Source code for qutip.visualization

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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

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 _pauli_basis, to_super
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 _isqubitdims(dims):
    """Checks whether all entries in a dims list are integer powers of 2.

    Parameters
    ----------
    dims : nested list of ints
        Dimensions to be checked.

    Returns
    -------
    isqubitdims : bool
        True if and only if every member of the flattened dims
        list is an integer power of 2.
    """
    return all([
        2**np.floor(np.log2(dim)) == dim
        for dim in flatten(dims)
    ])


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 = to_super(rho) nq = int(np.log2(sqobj.shape[0]) / 2) B = _pauli_basis(nq) / np.sqrt(2**nq) # To do this, we have to hack a bit and force the dims to match, # since the _pauli_basis function makes different assumptions # about indices than we need here. B.dims = sqobj.dims sqobj = B.dag() * sqobj * B W = sqobj.full() # 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