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__all__ = ['wigner', 'qfunc', 'spin_q_function', 'spin_wigner']
import numpy as np
from scipy import (zeros, array, arange, exp, real, conj, pi,
copy, sqrt, meshgrid, size, polyval, fliplr, conjugate,
cos, sin)
import scipy.sparse as sp
import scipy.fftpack as ft
import scipy.linalg as la
from scipy.special import genlaguerre
from scipy.special import binom
from scipy.special import sph_harm
from qutip.qobj import Qobj, isket, isoper
from qutip.states import ket2dm
from qutip.parallel import parfor
from qutip.utilities import clebsch
from scipy.special import factorial
from qutip.cy.sparse_utils import _csr_get_diag
[docs]def wigner(psi, xvec, yvec, method='clenshaw', g=sqrt(2),
sparse=False, parfor=False):
"""Wigner function for a state vector or density matrix at points
`xvec + i * yvec`.
Parameters
----------
state : qobj
A state vector or density matrix.
xvec : array_like
x-coordinates at which to calculate the Wigner function.
yvec : array_like
y-coordinates at which to calculate the Wigner function. Does not
apply to the 'fft' method.
g : float
Scaling factor for `a = 0.5 * g * (x + iy)`, default `g = sqrt(2)`.
method : string {'clenshaw', 'iterative', 'laguerre', 'fft'}
Select method 'clenshaw' 'iterative', 'laguerre', or 'fft', where 'clenshaw'
and 'iterative' use an iterative method to evaluate the Wigner functions for density
matrices :math:`|m><n|`, while 'laguerre' uses the Laguerre polynomials
in scipy for the same task. The 'fft' method evaluates the Fourier
transform of the density matrix. The 'iterative' method is default, and
in general recommended, but the 'laguerre' method is more efficient for
very sparse density matrices (e.g., superpositions of Fock states in a
large Hilbert space). The 'clenshaw' method is the preferred method for
dealing with density matrices that have a large number of excitations
(>~50). 'clenshaw' is a fast and numerically stable method.
sparse : bool {False, True}
Tells the default solver whether or not to keep the input density
matrix in sparse format. As the dimensions of the density matrix
grow, setthing this flag can result in increased performance.
parfor : bool {False, True}
Flag for calculating the Laguerre polynomial based Wigner function
method='laguerre' in parallel using the parfor function.
Returns
-------
W : array
Values representing the Wigner function calculated over the specified
range [xvec,yvec].
yvex : array
FFT ONLY. Returns the y-coordinate values calculated via the Fourier
transform.
Notes
-----
The 'fft' method accepts only an xvec input for the x-coordinate.
The y-coordinates are calculated internally.
References
----------
Ulf Leonhardt,
Measuring the Quantum State of Light, (Cambridge University Press, 1997)
"""
if not (psi.type == 'ket' or psi.type == 'oper' or psi.type == 'bra'):
raise TypeError('Input state is not a valid operator.')
if method == 'fft':
return _wigner_fourier(psi, xvec, g)
if psi.type == 'ket' or psi.type == 'bra':
rho = ket2dm(psi)
else:
rho = psi
if method == 'iterative':
return _wigner_iterative(rho, xvec, yvec, g)
elif method == 'laguerre':
return _wigner_laguerre(rho, xvec, yvec, g, parfor)
elif method == 'clenshaw':
return _wigner_clenshaw(rho, xvec, yvec, g, sparse=sparse)
else:
raise TypeError(
"method must be either 'iterative', 'laguerre', or 'fft'.")
def _wigner_iterative(rho, xvec, yvec, g=sqrt(2)):
"""
Using an iterative method to evaluate the wigner functions for the Fock
state :math:`|m><n|`.
The Wigner function is calculated as
:math:`W = \sum_{mn} \\rho_{mn} W_{mn}` where :math:`W_{mn}` is the Wigner
function for the density matrix :math:`|m><n|`.
In this implementation, for each row m, Wlist contains the Wigner functions
Wlist = [0, ..., W_mm, ..., W_mN]. As soon as one W_mn Wigner function is
calculated, the corresponding contribution is added to the total Wigner
function, weighted by the corresponding element in the density matrix
:math:`rho_{mn}`.
