def BCFIM(x, p, rho, drho, M=[], eps=1e-8):
r"""
Calculation of the Bayesian classical Fisher information (BCFI) and the
Bayesian classical Fisher information matrix (BCFIM) of the form
\begin{align}
\mathcal{I}_{\mathrm{Bayes}}=\int p(\textbf{x})\mathcal{I}\mathrm{d}\textbf{x}
\end{align}
with $\mathcal{I}$ the CFIM and $p(\textbf{x})$ the prior distribution.
Parameters
----------
> **x:** `list`
-- The regimes of the parameters for the integral.
> **p:** `multidimensional array`
-- The prior distribution.
> **rho:** `multidimensional list`
-- Parameterized density matrix.
> **drho:** `multidimensional list`
-- Derivatives of the parameterized density matrix (rho) with respect to the unknown
parameters to be estimated.
> **M:** `list of matrices`
-- A set of positive operator-valued measure (POVM). The default measurement
is a set of rank-one symmetric informationally complete POVM (SIC-POVM).
> **eps:** `float`
-- Machine epsilon.
Returns
----------
**BCFI or BCFIM:** `float or matrix`
-- For single parameter estimation (the length of x is equal to one), the output
is BCFI and for multiparameter estimation (the length of x is more than one),
it returns BCFIM.
**Note:**
SIC-POVM is calculated by the Weyl-Heisenberg covariant SIC-POVM fiducial state
which can be downloaded from [here](http://www.physics.umb.edu/Research/QBism/
solutions.html).
"""
para_num = len(x)
if para_num == 1:
#### single parameter scenario ####
if M == []:
M = SIC(len(rho[0]))
else:
if type(M) != list:
raise TypeError("Please make sure M is a list!")
p_num = len(p)
if type(drho[0]) == list:
drho = [drho[i][0] for i in range(p_num)]
p_num = len(p)
F_tp = np.zeros(p_num)
for m in range(p_num):
F_tp[m] = CFIM(rho[m], [drho[m]], M=M, eps=eps)
arr = [p[i] * F_tp[i] for i in range(p_num)]
return simps(arr, x[0])
else:
#### multiparameter scenario ####
p_shape = np.shape(p)
p_ext = extract_ele(p, para_num)
rho_ext = extract_ele(rho, para_num)
drho_ext = extract_ele(drho, para_num)
p_list, rho_list, drho_list = [], [], []
for p_ele, rho_ele, drho_ele in zip(p_ext, rho_ext, drho_ext):
p_list.append(p_ele)
rho_list.append(rho_ele)
drho_list.append(drho_ele)
dim = len(rho_list[0])
if M == []:
M = SIC(dim)
else:
if type(M) != list:
raise TypeError("Please make sure M is a list!")
F_list = [[[0.0 for i in range(len(p_list))] for j in range(para_num)]
for k in range(para_num)]
for i in range(len(p_list)):
F_tp = CFIM(rho_list[i], drho_list[i], M=M, eps=eps)
for pj in range(para_num):
for pk in range(para_num):
F_list[pj][pk][i] = F_tp[pj][pk]
BCFIM_res = np.zeros([para_num, para_num])
for para_i in range(0, para_num):
for para_j in range(para_i, para_num):
F_ij = np.array(F_list[para_i][para_j]).reshape(p_shape)
arr = p * F_ij
for si in reversed(range(para_num)):
arr = simps(arr, x[si])
BCFIM_res[para_i][para_j] = arr
BCFIM_res[para_j][para_i] = arr
return BCFIM_res
def QVTB(x, p, dp, rho, drho, LDtype="SLD", eps=1e-8):
r"""
Calculation of the Bayesian version of quantum Cramer-Rao bound introduced
by Van Trees (QVTB). The covariance matrix with a prior distribution p(\textbf{x})
is defined as
\begin{align}
\mathrm{cov}(\hat{\textbf{x}},\{\Pi_y\})=\int p(\textbf{x})\sum_y\mathrm{Tr}
(\rho\Pi_y)(\hat{\textbf{x}}-\textbf{x})(\hat{\textbf{x}}-\textbf{x})^{\mathrm{T}}
\mathrm{d}\textbf{x}
\end{align}
where $\textbf{x}=(x_0,x_1,\dots)^{\mathrm{T}}$ are the unknown parameters to be estimated
and the integral $\int\mathrm{d}\textbf{x}:=\iiint\mathrm{d}x_0\mathrm{d}x_1\cdots$.
