Source code for qutip.distributions

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"""
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.misc import factorial
from scipy.special import hermite

from qutip.qobj import isket
from qutip.wigner import wigner, qfunc
from qutip.states import ket2dm, state_number_index

try:
    import matplotlib as mpl
    import matplotlib.pyplot as plt
    from mpl_toolkits.mplot3d import Axes3D
except:
    pass


[docs]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
[docs] 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
[docs] 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]])
[docs] 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]])
[docs]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])
[docs]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])
[docs]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)
[docs] 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)
[docs] 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
[docs] 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
[docs]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)
[docs] 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]
[docs]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)
[docs] 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]