"""
M = np.prod(rho.shape[0])
X, Y = meshgrid(xvec, yvec)
A = 0.5 * g * (X + 1.0j * Y)
Wlist = array([zeros(np.shape(A), dtype=complex) for k in range(M)])
Wlist[0] = exp(-2.0 * abs(A) ** 2) / pi
W = real(rho[0, 0]) * real(Wlist[0])
for n in range(1, M):
Wlist[n] = (2.0 * A * Wlist[n - 1]) / sqrt(n)
W += 2 * real(rho[0, n] * Wlist[n])
for m in range(1, M):
temp = copy(Wlist[m])
Wlist[m] = (2 * conj(A) * temp - sqrt(m) * Wlist[m - 1]) / sqrt(m)
# Wlist[m] = Wigner function for |m><m|
W += real(rho[m, m] * Wlist[m])
for n in range(m + 1, M):
temp2 = (2 * A * Wlist[n - 1] - sqrt(m) * temp) / sqrt(n)
temp = copy(Wlist[n])
Wlist[n] = temp2
# Wlist[n] = Wigner function for |m><n|
W += 2 * real(rho[m, n] * Wlist[n])
return 0.5 * W * g ** 2
def _wigner_laguerre(rho, xvec, yvec, g, parallel):
"""
Using Laguerre polynomials from scipy to evaluate the Wigner function for
the density matrices :math:`|m><n|`, :math:`W_{mn}`. The total Wigner
function is calculated as :math:`W = \sum_{mn} \\rho_{mn} W_{mn}`.
"""
M = np.prod(rho.shape[0])
X, Y = meshgrid(xvec, yvec)
A = 0.5 * g * (X + 1.0j * Y)
W = zeros(np.shape(A))
# compute wigner functions for density matrices |m><n| and
# weight by all the elements in the density matrix
B = 4 * abs(A) ** 2
if sp.isspmatrix_csr(rho.data):
# for compress sparse row matrices
if parallel:
iterator = (
(m, rho, A, B) for m in range(len(rho.data.indptr) - 1))
W1_out = parfor(_par_wig_eval, iterator)
W += sum(W1_out)
else:
for m in range(len(rho.data.indptr) - 1):
for jj in range(rho.data.indptr[m], rho.data.indptr[m + 1]):
n = rho.data.indices[jj]
if m == n:
W += real(rho[m, m] * (-1) ** m * genlaguerre(m, 0)(B))
elif n > m:
W += 2.0 * real(rho[m, n] * (-1) ** m *
(2 * A) ** (n - m) *
sqrt(factorial(m) / factorial(n)) *
genlaguerre(m, n - m)(B))
else:
# for dense density matrices
B = 4 * abs(A) ** 2
for m in range(M):
if abs(rho[m, m]) > 0.0:
W += real(rho[m, m] * (-1) ** m * genlaguerre(m, 0)(B))
for n in range(m + 1, M):
if abs(rho[m, n]) > 0.0:
W += 2.0 * real(rho[m, n] * (-1) ** m *
(2 * A) ** (n - m) *
sqrt(factorial(m) / factorial(n)) *
genlaguerre(m, n - m)(B))
return 0.5 * W * g ** 2 * np.exp(-B / 2) / pi
def _par_wig_eval(args):
"""
Private function for calculating terms of Laguerre Wigner function
using parfor.
"""
m, rho, A, B = args
W1 = zeros(np.shape(A))
for jj in range(rho.data.indptr[m], rho.data.indptr[m + 1]):
n = rho.data.indices[jj]
if m == n:
W1 += real(rho[m, m] * (-1) ** m * genlaguerre(m, 0)(B))
elif n > m:
W1 += 2.0 * real(rho[m, n] * (-1) ** m *
(2 * A) ** (n - m) *
sqrt(factorial(m) / factorial(n)) *
genlaguerre(m, n - m)(B))
return W1
def _wigner_fourier(psi, xvec, g=np.sqrt(2)):
"""
Evaluate the Wigner function via the Fourier transform.
"""
if psi.type == 'bra':
psi = psi.dag()
if psi.type == 'ket':
return _psi_wigner_fft(psi.full(), xvec, g)
elif psi.type == 'oper':
eig_vals, eig_vecs = la.eigh(psi.full())
W = 0
for ii in range(psi.shape[0]):
W1, yvec = _psi_wigner_fft(
np.reshape(eig_vecs[:, ii], (psi.shape[0], 1)), xvec, g)
W += eig_vals[ii] * W1
return W, yvec
def _psi_wigner_fft(psi, xvec, g=sqrt(2)):
"""
FFT method for a single state vector. Called multiple times when the
input is a density matrix.
"""
n = len(psi)
A = _osc_eigen(n, xvec * g / np.sqrt(2))
xpsi = np.dot(psi.T, A)
W, yvec = _wigner_fft(xpsi, xvec * g / np.sqrt(2))
return (0.5 * g ** 2) * np.real(W.T), yvec * np.sqrt(2) / g
def _wigner_fft(psi, xvec):
"""
Evaluates the Fourier transformation of a given state vector.