$\{\Pi_y\}$ is a set of positive operator-valued measure (POVM) and $\rho$ represent
the parameterized density matrix.
\begin{align}
\mathrm{cov}(\hat{\textbf{x}},\{\Pi_y\})\geq \left(\mathcal{I}_{\mathrm{prior}}
+\mathcal{F}_{\mathrm{Bayes}}\right)^{-1},
\end{align}
where $\mathcal{I}_{\mathrm{prior}}=\int p(\textbf{x})\mathcal{I}_{p}\mathrm{d}\textbf{x}$ is
the CFIM for $p(\textbf{x})$ and $\mathcal{F}_{\mathrm{Bayes}}=\int p(\textbf{x})\mathcal{F}
\mathrm{d}\textbf{x}$ is the average QFIM of all types.
Parameters
----------
> **x:** `list`
-- The regimes of the parameters for the integral.
> **p:** multidimensional array
-- The prior distribution.
> **dp:** `list`
-- Derivatives of the prior distribution with respect to the unknown parameters to to
estimated. For example, dp[0] is the derivative vector with respect to the first
parameter.
> **rho:** `multidimensional list`
-- Parameterized density matrix.
> **drho:** `multidimensional list`
-- Derivatives of the parameterized density matrix (rho) with respect to the unknown
parameters to be estimated.
> **LDtype:** `string`
-- Types of QFI (QFIM) can be set as the objective function. Options are:
"SLD" (default) -- QFI (QFIM) based on symmetric logarithmic derivative (SLD).
"RLD" -- QFI (QFIM) based on right logarithmic derivative (RLD).
"LLD" -- QFI (QFIM) based on left logarithmic derivative (LLD).
> **eps:** `float`
-- Machine epsilon.
Returns
----------
**QVTB:** `float or matrix`
-- For single parameter estimation (the length of x is equal to one), the
output is a float and for multiparameter estimation (the length of x is
more than one), it returns a matrix.
"""
para_num = len(x)
p_num = len(p)
if para_num == 1:
if type(drho[0]) == list:
drho = [drho[i][0] for i in range(p_num)]
if type(dp[0]) == list or type(dp[0]) == np.ndarray:
dp = [dp[i][0] for i in range(p_num)]
F_tp = np.zeros(p_num)
for m in range(p_num):
F_tp[m] = QFIM(rho[m], [drho[m]], LDtype=LDtype, eps=eps)
arr1 = [np.real(dp[i] * dp[i] / p[i]) for i in range(p_num)]
I = simps(arr1, x[0])
arr2 = [np.real(F_tp[j] * p[j]) for j in range(p_num)]
F = simps(arr2, x[0])
return 1.0 / (I + F)
else:
#### multiparameter scenario ####
p_shape = np.shape(p)
p_ext = extract_ele(p, para_num)
dp_ext = extract_ele(dp, para_num)
rho_ext = extract_ele(rho, para_num)
drho_ext = extract_ele(drho, para_num)
p_list, dp_list, rho_list, drho_list, = (
[],
[],
[],
[],
)
for p_ele, dp_ele, rho_ele, drho_ele in zip(p_ext, dp_ext, rho_ext,
drho_ext):
p_list.append(p_ele)
dp_list.append(dp_ele)
rho_list.append(rho_ele)
drho_list.append(drho_ele)
F_list = [[[0.0 for i in range(len(p_list))] for j in range(para_num)]
for k in range(para_num)]
I_list = [[[0.0 for i in range(len(p_list))] for j in range(para_num)]
for k in range(para_num)]
for i in range(len(p_list)):
F_tp = QFIM(rho_list[i], drho_list[i], LDtype=LDtype, eps=eps)
for pj in range(para_num):
for pk in range(para_num):
F_list[pj][pk][i] = F_tp[pj][pk]
I_list[pj][pk][i] = (dp_list[i][pj] * dp_list[i][pk] /
p_list[i]**2)
F_res = np.zeros([para_num, para_num])
I_res = np.zeros([para_num, para_num])
for para_i in range(0, para_num):
for para_j in range(para_i, para_num):
F_ij = np.array(F_list[para_i][para_j]).reshape(p_shape)
I_ij = np.array(I_list[para_i][para_j]).reshape(p_shape)
arr1 = p * F_ij
arr2 = p * I_ij
for si in reversed(range(para_num)):
arr1 = simps(arr1, x[si])
arr2 = simps(arr2, x[si])
F_res[para_i][para_j] = arr1
F_res[para_j][para_i] = arr1
I_res[para_i][para_j] = arr2
I_res[para_j][para_i] = arr2
return np.linalg.pinv(F_res + I_res)
def BQCRB(x, p, dp, rho, drho, b=[], db=[], btype=1, LDtype="SLD", eps=1e-8):
r"""
Calculation of the Bayesian quantum Cramer-Rao bound (BQCRB). The covariance matrix
with a prior distribution $p(\textbf{x})$ is defined as
\begin{align}
\mathrm{cov}(\hat{\textbf{x}},\{\Pi_y\})=\int p(\textbf{x})\sum_y\mathrm{Tr}
(\rho\Pi_y)(\hat{\textbf{x}}-\textbf{x})(\hat{\textbf{x}}-\textbf{x})^{\mathrm{T}}
\mathrm{d}\textbf{x}
\end{align}
where $\textbf{x}=(x_0,x_1,\dots)^{\mathrm{T}}$ are the unknown parameters to be estimated
and the integral $\int\mathrm{d}\textbf{x}:=\iiint\mathrm{d}x_0\mathrm{d}x_1\cdots$.
