forked from qutip/qutip
-
Notifications
You must be signed in to change notification settings - Fork 1
/
wigner.py
262 lines (204 loc) · 8.06 KB
/
wigner.py
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
46
47
48
49
50
51
52
53
54
55
56
57
58
59
60
61
62
63
64
65
66
67
68
69
70
71
72
73
74
75
76
77
78
79
80
81
82
83
84
85
86
87
88
89
90
91
92
93
94
95
96
97
98
99
100
101
102
103
104
105
106
107
108
109
110
111
112
113
114
115
116
117
118
119
120
121
122
123
124
125
126
127
128
129
130
131
132
133
134
135
136
137
138
139
140
141
142
143
144
145
146
147
148
149
150
151
152
153
154
155
156
157
158
159
160
161
162
163
164
165
166
167
168
169
170
171
172
173
174
175
176
177
178
179
180
181
182
183
184
185
186
187
188
189
190
191
192
193
194
195
196
197
198
199
200
201
202
203
204
205
206
207
208
209
210
211
212
213
214
215
216
217
218
219
220
221
222
223
224
225
226
227
228
229
230
231
232
233
234
235
236
237
238
239
240
241
242
243
244
245
246
247
248
249
250
251
252
253
254
255
256
257
#This file is part of QuTIP.
#
# QuTIP is free software: you can redistribute it and/or modify
# it under the terms of the GNU General Public License as published by
# the Free Software Foundation, either version 3 of the License, or
# (at your option) any later version.
#
# QuTIP is distributed in the hope that it will be useful,
# but WITHOUT ANY WARRANTY; without even the implied warranty of
# MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the
# GNU General Public License for more details.
#
# You should have received a copy of the GNU General Public License
# along with QuTIP. If not, see <http://www.gnu.org/licenses/>.
#
# Copyright (C) 2011-2012, Paul D. Nation & Robert J. Johansson
#
###########################################################################
import numpy
from scipy import zeros,array,arange,exp,real,imag,conj,copy,sqrt,meshgrid,size,polyval,fliplr,conjugate
import scipy.sparse as sp
import scipy.linalg as la
from scipy.special import genlaguerre
from qutip.tensor import tensor
from qutip.Qobj import *
from qutip.states import *
from qutip.istests import *
try:#for scipy v <= 0.90
from scipy import factorial
except:#for scipy v >= 0.10
from scipy.misc import factorial
def wigner(psi, xvec, yvec, g=sqrt(2), method='iterative'):
"""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.
g : float
Scaling factor for `a = 0.5 * g * (x + iy)`, default `g = sqrt(2)`.
method : string {'iterative', 'laguerre'}
Select method 'iterative' or 'laguerre', where 'iterative' use a
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 '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).
Returns
-------
W : array
Values representing the Wigner function calculated over the specified
range [xvec,yvec].
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 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)
else:
raise TypeError("method must be either 'iterative' or 'laguerre'")
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 = prod(rho.shape[0])
X,Y = meshgrid(xvec, yvec)
A = 0.5 * g * (X + 1.0j * Y)
Wlist = array([zeros(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=sqrt(2)):
"""
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 = prod(rho.shape[0])
X,Y = meshgrid(xvec, yvec)
A = 0.5 * g * (X + 1.0j * Y)
W = zeros(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
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
#-------------------------------------------------------------------------------
# Q FUNCTION
#
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 isoper(state):
ketflag = 0
elif isket(state):
ketflag = 1
else:
TypeError('Invalid state operand to qfunc.')
N = prod(state.dims)
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(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 = prod(psi.shape)
if isinstance(psi, Qobj):
psi = array(psi.trans().full())[0,:]
else:
psi = psi.T
qmat1 = abs(polyval(fliplr([psi/sqrt(factorial(arange(0, n)))])[0], conjugate(alpha_mat))) ** 2;
qmat1 = real(qmat1) * exp(-abs(alpha_mat)**2) / pi;
return qmat1