Returns the corresponding density matrix and range
"""
n = 2*len(psi.T)
r1 = np.concatenate((np.array([[0]]),
np.fliplr(psi.conj()),
np.zeros((1, n//2 - 1))), axis=1)
r2 = np.concatenate((np.array([[0]]), psi,
np.zeros((1, n//2 - 1))), axis=1)
w = la.toeplitz(np.zeros((n//2, 1)), r1) * \
np.flipud(la.toeplitz(np.zeros((n//2, 1)), r2))
w = np.concatenate((w[:, n//2:n], w[:, 0:n//2]), axis=1)
w = ft.fft(w)
w = np.real(np.concatenate((w[:, 3*n//4:n+1], w[:, 0:n//4]), axis=1))
p = np.arange(-n/4, n/4)*np.pi / (n*(xvec[1] - xvec[0]))
w = w / (p[1] - p[0]) / n
return w, p
def _osc_eigen(N, pnts):
"""
Vector of and N-dim oscillator eigenfunctions evaluated
at the points in pnts.
"""
pnts = np.asarray(pnts)
lpnts = len(pnts)
A = np.zeros((N, lpnts))
A[0, :] = np.exp(-pnts ** 2 / 2.0) / pi ** 0.25
if N == 1:
return A
else:
A[1, :] = np.sqrt(2) * pnts * A[0, :]
for k in range(2, N):
A[k, :] = np.sqrt(2.0 / k) * pnts * A[k - 1, :] - \
np.sqrt((k - 1.0) / k) * A[k - 2, :]
return A
def _wigner_clenshaw(rho, xvec, yvec, g=sqrt(2), sparse=False):
"""
Using Clenshaw summation - numerically stable and efficient
iterative algorithm to evaluate polynomial series.
The Wigner function is calculated as
:math:`W = e^(-0.5*x^2)/pi * \sum_{L} c_L (2x)^L / sqrt(L!)` where
:math:`c_L = \sum_n \\rho_{n,L+n} LL_n^L` where
:math:`LL_n^L = (-1)^n sqrt(L!n!/(L+n)!) LaguerreL[n,L,x]`
"""
M = np.prod(rho.shape[0])
X,Y = np.meshgrid(xvec, yvec)
#A = 0.5 * g * (X + 1.0j * Y)
A2 = g * (X + 1.0j * Y) #this is A2 = 2*A
B = np.abs(A2)
B *= B
w0 = (2*rho.data[0,-1])*np.ones_like(A2)
L = M-1
#calculation of \sum_{L} c_L (2x)^L / sqrt(L!)
#using Horner's method
if not sparse:
rho = rho.full() * (2*np.ones((M,M)) - np.diag(np.ones(M)))
while L > 0:
L -= 1
#here c_L = _wig_laguerre_val(L, B, np.diag(rho, L))
w0 = _wig_laguerre_val(L, B, np.diag(rho, L)) + w0 * A2 * (L+1)**-0.5
else:
while L > 0:
L -= 1
diag = _csr_get_diag(rho.data.data,rho.data.indices,
rho.data.indptr,L)
if L != 0:
diag *= 2
#here c_L = _wig_laguerre_val(L, B, np.diag(rho, L))
w0 = _wig_laguerre_val(L, B, diag) + w0 * A2 * (L+1)**-0.5
return w0.real * np.exp(-B*0.5) * (g*g*0.5 / pi)
def _wig_laguerre_val(L, x, c):
"""
this is evaluation of polynomial series inspired by hermval from numpy.
Returns polynomial series
\sum_n b_n LL_n^L,
where
LL_n^L = (-1)^n sqrt(L!n!/(L+n)!) LaguerreL[n,L,x]
The evaluation uses Clenshaw recursion
"""
if len(c) == 1:
y0 = c[0]
y1 = 0
elif len(c) == 2:
y0 = c[0]
y1 = c[1]
else:
k = len(c)
y0 = c[-2]
y1 = c[-1]
for i in range(3, len(c) + 1):
k -= 1
y0, y1 = c[-i] - y1 * (float((k - 1)*(L + k - 1))/((L+k)*k))**0.5, \
y0 - y1 * ((L + 2*k -1) - x) * ((L+k)*k)**-0.5
return y0 - y1 * ((L + 1) - x) * (L + 1)**-0.5
# -----------------------------------------------------------------------------
# Q FUNCTION
#
[docs]def qfunc(state, xvec, yvec, g=sqrt(2)):
"""Q-function of a given state vector or density matrix
at points `xvec + i * yvec`.