$\{\Pi_y\}$ is a set of positive operator-valued measure (POVM) and $\rho$ represent
the parameterized density matrix.
This function calculates three types of the BQCRB. The first one is
\begin{align}
\mathrm{cov}(\hat{\textbf{x}},\{\Pi_y\})\geq\int p(\textbf{x})\left(B\mathcal{F}^{-1}B
+\textbf{b}\textbf{b}^{\mathrm{T}}\right)\mathrm{d}\textbf{x},
\end{align}
where $\textbf{b}$ and $\textbf{b}'$ are the vectors of biase and its derivatives on parameters.
$B$ is a diagonal matrix with the $i$th entry $B_{ii}=1+[\textbf{b}']_{i}$ and $\mathcal{F}$
is the QFIM for all types.
The second one is
\begin{align}
\mathrm{cov}(\hat{\textbf{x}},\{\Pi_y\})\geq \mathcal{B}\,\mathcal{F}_{\mathrm{Bayes}}^{-1}\,
\mathcal{B}+\int p(\textbf{x})\textbf{b}\textbf{b}^{\mathrm{T}}\mathrm{d}\textbf{x},
\end{align}
where $\mathcal{B}=\int p(\textbf{x})B\mathrm{d}\textbf{x}$ is the average of $B$ and
$\mathcal{F}_{\mathrm{Bayes}}=\int p(\textbf{x})\mathcal{F}\mathrm{d}\textbf{x}$ is
the average QFIM.
The third one is
\begin{align}
\mathrm{cov}(\hat{\textbf{x}},\{\Pi_y\})\geq \int p(\textbf{x})
\mathcal{G}\left(\mathcal{I}_p+\mathcal{F}\right)^{-1}\mathcal{G}^{\mathrm{T}}\mathrm{d}\textbf{x}
\end{align}
with $[\mathcal{I}_{p}]_{ab}:=[\partial_a \ln p(\textbf{x})][\partial_b \ln p(\textbf{x})]$ and
$\mathcal{G}_{ab}:=[\partial_b\ln p(\textbf{x})][\textbf{b}]_a+B_{aa}\delta_{ab}$.
Parameters
----------
> **x:** `list`
-- The regimes of the parameters for the integral.
> **p:** `multidimensional array`
-- The prior distribution.
> **rho:** `multidimensional list`
-- Parameterized density matrix.
> **drho:** `multidimensional list`
-- Derivatives of the parameterized density matrix (rho) with respect to the unknown
parameters to be estimated.
> **b:** `list`
-- Vector of biases of the form $\textbf{b}=(b(x_0),b(x_1),\dots)^{\mathrm{T}}$.
> **db:** `list`
-- Derivatives of b with respect to the unknown parameters to be estimated, It should be
expressed as $\textbf{b}'=(\partial_0 b(x_0),\partial_1 b(x_1),\dots)^{\mathrm{T}}$.
> **btype:** `int (1, 2, 3)`
-- Types of the BQCRB. Options are:
1 (default) -- It means to calculate the first type of the BQCRB.
2 -- It means to calculate the second type of the BQCRB.
3 -- It means to calculate the third type of the BCRB.
> **LDtype:** `string`
-- Types of QFI (QFIM) can be set as the objective function. Options are:
"SLD" (default) -- QFI (QFIM) based on symmetric logarithmic derivative (SLD).
"RLD" -- QFI (QFIM) based on right logarithmic derivative (RLD).
"LLD" -- QFI (QFIM) based on left logarithmic derivative (LLD).
> **eps:** `float`
-- Machine epsilon.
Returns
----------
**BQCRB:** `float or matrix`
-- For single parameter estimation (the length of x is equal to one), the
output is a float and for multiparameter estimation (the length of x is
more than one), it returns a matrix.
