"""
This module provides classes and functions for working with spatial
distributions, such as Wigner distributions, etc.
.. note::
Experimental.
"""
__all__ = ['Distribution', 'HarmonicOscillatorWaveFunction',
'HarmonicOscillatorProbabilityFunction']
import numpy as np
from numpy.typing import ArrayLike
from numpy import pi, exp, sqrt
from scipy.special import hermite, factorial
from . import isket, ket2dm, state_number_index, Qobj
from .wigner import wigner, qfunc
from ._distributions import psi_n_single_fock_multiple_position_complex
try:
import matplotlib as mpl
import matplotlib.pyplot as plt
from matplotlib.figure import Figure
from matplotlib.axes import Axes
from matplotlib.colors import Colormap
from mpl_toolkits.mplot3d import Axes3D
except:
# Type when matplotlib is not installed
from typing import Any
Figure = Any
Axes = Any
Colormap = Any
[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: ArrayLike = None, xvecs: list = [],
xlabels: list = []):
self.data = data
self.xvecs = xvecs
self.xlabels = xlabels
[docs]
def visualize(self, fig: Figure = None, ax: Axes = None,
figsize: tuple = (8, 6), colorbar: bool = True,
cmap: Colormap = None, style: str = "colormap",
show_xlabel: bool = True, show_ylabel: bool = True) -> tuple[Figure, Axes]:
"""
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'.
show_xlabel : bool
Whether or not the xlabel is shown.
show_ylabel : bool
Whether or not the ylabel is shown.
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.colormaps['RdBu']
lim = abs(self.data.real).max()
cf = ax.contourf(self.xvecs[0], self.xvecs[1], self.data.real, 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.colormaps['RdBu']
lim = abs(self.data.real).max()
X, Y = np.meshgrid(self.xvecs[0], self.xvecs[1])
s = ax.plot_surface(X, Y, self.data.real,
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.real)
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]])
class TwoModeQuadratureCorrelation(Distribution):
"""A class for representing the probability distribution for
quadrature measurement outcomes given a two-mode wavefunction
or density matrix.
Parameters
----------
state : Qobj, default : None
A quantum state (wavefunction or density matrix) from which the
distribution is generated.
theta1 : float, default : 0.0
Angle for the first coordinate.
theta2 : float, default : 0.0
Angle for the second coordinate.
extent : ArrayLike, default : [[-5, 5], [-5, 5]]
List of arrays with the bounds [a, b] for each coordinate.
steps : int, default : 250
The number of data points generated between the bounds for
each coordinate.
"""
def __init__(self, state: Qobj = None, theta1: float = 0.0, theta2: float = 0.0,
extent: ArrayLike = [[-5, 5], [-5, 5]], steps: int = 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
if state:
self.update(state)
def update(self, state: Qobj):
"""Calculates the probability distribution for quadrature measurement
outcomes.
Parameters
----------
state : Qobj
A quantum state (wavefunction or density matrix) from which the
distribution is generated.
"""
if isket(state):
self.update_psi(state)
else:
self.update_rho(state)
def update_psi(self, psi: Qobj):
"""Calculates the probability distribution for quadrature measurement
outcomes given a two-mode wavefunction.
Parameters
----------
state : Qobj
A wavefunction from which the distribution is generated.
"""
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.full()[i, 0]
self.data = abs(p) ** 2
def update_rho(self, rho: Qobj):
"""Calculates the probability distribution for quadrature measurement
outcomes given a two-mode density matrix.
Parameters
----------
state : Qobj
A density matrix from which the distribution is generated.
"""
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.full()[i, j]
self.data = p
[docs]
class HarmonicOscillatorWaveFunction(Distribution):
"""Calculates and represents the wave function of
a quantum harmonic oscillator.
The `HarmonicOscillatorWaveFunction` class computes
the spatial distribution of the wave function for a quantum
harmonic oscillator given a set of state coefficients (`psi`).
By extending the `Distribution` base class, this class
provides specialized attributes and methods tailored for modeling
the harmonic oscillator's wave function.This implementation leverages
the Cython function `psi_n_single_fock_multiple_position_complex`from the
`_distributions.pyx` module to efficiently compute the wave function's
contribution for each Fock state across spatial coordinates using an
optimized recurrence relation.
Parameters
----------
psi : array_like, optional
Coefficients for each harmonic oscillator state (Fock state) to
calculate the wave function. Defaults to None, in which case the
wave function is not initialized until `update` is called.
omega : float, optional
The angular frequency of the harmonic oscillator. Defaults to 1.0.
extent : list, optional
A list with two elements that defines the range of the spatial
dimension for calculating the wave function. Defaults to [-5, 5].
steps : int, optional
Number of points used to discretize the spatial range defined by
`extent`. Higher values increase resolution but may slow down
computations. Defaults to 250.
Attributes
----------
xvecs : list of arrays
A list containing arrays that represent the spatial
coordinates over which the wave function is calculated.
xlabels : list of str
A list of labels for each spatial coordinate, in this case with
one element representing the x-axis.
omega : float
The angular frequency of the harmonic oscillator, stored as an
attribute for use in wave function calculations.
data : np.ndarray of complex numbers
The calculated wave function values across the spatial range.
Populated when `update` is called.
Methods
-------
update(psi)
Calculates and updates the wave function values for the harmonic
oscillator based on the provided state coefficients, `psi`.
References
----------
- Pérez-Jordá, J. M. (2017). On the recursive solution of the quantum
harmonic oscillator. *European Journal of Physics*, 39(1),
015402. doi:10.1088/1361-6404/aa9584
- *Fast-Wave*: High-performance wave function calculations for quantum
harmonic oscillators.Available at:
https://github.com/fobos123deimos/fast-wave
"""
def __init__(self, psi: ArrayLike = None, omega: float = 1.0,
extent: list = [-5, 5], steps: int = 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: Qobj):
"""Calculate the wavefunction for the given state of an harmonic
oscillator.
Parameters
----------
psi : Qobj
A quantum state from which the distribution is generated.
"""
self.data = np.zeros(len(self.xvecs[0]), dtype=complex)
N = psi.shape[0]
for n in range(N):
self.data += (
psi_n_single_fock_multiple_position_complex(
n, self.xvecs[0].astype(complex)
) * psi[n, 0]
)
self.data *= pow(self.omega, 0.25)
[docs]
class HarmonicOscillatorProbabilityFunction(Distribution):
"""A class for representing the probability distribution of a quantum
harmonic oscillator given a density matrix.
Parameters
----------
rho : qobj, default : None
Density matrix for composite quantum systems.
omega : float, default : 1.0
The angular frequency of the harmonic oscillator.
extent : list, default : [[-5, 5], [-5, 5]]
List of arrays with the bounds [a, b] for each coordinate.
steps : int, default : 250
The number of data points generated between the bounds for
each coordinate.
"""
def __init__(self, rho: Qobj = None, omega: float = 1.0,
extent: list = [-5, 5], steps: int = 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: Qobj):
"""Calculates the probability function for the given state of an
harmonic oscillator (as density matrix).
Parameters
----------
state : Qobj
A density matrix from which the distribution is generated.
"""
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.full()[m, n]