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bloch.py
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bloch.py
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# 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.
###############################################################################
import os
from numpy import (ndarray, array, linspace, pi, outer, cos, sin, ones, size,
sqrt, real, mod, append, ceil, arange)
import matplotlib.pyplot as plt
from mpl_toolkits.mplot3d import Axes3D
from matplotlib.patches import FancyArrowPatch
from mpl_toolkits.mplot3d import proj3d
from qutip.qobj import Qobj
from qutip.expect import expect
from qutip.operators import sigmax, sigmay, sigmaz
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)
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
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
def clear(self):
"""Resets Bloch sphere data sets to empty.
"""
self.points = []
self.vectors = []
self.point_style = []
self.annotations = []
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')
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)
def add_vectors(self, vectors):
"""Add a list of vectors to Bloch sphere.
Parameters
----------
vectors : array/list
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)
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})
def make_sphere(self):
"""
Plots Bloch sphere and data sets.
"""
self.render(self.fig, self.axes)
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 *
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)
def show(self):
"""
Display Bloch sphere and corresponding data sets.
"""
self.render(self.fig, self.axes)
if self.fig:
plt.show(self.fig)
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:
plt.savefig(os.getcwd() + "/" + str(dirc) + '/bloch_' +
str(self.savenum) + '.' + format)
else:
plt.savefig(os.getcwd() + '/bloch_' + str(self.savenum) +
'.' + format)
else:
plt.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)