# This file is part of QuTiP: Quantum Toolbox in Python.
#
# Copyright (c) 2011 and later, Paul D. Nation and Robert J. Johansson.
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"""
Functions for visualizing results of quantum dynamics simulations,
visualizations of quantum states and processes.
"""
__all__ = ['hinton', 'sphereplot', 'energy_level_diagram',
'plot_energy_levels', 'fock_distribution',
'plot_fock_distribution', 'wigner_fock_distribution',
'plot_wigner_fock_distribution', 'plot_wigner',
'plot_expectation_values', 'plot_spin_distribution_2d',
'plot_spin_distribution_3d', 'plot_qubism', 'plot_schmidt',
'complex_array_to_rgb', 'matrix_histogram',
'matrix_histogram_complex', 'sphereplot']
import warnings
import itertools as it
import numpy as np
from numpy import pi, array, sin, cos, angle, log2, sqrt
try:
import matplotlib.pyplot as plt
import matplotlib as mpl
from matplotlib import cm
from mpl_toolkits.mplot3d import Axes3D
except:
pass
from qutip.qobj import Qobj, isket
from qutip.states import ket2dm
from qutip.wigner import wigner
from qutip.tensor import tensor
from qutip.matplotlib_utilities import complex_phase_cmap
from qutip.superoperator import vector_to_operator
from qutip.superop_reps import to_super, _super_to_superpauli, _isqubitdims, _pauli_basis
from qutip.tensor import flatten
from qutip import settings
# Adopted from the SciPy Cookbook.
def _blob(x, y, w, w_max, area, cmap=None):
"""
Draws a square-shaped blob with the given area (< 1) at
the given coordinates.
"""
hs = np.sqrt(area) / 2
xcorners = array([x - hs, x + hs, x + hs, x - hs])
ycorners = array([y - hs, y - hs, y + hs, y + hs])
plt.fill(xcorners, ycorners,
color=cmap(int((w + w_max) * 256 / (2 * w_max))))
def _cb_labels(left_dims):
"""Creates plot labels for matrix elements in the computational basis.
Parameters
----------
left_dims : flat list of ints
Dimensions of the left index of a density operator. E. g.
[2, 3] for a qubit tensored with a qutrit.
Returns
-------
left_labels, right_labels : lists of strings
Labels for the left and right indices of a density operator
(kets and bras, respectively).
"""
# FIXME: assumes dims, such that we only need left_dims == dims[0].
basis_labels = list(map(",".join, it.product(*[
map(str, range(dim))
for dim in left_dims
])))
return [
map(fmt.format, basis_labels) for fmt in
(
r"$|{}\rangle$",
r"$\langle{}|$"
)
]
# Adopted from the SciPy Cookbook.
[docs]def hinton(rho, xlabels=None, ylabels=None, title=None, ax=None, cmap=None,
label_top=True):
"""Draws a Hinton diagram for visualizing a density matrix or superoperator.
Parameters
----------
rho : qobj
Input density matrix or superoperator.
xlabels : list of strings or False
list of x labels
ylabels : list of strings or False
list of y labels
title : string
title of the plot (optional)
ax : a matplotlib axes instance
The axes context in which the plot will be drawn.
cmap : a matplotlib colormap instance
Color map to use when plotting.
label_top : bool
If True, x-axis labels will be placed on top, otherwise
they will appear below the plot.
Returns
-------
fig, ax : tuple
A tuple of the matplotlib figure and axes instances used to produce
the figure.
Raises
------
ValueError
Input argument is not a quantum object.
"""
# Apply default colormaps.
# TODO: abstract this away into something that makes default
# colormaps.
cmap = (
(cm.Greys_r if settings.colorblind_safe else cm.RdBu)
if cmap is None else cmap
)
# Extract plotting data W from the input.
if isinstance(rho, Qobj):
if rho.isoper:
W = rho.full()
# Create default labels if none are given.
if xlabels is None or ylabels is None:
labels = _cb_labels(rho.dims[0])
xlabels = xlabels if xlabels is not None else list(labels[0])
ylabels = ylabels if ylabels is not None else list(labels[1])
elif rho.isoperket:
W = vector_to_operator(rho).full()
elif rho.isoperbra:
W = vector_to_operator(rho.dag()).full()
elif rho.issuper:
if not _isqubitdims(rho.dims):
raise ValueError("Hinton plots of superoperators are "
"currently only supported for qubits.")
# Convert to a superoperator in the Pauli basis,
# so that all the elements are real.
sqobj = _super_to_superpauli(rho)
nq = int(log2(sqobj.shape[0]) / 2)
W = sqobj.full().T
# Create default labels, too.
if (xlabels is None) or (ylabels is None):
labels = list(map("".join, it.product("IXYZ", repeat=nq)))
xlabels = xlabels if xlabels is not None else labels
ylabels = ylabels if ylabels is not None else labels
else:
raise ValueError(
"Input quantum object must be an operator or superoperator."
