# This file is part of QuTiP: Quantum Toolbox in Python.
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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', 'plot_wigner_sphere']
import warnings
import itertools as it
import numpy as np
from numpy import pi, array, sin, cos, angle, log2, sqrt
from packaging.version import parse as parse_version
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
try:
import matplotlib.pyplot as plt
import matplotlib as mpl
from matplotlib import cm
from mpl_toolkits.mplot3d import Axes3D
# Define a custom _axes3D function based on the matplotlib version.
# The auto_add_to_figure keyword is new for matplotlib>=3.4.
if parse_version(mpl.__version__) >= parse_version('3.4'):
def _axes3D(fig, *args, **kwargs):
ax = Axes3D(fig, *args, auto_add_to_figure=False, **kwargs)
return fig.add_axes(ax)
else:
def _axes3D(*args, **kwargs):
return Axes3D(*args, **kwargs)
except:
pass
[docs]def plot_wigner_sphere(fig, ax, wigner, reflections):
"""Plots a coloured Bloch sphere.
Parameters
----------
fig : :obj:`matplotlib.figure.Figure`
An instance of :obj:`~matplotlib.figure.Figure`.
ax : :obj:`matplotlib.axes.Axes`
An axes instance in the given figure.
wigner : list of float
The wigner transformation at `steps` different theta and phi.
reflections : bool
If the reflections of the sphere should be plotted as well.
Notes
------
Special thanks to Russell P Rundle for writing this function.
"""
ax.set_xlabel("x")
ax.set_ylabel("y")
ax.set_zlabel("z")
steps = len(wigner)
theta = np.linspace(0, np.pi, steps)
phi = np.linspace(0, 2 * np.pi, steps)
x = np.outer(np.sin(theta), np.cos(phi))
y = np.outer(np.sin(theta), np.sin(phi))
z = np.outer(np.cos(theta), np.ones(steps))
wigner = np.real(wigner)
wigner_max = np.real(np.amax(np.abs(wigner)))
wigner_c1 = cm.seismic_r((wigner + wigner_max) / (2 * wigner_max))
# Plot coloured Bloch sphere:
ax.plot_surface(x, y, z, facecolors=wigner_c1, vmin=-wigner_max,
vmax=wigner_max, rcount=steps, ccount=steps, linewidth=0,
zorder=0.5, antialiased=None)
if reflections:
wigner_c2 = cm.seismic_r((wigner[0:steps, 0:steps]+wigner_max) /
(2*wigner_max)) # bottom
wigner_c3 = cm.seismic_r((wigner[0:steps, 0:steps]+wigner_max) /
(2*wigner_max)) # side
wigner_c4 = cm.seismic_r((wigner[0:steps, 0:steps]+wigner_max) /
(2*wigner_max)) # back
# Plot bottom reflection:
ax.plot_surface(x[0:steps, 0:steps], y[0:steps, 0:steps],
-1.5*np.ones((steps, steps)), facecolors=wigner_c2,
vmin=-wigner_max, vmax=wigner_max, rcount=steps/2,
ccount=steps/2, linewidth=0, zorder=0.5,
antialiased=False)
# Plot side reflection:
ax.plot_surface(-1.5*np.ones((steps, steps)), y[0:steps, 0:steps],
z[0:steps, 0:steps], facecolors=wigner_c3,
vmin=-wigner_max, vmax=wigner_max, rcount=steps/2,
ccount=steps/2, linewidth=0, zorder=0.5,
antialiased=False)
# Plot back reflection:
ax.plot_surface(x[0:steps, 0:steps], 1.5*np.ones((steps, steps)),
z[0:steps, 0:steps], facecolors=wigner_c4,
vmin=-wigner_max, vmax=wigner_max, rcount=steps/2,
ccount=steps/2, linewidth=0, zorder=0.5,
antialiased=False)
# Create colourbar:
m = cm.ScalarMappable(cmap=cm.seismic_r)
m.set_array([-wigner_max, wigner_max])
plt.colorbar(m, shrink=0.5, aspect=10)
plt.show()
# Adopted from the SciPy Cookbook.
def _blob(x, y, w, w_max, area, cmap=None, ax=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])
if ax is not None:
handle = ax
else:
handle = plt
handle.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.array(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, ax=ax)
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, ax=ax)
# 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)
xtics = 0.5 + np.arange(width)
# x axis
ax.xaxis.set_major_locator(plt.FixedLocator(xtics))
if xlabels:
nxlabels = len(xlabels)
if nxlabels != len(xtics):
raise ValueError(f"got {nxlabels} xlabels but needed {len(xtics)}")
ax.set_xticklabels(xlabels)
if label_top:
ax.xaxis.tick_top()
ax.tick_params(axis='x', labelsize=14)
# y axis
ytics = 0.5 + np.arange(height)
ax.yaxis.set_major_locator(plt.FixedLocator(ytics))
if ylabels:
nylabels = len(ylabels)
if nylabels != len(ytics):
raise ValueError(f"got {nylabels} ylabels but needed {len(ytics)}")
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 isinstance(limits, 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
xtics = -0.5 + np.arange(M.shape[0])
ax.axes.w_xaxis.set_major_locator(plt.FixedLocator(xtics))
if xlabels:
nxlabels = len(xlabels)
if nxlabels != len(xtics):
raise ValueError(f"got {nxlabels} xlabels but needed {len(xtics)}")
ax.set_xticklabels(xlabels)
ax.tick_params(axis='x', labelsize=14)
# y axis
ytics = -0.5 + np.arange(M.shape[1])
ax.axes.w_yaxis.set_major_locator(plt.FixedLocator(ytics))
if ylabels:
nylabels = len(ylabels)
if nylabels != len(ytics):
raise ValueError(f"got {nylabels} ylabels but needed {len(ytics)}")
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
xtics = -0.5 + np.arange(M.shape[0])
ax.axes.w_xaxis.set_major_locator(plt.FixedLocator(xtics))
if xlabels:
nxlabels = len(xlabels)
if nxlabels != len(xtics):
raise ValueError(f"got {nxlabels} xlabels but needed {len(xtics)}")
ax.set_xticklabels(xlabels)
ax.tick_params(axis='x', labelsize=12)
# y axis
ytics = -0.5 + np.arange(M.shape[1])
ax.axes.w_yaxis.set_major_locator(plt.FixedLocator(ytics))
if ylabels:
nylabels = len(ylabels)
if nylabels != len(ytics):
raise ValueError(f"got {nylabels} ylabels but needed {len(ytics)}")
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)
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()
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), 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=(6, 6),
cmap=None, alpha_max=7.5, colorbar=False,
method='clenshaw', 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 {'clenshaw', '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 isinstance(W0, 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 2k particles and the rest.
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.
Notes
-----
See also [1]_.
References
----------
.. [1] J. Rodriguez-Laguna, P. Migdal, M. Ibanez Berganza, M. Lewenstein
and G. Sierra, *Qubism: self-similar visualization of many-body
wavefunctions*, `New J. Phys. 14 053028
<http://dx.doi.org/10.1088/1367-2630/14/5/053028>`_, arXiv:1112.3560
(2012), open access.
"""
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