qutip/distributions.py
"""
This module provides classes and functions for working with spatial
distributions, such as Wigner distributions, etc.
.. note::
Experimental.
"""
__all__ = ['Distribution', 'WignerDistribution', 'QDistribution',
'TwoModeQuadratureCorrelation',
'HarmonicOscillatorWaveFunction',
'HarmonicOscillatorProbabilityFunction']
import numpy as np
from numpy import pi, exp, sqrt
from scipy.special import hermite, factorial
from . import isket, ket2dm, state_number_index
from .wigner import wigner, qfunc
try:
import matplotlib as mpl
import matplotlib.pyplot as plt
from mpl_toolkits.mplot3d import Axes3D
except:
pass
class Distribution:
"""A class for representation spatial distribution functions.
The Distribution class can be used to prepresent spatial distribution
functions of arbitray dimension (although only 1D and 2D distributions
are used so far).
It is indented as a base class for specific distribution function, and
provide implementation of basic functions that are shared among all
Distribution functions, such as visualization, calculating marginal
distributions, etc.
Parameters
----------
data : array_like
Data for the distribution. The dimensions must match the lengths of
the coordinate arrays in xvecs.
xvecs : list
List of arrays that spans the space for each coordinate.
xlabels : list
List of labels for each coordinate.
"""
def __init__(self, data=None, xvecs=[], xlabels=[]):
self.data = data
self.xvecs = xvecs
self.xlabels = xlabels
def visualize(self, fig=None, ax=None, figsize=(8, 6),
colorbar=True, cmap=None, style="colormap",
show_xlabel=True, show_ylabel=True):
"""
Visualize the data of the distribution in 1D or 2D, depending
on the dimensionality of the underlaying distribution.
Parameters:
fig : matplotlib Figure instance
If given, use this figure instance for the visualization,
ax : matplotlib Axes instance
If given, render the visualization using this axis instance.
figsize : tuple
Size of the new Figure instance, if one needs to be created.
colorbar: Bool
Whether or not the colorbar (in 2D visualization) should be used.
cmap: matplotlib colormap instance
If given, use this colormap for 2D visualizations.
style : string
Type of visualization: 'colormap' (default) or 'surface'.
Returns
-------
fig, ax : tuple
A tuple of matplotlib figure and axes instances.
"""
n = len(self.xvecs)
if n == 2:
if style == "colormap":
return self.visualize_2d_colormap(fig=fig, ax=ax,
figsize=figsize,
colorbar=colorbar,
cmap=cmap,
show_xlabel=show_xlabel,
show_ylabel=show_ylabel)
else:
return self.visualize_2d_surface(fig=fig, ax=ax,
figsize=figsize,
colorbar=colorbar,
cmap=cmap,
show_xlabel=show_xlabel,
show_ylabel=show_ylabel)
elif n == 1:
return self.visualize_1d(fig=fig, ax=ax, figsize=figsize,
show_xlabel=show_xlabel,
show_ylabel=show_ylabel)
else:
raise NotImplementedError("Distribution visualization in " +
"%d dimensions is not implemented." % n)
def visualize_2d_colormap(self, fig=None, ax=None, figsize=(8, 6),
colorbar=True, cmap=None,
show_xlabel=True, show_ylabel=True):
if not fig and not ax:
fig, ax = plt.subplots(1, 1, figsize=figsize)
if cmap is None:
cmap = mpl.cm.get_cmap('RdBu')
lim = abs(self.data).max()
cf = ax.contourf(self.xvecs[0], self.xvecs[1], self.data, 100,
norm=mpl.colors.Normalize(-lim, lim),
cmap=cmap)
if show_xlabel:
ax.set_xlabel(self.xlabels[0], fontsize=12)