)
else:
W = rho
if ax is None:
fig, ax = plt.subplots(1, 1, figsize=(8, 6))
else:
fig = None
if not (xlabels or ylabels):
ax.axis('off')
ax.axis('equal')
ax.set_frame_on(False)
height, width = W.shape
w_max = 1.25 * max(abs(np.diag(np.matrix(W))))
if w_max <= 0.0:
w_max = 1.0
ax.fill(array([0, width, width, 0]), array([0, 0, height, height]),
color=cmap(128))
for x in range(width):
for y in range(height):
_x = x + 1
_y = y + 1
if np.real(W[x, y]) > 0.0:
_blob(_x - 0.5, height - _y + 0.5, abs(W[x,
y]), w_max, min(1, abs(W[x, y]) / w_max), cmap=cmap)
else:
_blob(_x - 0.5, height - _y + 0.5, -abs(W[
x, y]), w_max, min(1, abs(W[x, y]) / w_max), cmap=cmap)
# color axis
norm = mpl.colors.Normalize(-abs(W).max(), abs(W).max())
cax, kw = mpl.colorbar.make_axes(ax, shrink=0.75, pad=.1)
mpl.colorbar.ColorbarBase(cax, norm=norm, cmap=cmap)
# x axis
ax.xaxis.set_major_locator(plt.IndexLocator(1, 0.5))
if xlabels:
ax.set_xticklabels(xlabels)
if label_top:
ax.xaxis.tick_top()
ax.tick_params(axis='x', labelsize=14)
# y axis
ax.yaxis.set_major_locator(plt.IndexLocator(1, 0.5))
if ylabels:
ax.set_yticklabels(list(reversed(ylabels)))
ax.tick_params(axis='y', labelsize=14)
return fig, ax
[docs]def sphereplot(theta, phi, values, fig=None, ax=None, save=False):
"""Plots a matrix of values on a sphere
Parameters
----------
theta : float
Angle with respect to z-axis
phi : float
Angle in x-y plane
values : array
Data set to be plotted
fig : a matplotlib Figure instance
The Figure canvas in which the plot will be drawn.
ax : a matplotlib axes instance
The axes context in which the plot will be drawn.
save : bool {False , True}
Whether to save the figure or not
Returns
-------
fig, ax : tuple
A tuple of the matplotlib figure and axes instances used to produce
the figure.
"""
if fig is None or ax is None:
fig = plt.figure()
ax = Axes3D(fig)
thetam, phim = np.meshgrid(theta, phi)
xx = sin(thetam) * cos(phim)
yy = sin(thetam) * sin(phim)
zz = cos(thetam)
r = array(abs(values))
ph = angle(values)
# normalize color range based on phase angles in list ph
nrm = mpl.colors.Normalize(ph.min(), ph.max())
# plot with facecolors set to cm.jet colormap normalized to nrm
ax.plot_surface(r * xx, r * yy, r * zz, rstride=1, cstride=1,
facecolors=cm.jet(nrm(ph)), linewidth=0)
# create new axes on plot for colorbar and shrink it a bit.
# pad shifts location of bar with repsect to the main plot
cax, kw = mpl.colorbar.make_axes(ax, shrink=.66, pad=.02)
# create new colorbar in axes cax with cm jet and normalized to nrm like
# our facecolors
cb1 = mpl.colorbar.ColorbarBase(cax, cmap=cm.jet, norm=nrm)
# add our colorbar label
cb1.set_label('Angle')
if save:
plt.savefig("sphereplot.png")
return fig, ax
[docs]def matrix_histogram(M, xlabels=None, ylabels=None, title=None, limits=None,
colorbar=True, fig=None, ax=None):
"""
Draw a histogram for the matrix M, with the given x and y labels and title.
Parameters
----------
M : Matrix of Qobj
The matrix to visualize
xlabels : list of strings
list of x labels
ylabels : list of strings
list of y labels
title : string
title of the plot (optional)
limits : list/array with two float numbers
The z-axis limits [min, max] (optional)
ax : a matplotlib axes instance
The axes context in which the plot will be drawn.
Returns
-------
fig, ax : tuple
A tuple of the matplotlib figure and axes instances used to produce
the figure.
Raises
------
ValueError
Input argument is not valid.
"""
if isinstance(M, Qobj):
# extract matrix data from Qobj
M = M.full()
n = np.size(M)
xpos, ypos = np.meshgrid(range(M.shape[0]), range(M.shape[1]))
xpos = xpos.T.flatten() - 0.5
ypos = ypos.T.flatten() - 0.5
zpos = np.zeros(n)
dx = dy = 0.8 * np.ones(n)
dz = np.real(M.flatten())
if limits and type(limits) is list and len(limits) == 2:
z_min = limits[0]
z_max = limits[1]
else:
z_min = min(dz)
z_max = max(dz)
if z_min == z_max:
z_min -= 0.1
z_max += 0.1
norm = mpl.colors.Normalize(z_min, z_max)
cmap = cm.get_cmap('jet') # Spectral
colors = cmap(norm(dz))
if ax is None:
fig = plt.figure()
ax = Axes3D(fig, azim=-35, elev=35)
ax.bar3d(xpos, ypos, zpos, dx, dy, dz, color=colors)
if title and fig:
ax.set_title(title)
# x axis
ax.axes.w_xaxis.set_major_locator(plt.IndexLocator(1, -0.5))
if xlabels:
ax.set_xticklabels(xlabels)
ax.tick_params(axis='x', labelsize=14)
# y axis
ax.axes.w_yaxis.set_major_locator(plt.IndexLocator(1, -0.5))
if ylabels:
ax.set_yticklabels(ylabels)
ax.tick_params(axis='y', labelsize=14)
# z axis
ax.axes.w_zaxis.set_major_locator(plt.IndexLocator(1, 0.5))
ax.set_zlim3d([min(z_min, 0), z_max])
# color axis
if colorbar:
cax, kw = mpl.colorbar.make_axes(ax, shrink=.75, pad=.0)
mpl.colorbar.ColorbarBase(cax, cmap=cmap, norm=norm)
return fig, ax
[docs]def matrix_histogram_complex(M, xlabels=None, ylabels=None,
title=None, limits=None, phase_limits=None,
colorbar=True, fig=None, ax=None,
threshold=None):
"""
Draw a histogram for the amplitudes of matrix M, using the argument
of each element for coloring the bars, with the given x and y labels
and title.