Parameters
----------
state : qobj
A state vector or density matrix.
xvec : array_like
x-coordinates at which to calculate the Wigner function.
yvec : array_like
y-coordinates at which to calculate the Wigner function.
g : float
Scaling factor for `a = 0.5 * g * (x + iy)`, default `g = sqrt(2)`.
Returns
--------
Q : array
Values representing the Q-function calculated over the specified range
[xvec,yvec].
"""
X, Y = meshgrid(xvec, yvec)
amat = 0.5 * g * (X + Y * 1j)
if not (isoper(state) or isket(state)):
raise TypeError('Invalid state operand to qfunc.')
qmat = zeros(size(amat))
if isket(state):
qmat = _qfunc_pure(state, amat)
elif isoper(state):
d, v = la.eig(state.full())
# d[i] = eigenvalue i
# v[:,i] = eigenvector i
qmat = zeros(np.shape(amat))
for k in arange(0, len(d)):
qmat1 = _qfunc_pure(v[:, k], amat)
qmat += real(d[k] * qmat1)
qmat = 0.25 * qmat * g ** 2
return qmat
#
# Q-function for a pure state: Q = |<alpha|psi>|^2 / pi
#
# |psi> = the state in fock basis
# |alpha> = the coherent state with amplitude alpha
#
def _qfunc_pure(psi, alpha_mat):
"""
Calculate the Q-function for a pure state.
"""
n = np.prod(psi.shape)
if isinstance(psi, Qobj):
psi = psi.full().flatten()
else:
psi = psi.T
qmat = abs(polyval(fliplr([psi / sqrt(factorial(arange(n)))])[0],
conjugate(alpha_mat))) ** 2
return real(qmat) * exp(-abs(alpha_mat) ** 2) / pi
# -----------------------------------------------------------------------------
# PSEUDO DISTRIBUTION FUNCTIONS FOR SPINS
#
[docs]def spin_q_function(rho, theta, phi):
"""Husimi Q-function for spins.
Parameters
----------
state : qobj
A state vector or density matrix for a spin-j quantum system.
theta : array_like
theta-coordinates at which to calculate the Q function.
phi : array_like
phi-coordinates at which to calculate the Q function.
Returns
-------
Q, THETA, PHI : 2d-array
Values representing the spin Q function at the values specified
by THETA and PHI.
"""
if rho.type == 'bra':
rho = rho.dag()
if rho.type == 'ket':
rho = ket2dm(rho)
J = rho.shape[0]
j = int((J - 1) / 2)
THETA, PHI = meshgrid(theta, phi)
Q = np.zeros_like(THETA, dtype=complex)
for m1 in range(-j, j+1):
Q += binom(2*j, j+m1) * cos(THETA/2) ** (2*(j-m1)) * sin(THETA/2) ** (2*(j+m1)) * \
rho.data[int(j-m1), int(j-m1)]
for m2 in range(m1+1, j+1):
Q += (sqrt(binom(2*j, j+m1)) * sqrt(binom(2*j, j+m2)) *
cos(THETA/2) ** (2*j-m1-m2) * sin(THETA/2) ** (2*j+m1+m2)) * \
(exp(1j * (m2-m1) * PHI) * rho.data[int(j-m1), int(j-m2)] +
exp(1j * (m1-m2) * PHI) * rho.data[int(j-m2), int(j-m1)])
return Q.real, THETA, PHI
def _rho_kq(rho, j, k, q):
v = 0j
for m1 in range(-j, j+1):
for m2 in range(-j, j+1):
v += (-1)**(j - m1 - q) * clebsch(j, j, k, m1, -m2,
q) * rho.data[m1 + j, m2 + j]
return v
[docs]def spin_wigner(rho, theta, phi):
"""Wigner function for spins on the Bloch sphere.
Parameters
----------
state : qobj
A state vector or density matrix for a spin-j quantum system.
theta : array_like
theta-coordinates at which to calculate the Q function.
phi : array_like
phi-coordinates at which to calculate the Q function.
Returns
-------
W, THETA, PHI : 2d-array
Values representing the spin Wigner function at the values specified
by THETA and PHI.
Notes
-----
Experimental.
"""
if rho.type == 'bra':
rho = rho.dag()
if rho.type == 'ket':
rho = ket2dm(rho)
J = rho.shape[0]
j = int((J - 1) / 2)
THETA, PHI = meshgrid(theta, phi)
W = np.zeros_like(THETA, dtype=complex)
for k in range(int(2 * j)+1):
for q in range(-k, k+1):
W += _rho_kq(rho, j, k, q) * sph_harm(q, k, PHI, THETA)
return W, THETA, PHI