if show_ylabel:
ax.set_ylabel(self.xlabels[1], fontsize=12)
if colorbar:
cb = fig.colorbar(cf, ax=ax)
return fig, ax
def visualize_2d_surface(self, fig=None, ax=None, figsize=(8, 6),
colorbar=True, cmap=None,
show_xlabel=True, show_ylabel=True):
if not fig and not ax:
fig = plt.figure(figsize=figsize)
ax = Axes3D(fig, azim=-62, elev=25)
if cmap is None:
cmap = mpl.cm.get_cmap('RdBu')
lim = abs(self.data).max()
X, Y = np.meshgrid(self.xvecs[0], self.xvecs[1])
s = ax.plot_surface(X, Y, self.data,
norm=mpl.colors.Normalize(-lim, lim),
rstride=5, cstride=5, cmap=cmap, lw=0.1)
if show_xlabel:
ax.set_xlabel(self.xlabels[0], fontsize=12)
if show_ylabel:
ax.set_ylabel(self.xlabels[1], fontsize=12)
if colorbar:
cb = fig.colorbar(s, ax=ax, shrink=0.5)
return fig, ax
def visualize_1d(self, fig=None, ax=None, figsize=(8, 6),
show_xlabel=True, show_ylabel=True):
if not fig and not ax:
fig, ax = plt.subplots(1, 1, figsize=figsize)
p = ax.plot(self.xvecs[0], self.data)
if show_xlabel:
ax.set_xlabel(self.xlabels[0], fontsize=12)
if show_ylabel:
ax.set_ylabel("Marginal distribution", fontsize=12)
return fig, ax
def marginal(self, dim=0):
"""
Calculate the marginal distribution function along the dimension
`dim`. Return a new Distribution instance describing this reduced-
dimensionality distribution.
Parameters
----------
dim : int
The dimension (coordinate index) along which to obtain the
marginal distribution.
Returns
-------
d : Distributions
A new instances of Distribution that describes the marginal
distribution.
"""
return Distribution(data=self.data.mean(axis=dim),
xvecs=[self.xvecs[dim]],
xlabels=[self.xlabels[dim]])
def project(self, dim=0):
"""
Calculate the projection (max value) distribution function along the
dimension `dim`. Return a new Distribution instance describing this
reduced-dimensionality distribution.
Parameters
----------
dim : int
The dimension (coordinate index) along which to obtain the
projected distribution.
Returns
-------
d : Distributions
A new instances of Distribution that describes the projection.
"""
return Distribution(data=self.data.max(axis=dim),
xvecs=[self.xvecs[dim]],
xlabels=[self.xlabels[dim]])
class WignerDistribution(Distribution):
def __init__(self, rho=None, extent=[[-5, 5], [-5, 5]], steps=250):
self.xvecs = [np.linspace(extent[0][0], extent[0][1], steps),
np.linspace(extent[1][0], extent[1][1], steps)]
self.xlabels = [r'$\rm{Re}(\alpha)$', r'$\rm{Im}(\alpha)$']
if rho:
self.update(rho)
def update(self, rho):
self.data = wigner(rho, self.xvecs[0], self.xvecs[1])
class QDistribution(Distribution):
def __init__(self, rho=None, extent=[[-5, 5], [-5, 5]], steps=250):
self.xvecs = [np.linspace(extent[0][0], extent[0][1], steps),
np.linspace(extent[1][0], extent[1][1], steps)]
self.xlabels = [r'$\rm{Re}(\alpha)$', r'$\rm{Im}(\alpha)$']
if rho:
self.update(rho)
def update(self, rho):
self.data = qfunc(rho, self.xvecs[0], self.xvecs[1])
class TwoModeQuadratureCorrelation(Distribution):
def __init__(self, state=None, theta1=0.0, theta2=0.0,
extent=[[-5, 5], [-5, 5]], steps=250):
self.xvecs = [np.linspace(extent[0][0], extent[0][1], steps),
np.linspace(extent[1][0], extent[1][1], steps)]
self.xlabels = [r'$X_1(\theta_1)$', r'$X_2(\theta_2)$']
self.theta1 = theta1
self.theta2 = theta2
self.update(state)
def update(self, state):
"""
calculate probability distribution for quadrature measurement