Parameters
----------
M : Matrix of Qobj
The matrix to visualize
xlabels : list of strings
list of x labels
ylabels : list of strings
list of y labels
title : string
title of the plot (optional)
limits : list/array with two float numbers
The z-axis limits [min, max] (optional)
phase_limits : list/array with two float numbers
The phase-axis (colorbar) limits [min, max] (optional)
ax : a matplotlib axes instance
The axes context in which the plot will be drawn.
threshold: float (None)
Threshold for when bars of smaller height should be transparent. If
not set, all bars are colored according to the color map.
Returns
-------
fig, ax : tuple
A tuple of the matplotlib figure and axes instances used to produce
the figure.
Raises
------
ValueError
Input argument is not valid.
"""
if isinstance(M, Qobj):
# extract matrix data from Qobj
M = M.full()
n = np.size(M)
xpos, ypos = np.meshgrid(range(M.shape[0]), range(M.shape[1]))
xpos = xpos.T.flatten() - 0.5
ypos = ypos.T.flatten() - 0.5
zpos = np.zeros(n)
dx = dy = 0.8 * np.ones(n)
Mvec = M.flatten()
dz = abs(Mvec)
# make small numbers real, to avoid random colors
idx, = np.where(abs(Mvec) < 0.001)
Mvec[idx] = abs(Mvec[idx])
if phase_limits: # check that limits is a list type
phase_min = phase_limits[0]
phase_max = phase_limits[1]
else:
phase_min = -pi
phase_max = pi
norm = mpl.colors.Normalize(phase_min, phase_max)
cmap = complex_phase_cmap()
colors = cmap(norm(angle(Mvec)))
if threshold is not None:
colors[:, 3] = 1 * (dz > threshold)
if ax is None:
fig = plt.figure()
ax = Axes3D(fig, azim=-35, elev=35)
ax.bar3d(xpos, ypos, zpos, dx, dy, dz, color=colors)
if title and fig:
ax.set_title(title)
# x axis
ax.axes.w_xaxis.set_major_locator(plt.IndexLocator(1, -0.5))
if xlabels:
ax.set_xticklabels(xlabels)
ax.tick_params(axis='x', labelsize=12)
# y axis
ax.axes.w_yaxis.set_major_locator(plt.IndexLocator(1, -0.5))
if ylabels:
ax.set_yticklabels(ylabels)
ax.tick_params(axis='y', labelsize=12)
# z axis
if limits and isinstance(limits, list):
ax.set_zlim3d(limits)
else:
ax.set_zlim3d([0, 1]) # use min/max
# ax.set_zlabel('abs')
# color axis
if colorbar:
cax, kw = mpl.colorbar.make_axes(ax, shrink=.75, pad=.0)
cb = mpl.colorbar.ColorbarBase(cax, cmap=cmap, norm=norm)
cb.set_ticks([-pi, -pi / 2, 0, pi / 2, pi])
cb.set_ticklabels(
(r'$-\pi$', r'$-\pi/2$', r'$0$', r'$\pi/2$', r'$\pi$'))
cb.set_label('arg')
return fig, ax
[docs]def plot_energy_levels(H_list, N=0, labels=None, show_ylabels=False,
figsize=(8, 12), fig=None, ax=None):
"""
Plot the energy level diagrams for a list of Hamiltonians. Include
up to N energy levels. For each element in H_list, the energy
levels diagram for the cummulative Hamiltonian sum(H_list[0:n]) is plotted,
where n is the index of an element in H_list.
Parameters
----------
H_list : List of Qobj
A list of Hamiltonians.
labels : List of string
A list of labels for each Hamiltonian
show_ylabels : Bool (default False)
Show y labels to the left of energy levels of the initial
Hamiltonian.
N : int
The number of energy levels to plot
figsize : tuple (int,int)
The size of the figure (width, height).
fig : a matplotlib Figure instance
The Figure canvas in which the plot will be drawn.
ax : a matplotlib axes instance
The axes context in which the plot will be drawn.
Returns
-------
fig, ax : tuple
A tuple of the matplotlib figure and axes instances used to produce
the figure.
Raises
------
ValueError
Input argument is not valid.
