes.add_artist(a)
def plot_points(self):
# -X and Y data are switched for plotting purposes
for k in range(len(self.points)):
num = len(self.points[k][0])
dist = [sqrt(self.points[k][0][j] ** 2 +
self.points[k][1][j] ** 2 +
self.points[k][2][j] ** 2) for j in range(num)]
if any(abs(dist - dist[0]) / dist[0] > 1e-12):
# combine arrays so that they can be sorted together
zipped = list(zip(dist, range(num)))
zipped.sort() # sort rates from lowest to highest
dist, indperm = zip(*zipped)
indperm = array(indperm)
else:
indperm = arange(num)
if self.point_style[k] == 's':
self.axes.scatter(
real(self.points[k][1][indperm]),
- real(self.points[k][0][indperm]),
real(self.points[k][2][indperm]),
s=self.point_size[mod(k, len(self.point_size))],
alpha=1,
edgecolor='none',
zdir='z',
color=self.point_color[mod(k, len(self.point_color))],
marker=self.point_marker[mod(k, len(self.point_marker))])
elif self.point_style[k] == 'm':
pnt_colors = array(self.point_color *
int(ceil(num / float(len(self.point_color)))))
pnt_colors = pnt_colors[0:num]
pnt_colors = list(pnt_colors[indperm])
marker = self.point_marker[mod(k, len(self.point_marker))]
s = self.point_size[mod(k, len(self.point_size))]
self.axes.scatter(real(self.points[k][1][indperm]),
-real(self.points[k][0][indperm]),
real(self.points[k][2][indperm]),
s=s, alpha=1, edgecolor='none',
zdir='z', color=pnt_colors,
marker=marker)
elif self.point_style[k] == 'l':
color = self.point_color[mod(k, len(self.point_color))]
self.axes.plot(real(self.points[k][1]),
-real(self.points[k][0]),
real(self.points[k][2]),
alpha=0.75, zdir='z',
color=color)
def plot_annotations(self):
# -X and Y data are switched for plotting purposes
for annotation in self.annotations:
vec = annotation['position']
opts = {'fontsize': self.font_size,
'color': self.font_color,
'horizontalalignment': 'center',
'verticalalignment': 'center'}
opts.update(annotation['opts'])
self.axes.text(vec[1], -vec[0], vec[2],
annotation['text'], **opts)
[docs] def show(self):
"""
Display Bloch sphere and corresponding data sets.
"""
self.render(self.fig, self.axes)
if self.run_from_ipython():
if self._shown:
display(self.fig)
else:
self.fig.show()
self._shown = True
[docs] def save(self, name=None, format='png', dirc=None):
"""Saves Bloch sphere to file of type ``format`` in directory ``dirc``.
Parameters
----------
name : str
Name of saved image. Must include path and format as well.
i.e. '/Users/Paul/Desktop/bloch.png'
This overrides the 'format' and 'dirc' arguments.
format : str
Format of output image.
dirc : str
Directory for output images. Defaults to current working directory.
Returns
-------
File containing plot of Bloch sphere.
"""
self.render(self.fig, self.axes)
if dirc:
if not os.path.isdir(os.getcwd() + "/" + str(dirc)):
os.makedirs(os.getcwd() + "/" + str(dirc))
if name is None:
if dirc:
self.fig.savefig(os.getcwd() + "/" + str(dirc) + '/bloch_' +
str(self.savenum) + '.' + format)
else:
self.fig.savefig(os.getcwd() + '/bloch_' + str(self.savenum) +
'.' + format)
else:
self.fig.savefig(name)
self.savenum += 1
if self.fig:
plt.close(self.fig)
def _hide_tick_lines_and_labels(axis):
'''
Set visible property of ticklines and ticklabels of an axis to False
'''
for a in axis.get_ticklines() + axis.get_ticklabels():
a.set_visible(False)