outcomes given a two-mode wavefunction or density matrix
"""
if isket(state):
self.update_psi(state)
else:
self.update_rho(state)
def update_psi(self, psi):
"""
calculate probability distribution for quadrature measurement
outcomes given a two-mode wavefunction
"""
X1, X2 = np.meshgrid(self.xvecs[0], self.xvecs[1])
p = np.zeros((len(self.xvecs[0]), len(self.xvecs[1])), dtype=complex)
N = psi.dims[0][0]
for n1 in range(N):
kn1 = exp(-1j * self.theta1 * n1) / \
sqrt(sqrt(pi) * 2 ** n1 * factorial(n1)) * \
exp(-X1 ** 2 / 2.0) * np.polyval(hermite(n1), X1)
for n2 in range(N):
kn2 = exp(-1j * self.theta2 * n2) / \
sqrt(sqrt(pi) * 2 ** n2 * factorial(n2)) * \
exp(-X2 ** 2 / 2.0) * np.polyval(hermite(n2), X2)
i = state_number_index([N, N], [n1, n2])
p += kn1 * kn2 * psi.data[i, 0]
self.data = abs(p) ** 2
def update_rho(self, rho):
"""
calculate probability distribution for quadrature measurement
outcomes given a two-mode density matrix
"""
X1, X2 = np.meshgrid(self.xvecs[0], self.xvecs[1])
p = np.zeros((len(self.xvecs[0]), len(self.xvecs[1])), dtype=complex)
N = rho.dims[0][0]
M1 = np.zeros(
(N, N, len(self.xvecs[0]), len(self.xvecs[1])), dtype=complex)
M2 = np.zeros(
(N, N, len(self.xvecs[0]), len(self.xvecs[1])), dtype=complex)
for m in range(N):
for n in range(N):
M1[m, n] = exp(-1j * self.theta1 * (m - n)) / \
sqrt(pi * 2 ** (m + n) * factorial(n) * factorial(m)) * \
exp(-X1 ** 2) * np.polyval(
hermite(m), X1) * np.polyval(hermite(n), X1)
M2[m, n] = exp(-1j * self.theta2 * (m - n)) / \
sqrt(pi * 2 ** (m + n) * factorial(n) * factorial(m)) * \
exp(-X2 ** 2) * np.polyval(
hermite(m), X2) * np.polyval(hermite(n), X2)
for n1 in range(N):
for n2 in range(N):
i = state_number_index([N, N], [n1, n2])
for p1 in range(N):
for p2 in range(N):
j = state_number_index([N, N], [p1, p2])
p += M1[n1, p1] * M2[n2, p2] * rho.data[i, j]
self.data = p
class HarmonicOscillatorWaveFunction(Distribution):
def __init__(self, psi=None, omega=1.0, extent=[-5, 5], steps=250):
self.xvecs = [np.linspace(extent[0], extent[1], steps)]
self.xlabels = [r'$x$']
self.omega = omega
if psi:
self.update(psi)
def update(self, psi):
"""
Calculate the wavefunction for the given state of an harmonic
oscillator
"""
self.data = np.zeros(len(self.xvecs[0]), dtype=complex)
N = psi.shape[0]
for n in range(N):
k = pow(self.omega / pi, 0.25) / \
sqrt(2 ** n * factorial(n)) * \
exp(-self.xvecs[0] ** 2 / 2.0) * \
np.polyval(hermite(n), self.xvecs[0])
self.data += k * psi.data[n, 0]
class HarmonicOscillatorProbabilityFunction(Distribution):
def __init__(self, rho=None, omega=1.0, extent=[-5, 5], steps=250):
self.xvecs = [np.linspace(extent[0], extent[1], steps)]
self.xlabels = [r'$x$']
self.omega = omega
if rho:
self.update(rho)
def update(self, rho):
"""
Calculate the probability function for the given state of an harmonic
oscillator (as density matrix)
"""
if isket(rho):
rho = ket2dm(rho)
self.data = np.zeros(len(self.xvecs[0]), dtype=complex)
M, N = rho.shape
for m in range(M):
k_m = pow(self.omega / pi, 0.25) / \
sqrt(2 ** m * factorial(m)) * \
exp(-self.xvecs[0] ** 2 / 2.0) * \
np.polyval(hermite(m), self.xvecs[0])
for n in range(N):
k_n = pow(self.omega / pi, 0.25) / \
sqrt(2 ** n * factorial(n)) * \
exp(-self.xvecs[0] ** 2 / 2.0) * \
np.polyval(hermite(n), self.xvecs[0])
self.data += np.conjugate(k_n) * k_m * rho.data[m, n]