"""
if not isinstance(H_list, list):
raise ValueError("H_list must be a list of Qobj instances")
if not fig and not ax:
fig, ax = plt.subplots(1, 1, figsize=figsize)
H = H_list[0]
N = H.shape[0] if N == 0 else min(H.shape[0], N)
xticks = []
yticks = []
x = 0
evals0 = H.eigenenergies(eigvals=N) / (2 * np.pi)
for e_idx, e in enumerate(evals0[:N]):
ax.plot([x, x + 2], np.array([1, 1]) * e, 'b', linewidth=2)
yticks.append(e)
xticks.append(x + 1)
x += 2
for H1 in H_list[1:]:
H = H + H1
evals1 = H.eigenenergies() / (2 * np.pi)
for e_idx, e in enumerate(evals1[:N]):
ax.plot([x, x + 1], np.array([evals0[e_idx], e]), 'k:')
x += 1
for e_idx, e in enumerate(evals1[:N]):
ax.plot([x, x + 2], np.array([1, 1]) * e, 'b', linewidth=2)
xticks.append(x + 1)
x += 2
evals0 = evals1
ax.set_frame_on(False)
if show_ylabels:
yticks = np.unique(np.around(yticks, 1))
ax.set_yticks(yticks)
else:
ax.axes.get_yaxis().set_visible(False)
if labels:
ax.get_xaxis().tick_bottom()
ax.set_xticks(xticks)
ax.set_xticklabels(labels, fontsize=16)
else:
ax.axes.get_xaxis().set_visible(False)
return fig, ax
def energy_level_diagram(H_list, N=0, labels=None, show_ylabels=False,
figsize=(8, 12), fig=None, ax=None):
warnings.warn("Deprecated: Use plot_energy_levels")
return plot_energy_levels(H_list, N=N, labels=labels,
show_ylabels=show_ylabels,
figsize=figsize, fig=fig, ax=ax)
[docs]def plot_fock_distribution(rho, offset=0, fig=None, ax=None,
figsize=(8, 6), title=None, unit_y_range=True):
"""
Plot the Fock distribution for a density matrix (or ket) that describes
an oscillator mode.
Parameters
----------
rho : :class:`qutip.qobj.Qobj`
The density matrix (or ket) of the state to visualize.
fig : a matplotlib Figure instance
The Figure canvas in which the plot will be drawn.
ax : a matplotlib axes instance
The axes context in which the plot will be drawn.
title : string
An optional title for the figure.
figsize : (width, height)
The size of the matplotlib figure (in inches) if it is to be created
(that is, if no 'fig' and 'ax' arguments are passed).
Returns
-------
fig, ax : tuple
A tuple of the matplotlib figure and axes instances used to produce
the figure.
"""
if not fig and not ax:
fig, ax = plt.subplots(1, 1, figsize=figsize)
if isket(rho):
rho = ket2dm(rho)
N = rho.shape[0]
ax.bar(np.arange(offset, offset + N) - .4, np.real(rho.diag()),
color="green", alpha=0.6, width=0.8)
if unit_y_range:
ax.set_ylim(0, 1)
ax.set_xlim(-.5 + offset, N + offset)
ax.set_xlabel('Fock number', fontsize=12)
ax.set_ylabel('Occupation probability', fontsize=12)
if title:
ax.set_title(title)
return fig, ax
def fock_distribution(rho, offset=0, fig=None, ax=None,
figsize=(8, 6), title=None, unit_y_range=True):
warnings.warn("Deprecated: Use plot_fock_distribution")
return plot_fock_distribution(rho, offset=offset, fig=fig, ax=ax,
figsize=figsize, title=title,
unit_y_range=unit_y_range)
[docs]def plot_wigner(rho, fig=None, ax=None, figsize=(8, 4),
cmap=None, alpha_max=7.5, colorbar=False,
method='iterative', projection='2d'):
"""
Plot the the Wigner function for a density matrix (or ket) that describes
an oscillator mode.
Parameters
----------
rho : :class:`qutip.qobj.Qobj`
The density matrix (or ket) of the state to visualize.
fig : a matplotlib Figure instance
The Figure canvas in which the plot will be drawn.
ax : a matplotlib axes instance
The axes context in which the plot will be drawn.
figsize : (width, height)
The size of the matplotlib figure (in inches) if it is to be created
(that is, if no 'fig' and 'ax' arguments are passed).
cmap : a matplotlib cmap instance
The colormap.
alpha_max : float
The span of the x and y coordinates (both [-alpha_max, alpha_max]).
colorbar : bool
Whether (True) or not (False) a colorbar should be attached to the
Wigner function graph.
method : string {'iterative', 'laguerre', 'fft'}
The method used for calculating the wigner function. See the
documentation for qutip.wigner for details.
projection: string {'2d', '3d'}
Specify whether the Wigner function is to be plotted as a
contour graph ('2d') or surface plot ('3d').
Returns
-------
fig, ax : tuple
A tuple of the matplotlib figure and axes instances used to produce
the figure.
"""
if not fig and not ax:
if projection == '2d':
fig, ax = plt.subplots(1, 1, figsize=figsize)
elif projection == '3d':
fig = plt.figure(figsize=figsize)
ax = fig.add_subplot(1, 1, 1, projection='3d')
else:
raise ValueError('Unexpected value of projection keyword argument')
if isket(rho):
rho = ket2dm(rho)
xvec = np.linspace(-alpha_max, alpha_max, 200)
W0 = wigner(rho, xvec, xvec, method=method)
W, yvec = W0 if type(W0) is tuple else (W0, xvec)
wlim = abs(W).max()
if cmap is None:
cmap = cm.get_cmap('RdBu')
if projection == '2d':
cf = ax.contourf(xvec, yvec, W, 100,
norm=mpl.colors.Normalize(-wlim, wlim), cmap=cmap)
elif projection == '3d':
X, Y = np.meshgrid(xvec, xvec)
cf = ax.plot_surface(X, Y, W0, rstride=5, cstride=5, linewidth=0.5,
norm=mpl.colors.Normalize(-wlim, wlim), cmap=cmap)
else:
raise ValueError('Unexpected value of projection keyword argument.')
if xvec is not yvec:
ax.set_ylim(xvec.min(), xvec.max())
ax.set_xlabel(r'$\rm{Re}(\alpha)$', fontsize=12)
ax.set_ylabel(r'$\rm{Im}(\alpha)$', fontsize=12)
if colorbar:
fig.colorbar(cf, ax=ax)
ax.set_title("Wigner function", fontsize=12)
return fig, ax
[docs]def plot_wigner_fock_distribution(rho, fig=None, axes=None, figsize=(8, 4),
cmap=None, alpha_max=7.5, colorbar=False,
method='iterative', projection='2d'):
"""
Plot the Fock distribution and the Wigner function for a density matrix
(or ket) that describes an oscillator mode.
Parameters
----------
rho : :class:`qutip.qobj.Qobj`
The density matrix (or ket) of the state to visualize.
fig : a matplotlib Figure instance
The Figure canvas in which the plot will be drawn.
axes : a list of two matplotlib axes instances
The axes context in which the plot will be drawn.
figsize : (width, height)
The size of the matplotlib figure (in inches) if it is to be created
(that is, if no 'fig' and 'ax' arguments are passed).
cmap : a matplotlib cmap instance
The colormap.
alpha_max : float
The span of the x and y coordinates (both [-alpha_max, alpha_max]).
colorbar : bool
Whether (True) or not (False) a colorbar should be attached to the
Wigner function graph.
method : string {'iterative', 'laguerre', 'fft'}
The method used for calculating the wigner function. See the
documentation for qutip.wigner for details.
projection: string {'2d', '3d'}
Specify whether the Wigner function is to be plotted as a
contour graph ('2d') or surface plot ('3d').
Returns
-------
fig, ax : tuple
A tuple of the matplotlib figure and axes instances used to produce
the figure.
"""
if not fig and not axes:
if projection == '2d':
fig, axes = plt.subplots(1, 2, figsize=figsize)
elif projection == '3d':
fig = plt.figure(figsize=figsize)
axes = [fig.add_subplot(1, 2, 1),
fig.add_subplot(1, 2, 2, projection='3d')]
else:
raise ValueError('Unexpected value of projection keyword argument')
if isket(rho):
rho = ket2dm(rho)
plot_fock_distribution(rho, fig=fig, ax=axes[0])
plot_wigner(rho, fig=fig, ax=axes[1], figsize=figsize, cmap=cmap,
alpha_max=alpha_max, colorbar=colorbar, method=method,
projection=projection)
return fig, axes
def wigner_fock_distribution(rho, fig=None, axes=None, figsize=(8, 4),
cmap=None, alpha_max=7.5, colorbar=False,
method='iterative'):
warnings.warn("Deprecated: Use plot_wigner_fock_distribution")
return plot_wigner_fock_distribution(rho, fig=fig, axes=axes,
figsize=figsize, cmap=cmap,
alpha_max=alpha_max,
colorbar=colorbar,
method=method)
[docs]def plot_expectation_values(results, ylabels=[], title=None, show_legend=False,
fig=None, axes=None, figsize=(8, 4)):
"""
Visualize the results (expectation values) for an evolution solver.
`results` is assumed to be an instance of Result, or a list of Result
instances.
Parameters
----------
results : (list of) :class:`qutip.solver.Result`
List of results objects returned by any of the QuTiP evolution solvers.
ylabels : list of strings
The y-axis labels. List should be of the same length as `results`.
title : string
The title of the figure.
show_legend : bool
Whether or not to show the legend.
fig : a matplotlib Figure instance
The Figure canvas in which the plot will be drawn.
axes : a matplotlib axes instance
The axes context in which the plot will be drawn.
figsize : (width, height)
The size of the matplotlib figure (in inches) if it is to be created
(that is, if no 'fig' and 'ax' arguments are passed).
Returns
-------
fig, ax : tuple
A tuple of the matplotlib figure and axes instances used to produce
the figure.
"""
if not isinstance(results, list):
results = [results]
n_e_ops = max([len(result.expect) for result in results])
if not fig or not axes:
if not figsize:
figsize = (12, 3 * n_e_ops)
fig, axes = plt.subplots(n_e_ops, 1, sharex=True,
figsize=figsize, squeeze=False)
for r_idx, result in enumerate(results):
for e_idx, e in enumerate(result.expect):
axes[e_idx, 0].plot(result.times, e,
label="%s [%d]" % (result.solver, e_idx))
if title:
axes[0, 0].set_title(title)
axes[n_e_ops - 1, 0].set_xlabel("time", fontsize=12)
for n in range(n_e_ops):
if show_legend:
axes[n, 0].legend()
if ylabels:
axes[n, 0].set_ylabel(ylabels[n], fontsize=12)
return fig, axes
[docs]def plot_spin_distribution_2d(P, THETA, PHI,
fig=None, ax=None, figsize=(8, 8)):
"""
Plot a spin distribution function (given as meshgrid data) with a 2D
projection where the surface of the unit sphere is mapped on the unit disk.
Parameters
----------
P : matrix
Distribution values as a meshgrid matrix.
THETA : matrix
Meshgrid matrix for the theta coordinate.
PHI : matrix
Meshgrid matrix for the phi coordinate.
fig : a matplotlib figure instance
The figure canvas on which the plot will be drawn.
ax : a matplotlib axis instance
The axis context in which the plot will be drawn.
figsize : (width, height)
The size of the matplotlib figure (in inches) if it is to be created
(that is, if no 'fig' and 'ax' arguments are passed).
Returns
-------
fig, ax : tuple
A tuple of the matplotlib figure and axes instances used to produce
the figure.
"""
if not fig or not ax:
if not figsize:
figsize = (8, 8)
fig, ax = plt.subplots(1, 1, figsize=figsize)
Y = (THETA - pi / 2) / (pi / 2)
X = (pi - PHI) / pi * np.sqrt(cos(THETA - pi / 2))
if P.min() < -1e12:
cmap = cm.RdBu
else:
cmap = cm.RdYlBu
ax.pcolor(X, Y, P.real, cmap=cmap)
ax.set_xlabel(r'$\varphi$', fontsize=18)
ax.set_ylabel(r'$\theta$', fontsize=18)
ax.set_xticks([-1, 0, 1])
ax.set_xticklabels([r'$0$', r'$\pi$', r'$2\pi$'], fontsize=18)
ax.set_yticks([-1, 0, 1])
ax.set_yticklabels([r'$\pi$', r'$\pi/2$', r'$0$'], fontsize=18)
return fig, ax
[docs]def plot_spin_distribution_3d(P, THETA, PHI,
fig=None, ax=None, figsize=(8, 6)):
"""Plots a matrix of values on a sphere
Parameters
----------
P : matrix
Distribution values as a meshgrid matrix.
THETA : matrix
Meshgrid matrix for the theta coordinate.
PHI : matrix
Meshgrid matrix for the phi coordinate.
fig : a matplotlib figure instance
The figure canvas on which the plot will be drawn.
ax : a matplotlib axis instance
The axis context in which the plot will be drawn.
figsize : (width, height)
The size of the matplotlib figure (in inches) if it is to be created
(that is, if no 'fig' and 'ax' arguments are passed).
Returns
-------
fig, ax : tuple
A tuple of the matplotlib figure and axes instances used to produce
the figure.
"""
if fig is None or ax is None:
fig = plt.figure(figsize=figsize)
ax = Axes3D(fig, azim=-35, elev=35)
xx = sin(THETA) * cos(PHI)
yy = sin(THETA) * sin(PHI)
zz = cos(THETA)
if P.min() < -1e12:
cmap = cm.RdBu
norm = mpl.colors.Normalize(-P.max(), P.max())
else:
cmap = cm.RdYlBu
norm = mpl.colors.Normalize(P.min(), P.max())
ax.plot_surface(xx, yy, zz, rstride=1, cstride=1,
facecolors=cmap(norm(P)), linewidth=0)
cax, kw = mpl.colorbar.make_axes(ax, shrink=.66, pad=.02)
cb1 = mpl.colorbar.ColorbarBase(cax, cmap=cmap, norm=norm)
cb1.set_label('magnitude')
return fig, ax
#
# Qubism and other qubistic visualizations
#
def complex_array_to_rgb(X, theme='light', rmax=None):
"""
Makes an array of complex number and converts it to an array of [r, g, b],
where phase gives hue and saturation/value are given by the absolute value.
Especially for use with imshow for complex plots.
For more info on coloring, see:
Emilia Petrisor,
Visualizing complex-valued functions with Matplotlib and Mayavi
http://nbviewer.ipython.org/github/empet/Math/blob/master/DomainColoring.ipynb
Parameters
----------
X : array
Array (of any dimension) of complex numbers.
theme : 'light' (default) or 'dark'
Set coloring theme for mapping complex values into colors.
rmax : float
Maximal abs value for color normalization.
If None (default), uses np.abs(X).max().
Returns
-------
Y : array
Array of colors (of shape X.shape + (3,)).
"""
absmax = rmax or np.abs(X).max()
if absmax == 0.:
absmax = 1.
Y = np.zeros(X.shape + (3,), dtype='float')
Y[..., 0] = np.angle(X) / (2 * pi) % 1
if theme == 'light':
Y[..., 1] = np.clip(np.abs(X) / absmax, 0, 1)
Y[..., 2] = 1
elif theme == 'dark':
Y[..., 1] = 1
Y[..., 2] = np.clip(np.abs(X) / absmax, 0, 1)
Y = mpl.colors.hsv_to_rgb(Y)
return Y
def _index_to_sequence(i, dim_list):
"""
For a matrix entry with index i it returns state it corresponds to.
In particular, for dim_list=[2]*n it returns i written as a binary number.
Parameters
----------
i : int
Index in a matrix.
dim_list : list of int
List of dimensions of consecutive particles.
Returns
-------
seq : list
List of coordinates for each particle.
"""
res = []
j = i
for d in reversed(dim_list):
j, s = divmod(j, d)
res.append(s)
return list(reversed(res))
def _sequence_to_index(seq, dim_list):
"""
Inverse of _index_to_sequence.
Parameters
----------
seq : list of ints
List of coordinates for each particle.
dim_list : list of int
List of dimensions of consecutive particles.
Returns
-------
i : list
Index in a matrix.
"""
i = 0
for s, d in zip(seq, dim_list):
i *= d
i += s
return i
def _to_qubism_index_pair(i, dim_list, how='pairs'):
"""
For a matrix entry with index i
it returns x, y coordinates in qubism mapping.
Parameters
----------
i : int
Index in a matrix.
dim_list : list of int
List of dimensions of consecutive particles.
how : 'pairs' ('default'), 'pairs_skewed' or 'before_after'
Type of qubistic plot.
Returns
-------
x, y : tuple of ints
List of coordinates for each particle.
"""
seq = _index_to_sequence(i, dim_list)
if how == 'pairs':
y = _sequence_to_index(seq[::2], dim_list[::2])
x = _sequence_to_index(seq[1::2], dim_list[1::2])
elif how == 'pairs_skewed':
dim_list2 = dim_list[::2]
y = _sequence_to_index(seq[::2], dim_list2)
seq2 = [(b - a) % d for a, b, d in zip(seq[::2], seq[1::2], dim_list2)]
x = _sequence_to_index(seq2, dim_list2)
elif how == 'before_after':
# https://en.wikipedia.org/wiki/File:Ising-tartan.png
n = len(dim_list)
y = _sequence_to_index(reversed(seq[:(n // 2)]),
reversed(dim_list[:(n // 2)]))
x = _sequence_to_index(seq[(n // 2):], dim_list[(n // 2):])
else:
raise Exception("No such 'how'.")
return x, y
def _sequence_to_latex(seq, style='ket'):
"""
For a sequence of particle states generate LaTeX code.
Parameters
----------
seq : list of ints
List of coordinates for each particle.
style : 'ket' (default), 'bra' or 'bare'
Style of LaTeX (i.e. |01> or <01| or 01, respectively).
Returns
-------
latex : str
LaTeX output.
"""
if style == 'ket':
latex = "$\\left|{0}\\right\\rangle$"
elif style == 'bra':
latex = "$\\left\\langle{0}\\right|$"
elif style == 'bare':
latex = "${0}$"
else:
raise Exception("No such style.")
return latex.format("".join(map(str, seq)))
[docs]def plot_qubism(ket, theme='light', how='pairs',
grid_iteration=1, legend_iteration=0,
fig=None, ax=None, figsize=(6, 6)):
"""
Qubism plot for pure states of many qudits.
Works best for spin chains, especially with even number of particles
of the same dimension.
Allows to see entanglement between first 2*k particles and the rest.
More information:
J. Rodriguez-Laguna, P. Migdal,
M. Ibanez Berganza, M. Lewenstein, G. Sierra,
"Qubism: self-similar visualization of many-body wavefunctions",
New J. Phys. 14 053028 (2012), arXiv:1112.3560,
http://dx.doi.org/10.1088/1367-2630/14/5/053028 (open access)
Parameters
----------
ket : Qobj
Pure state for plotting.
theme : 'light' (default) or 'dark'
Set coloring theme for mapping complex values into colors.
See: complex_array_to_rgb.
how : 'pairs' (default), 'pairs_skewed' or 'before_after'
Type of Qubism plotting.
Options:
'pairs' - typical coordinates,
'pairs_skewed' - for ferromagnetic/antriferromagnetic plots,
'before_after' - related to Schmidt plot (see also: plot_schmidt).
grid_iteration : int (default 1)
Helper lines to be drawn on plot.
Show tiles for 2*grid_iteration particles vs all others.
legend_iteration : int (default 0) or 'grid_iteration' or 'all'
Show labels for first 2*legend_iteration particles.
Option 'grid_iteration' sets the same number of particles
as for grid_iteration.
Option 'all' makes label for all particles.
Typically it should be 0, 1, 2 or perhaps 3.
fig : a matplotlib figure instance
The figure canvas on which the plot will be drawn.
ax : a matplotlib axis instance
The axis context in which the plot will be drawn.
figsize : (width, height)
The size of the matplotlib figure (in inches) if it is to be created
(that is, if no 'fig' and 'ax' arguments are passed).
Returns
-------
fig, ax : tuple
A tuple of the matplotlib figure and axes instances used to produce
the figure.
"""
if not isket(ket):
raise Exception("Qubism works only for pure states, i.e. kets.")
# add for dm? (perhaps a separate function, plot_qubism_dm)
if not fig and not ax:
fig, ax = plt.subplots(1, 1, figsize=figsize)
dim_list = ket.dims[0]
n = len(dim_list)
# for odd number of particles - pixels are rectangular
if n % 2 == 1:
ket = tensor(ket, Qobj([1] * dim_list[-1]))
dim_list = ket.dims[0]
n += 1
ketdata = ket.full()
if how == 'pairs':
dim_list_y = dim_list[::2]
dim_list_x = dim_list[1::2]
elif how == 'pairs_skewed':
dim_list_y = dim_list[::2]
dim_list_x = dim_list[1::2]
if dim_list_x != dim_list_y:
raise Exception("For 'pairs_skewed' pairs " +
"of dimensions need to be the same.")
elif how == 'before_after':
dim_list_y = list(reversed(dim_list[:(n // 2)]))
dim_list_x = dim_list[(n // 2):]
else:
raise Exception("No such 'how'.")
size_x = np.prod(dim_list_x)
size_y = np.prod(dim_list_y)
qub = np.zeros([size_x, size_y], dtype=complex)
for i in range(ketdata.size):
qub[_to_qubism_index_pair(i, dim_list, how=how)] = ketdata[i, 0]
qub = qub.transpose()
quadrants_x = np.prod(dim_list_x[:grid_iteration])
quadrants_y = np.prod(dim_list_y[:grid_iteration])
ticks_x = [size_x // quadrants_x * i for i in range(1, quadrants_x)]
ticks_y = [size_y // quadrants_y * i for i in range(1, quadrants_y)]
ax.set_xticks(ticks_x)
ax.set_xticklabels([""] * (quadrants_x - 1))
ax.set_yticks(ticks_y)
ax.set_yticklabels([""] * (quadrants_y - 1))
theme2color_of_lines = {'light': '#000000',
'dark': '#FFFFFF'}
ax.grid(True, color=theme2color_of_lines[theme])
ax.imshow(complex_array_to_rgb(qub, theme=theme),
interpolation="none",
extent=(0, size_x, 0, size_y))
if legend_iteration == 'all':
label_n = n // 2
elif legend_iteration == 'grid_iteration':
label_n = grid_iteration
else:
try:
label_n = int(legend_iteration)
except:
raise Exception("No such option for legend_iteration keyword " +
"argument. Use 'all', 'grid_iteration' or an " +
"integer.")
if label_n:
if how == 'before_after':
dim_list_small = list(reversed(dim_list_y[-label_n:])) \
+ dim_list_x[:label_n]
else:
dim_list_small = []
for j in range(label_n):
dim_list_small.append(dim_list_y[j])
dim_list_small.append(dim_list_x[j])
scale_x = float(size_x) / np.prod(dim_list_x[:label_n])
shift_x = 0.5 * scale_x
scale_y = float(size_y) / np.prod(dim_list_y[:label_n])
shift_y = 0.5 * scale_y
bbox = ax.get_window_extent().transformed(
fig.dpi_scale_trans.inverted())
fontsize = 35 * bbox.width / np.prod(dim_list_x[:label_n]) / label_n
opts = {'fontsize': fontsize,
'color': theme2color_of_lines[theme],
'horizontalalignment': 'center',
'verticalalignment': 'center'}
for i in range(np.prod(dim_list_small)):
x, y = _to_qubism_index_pair(i, dim_list_small, how=how)
seq = _index_to_sequence(i, dim_list=dim_list_small)
ax.text(scale_x * x + shift_x,
size_y - (scale_y * y + shift_y),
_sequence_to_latex(seq),
**opts)
return fig, ax
[docs]def plot_schmidt(ket, splitting=None,
labels_iteration=(3, 2),
theme='light',
fig=None, ax=None, figsize=(6, 6)):
"""
Plotting scheme related to Schmidt decomposition.
Converts a state into a matrix (A_ij -> A_i^j),
where rows are first particles and columns - last.
See also: plot_qubism with how='before_after' for a similar plot.
Parameters
----------
ket : Qobj
Pure state for plotting.
splitting : int
Plot for a number of first particles versus the rest.
If not given, it is (number of particles + 1) // 2.
theme : 'light' (default) or 'dark'
Set coloring theme for mapping complex values into colors.
See: complex_array_to_rgb.
labels_iteration : int or pair of ints (default (3,2))
Number of particles to be shown as tick labels,
for first (vertical) and last (horizontal) particles, respectively.
fig : a matplotlib figure instance
The figure canvas on which the plot will be drawn.
ax : a matplotlib axis instance
The axis context in which the plot will be drawn.
figsize : (width, height)
The size of the matplotlib figure (in inches) if it is to be created
(that is, if no 'fig' and 'ax' arguments are passed).
Returns
-------
fig, ax : tuple
A tuple of the matplotlib figure and axes instances used to produce
the figure.
"""
if not isket(ket):
raise Exception("Schmidt plot works only for pure states, i.e. kets.")
if not fig and not ax:
fig, ax = plt.subplots(1, 1, figsize=figsize)
dim_list = ket.dims[0]
if splitting is None:
splitting = (len(dim_list) + 1) // 2
if isinstance(labels_iteration, int):
labels_iteration = labels_iteration, labels_iteration
ketdata = ket.full()
dim_list_y = dim_list[:splitting]
dim_list_x = dim_list[splitting:]
size_x = np.prod(dim_list_x)
size_y = np.prod(dim_list_y)
ketdata = ketdata.reshape((size_y, size_x))
dim_list_small_x = dim_list_x[:labels_iteration[1]]
dim_list_small_y = dim_list_y[:labels_iteration[0]]
quadrants_x = np.prod(dim_list_small_x)
quadrants_y = np.prod(dim_list_small_y)
ticks_x = [size_x / quadrants_x * (i + 0.5)
for i in range(quadrants_x)]
ticks_y = [size_y / quadrants_y * (quadrants_y - i - 0.5)
for i in range(quadrants_y)]
labels_x = [_sequence_to_latex(_index_to_sequence(i*size_x // quadrants_x,
dim_list=dim_list_x))
for i in range(quadrants_x)]
labels_y = [_sequence_to_latex(_index_to_sequence(i*size_y // quadrants_y,
dim_list=dim_list_y))
for i in range(quadrants_y)]
ax.set_xticks(ticks_x)
ax.set_xticklabels(labels_x)
ax.set_yticks(ticks_y)
ax.set_yticklabels(labels_y)
ax.set_xlabel("last particles")
ax.set_ylabel("first particles")
ax.imshow(complex_array_to_rgb(ketdata, theme=theme),
interpolation="none",
extent=(0, size_x, 0, size_y))
return fig, ax