"""
para_num = len(x)
if para_num == 1:
#### single parameter scenario ####
p_num = len(p)
if b == []:
b = np.zeros(p_num)
db = np.zeros(p_num)
if b != [] and db == []:
db = np.zeros(p_num)
if type(drho[0]) == list:
drho = [drho[i][0] for i in range(p_num)]
if type(b[0]) == list or type(b[0]) == np.ndarray:
b = b[0]
if type(db[0]) == list or type(db[0]) == np.ndarray:
db = db[0]
F_tp = np.zeros(p_num)
for m in range(p_num):
F_tp[m] = QFIM(rho[m], [drho[m]], LDtype=LDtype, eps=eps)
if btype == 1:
arr = [
p[i] * ((1 + db[i])**2 / F_tp[i] + b[i]**2)
for i in range(p_num)
]
F = simps(arr, x[0])
return F
elif btype == 2:
arr2 = [p[i] * F_tp[i] for i in range(p_num)]
F2 = simps(arr2, x[0])
arr2 = [p[j] * (1 + db[j]) for j in range(p_num)]
B = simps(arr2, x[0])
arr3 = [p[k] * b[k]**2 for k in range(p_num)]
bb = simps(arr3, x[0])
F = B**2 / F2 + bb
return F
elif btype == 3:
I_tp = [np.real(dp[i] * dp[i] / p[i]**2) for i in range(p_num)]
arr = [
p[j] * (dp[j] * b[j] / p[j] + (1 + db[j]))**2 /
(I_tp[j] + F_tp[j]) for j in range(p_num)
]
F = simps(arr, x[0])
return F
else:
raise NameError("NameError: btype should be choosen in {1, 2, 3}.")
else:
#### multiparameter scenario ####
if b == []:
b, db = [], []
for i in range(para_num):
b.append(np.zeros(len(x[i])))
db.append(np.zeros(len(x[i])))
if b != [] and db == []:
db = []
for i in range(para_num):
db.append(np.zeros(len(x[i])))
p_shape = np.shape(p)
p_ext = extract_ele(p, para_num)
dp_ext = extract_ele(dp, para_num)
rho_ext = extract_ele(rho, para_num)
drho_ext = extract_ele(drho, para_num)
b_pro = product(*b)
db_pro = product(*db)
p_list, dp_list, rho_list, drho_list = [], [], [], []
for p_ele, dp_ele, rho_ele, drho_ele in zip(p_ext, dp_ext, rho_ext,
drho_ext):
p_list.append(p_ele)
dp_list.append(dp_ele)
rho_list.append(rho_ele)
drho_list.append(drho_ele)
b_list, db_list = [], []
for b_ele, db_ele in zip(b_pro, db_pro):
b_list.append([b_ele[i] for i in range(para_num)])
db_list.append([db_ele[j] for j in range(para_num)])
if btype == 1:
F_list = [[[0.0 for i in range(len(p_list))]
for j in range(para_num)] for k in range(para_num)]
for i in range(len(p_list)):
F_tp = QFIM(rho_list[i], drho_list[i], LDtype=LDtype, eps=eps)
F_inv = np.linalg.pinv(F_tp)
B = np.diag([(1.0 + db_list[i][j]) for j in range(para_num)])
term1 = np.dot(B, np.dot(F_inv, B))
term2 = np.dot(
np.array(b_list[i]).reshape(para_num, 1),
np.array(b_list[i]).reshape(1, para_num),
)
for pj in range(para_num):
for pk in range(para_num):
F_list[pj][pk][i] = term1[pj][pk] + term2[pj][pk]
res = np.zeros([para_num, para_num])
for para_i in range(0, para_num):
for para_j in range(para_i, para_num):
F_ij = np.array(F_list[para_i][para_j]).reshape(p_shape)
arr = p * F_ij
for si in reversed(range(para_num)):
arr = simps(arr, x[si])
res[para_i][para_j] = arr
res[para_j][para_i] = arr
return res
elif btype == 2:
F_list = [[[0.0 for i in range(len(p_list))]
for j in range(para_num)] for k in range(para_num)]
B_list = [[[0.0 for i in range(len(p_list))]
for j in range(para_num)] for k in range(para_num)]
bb_list = [[[0.0 for i in range(len(p_list))]
for j in range(para_num)] for k in range(para_num)]
for i in range(len(p_list)):
F_tp = QFIM(rho_list[i], drho_list[i], LDtype=LDtype, eps=eps)
B_tp = np.diag([(1.0 + db_list[i][j])
for j in range(para_num)])
bb_tp = np.dot(
np.array(b_list[i]).reshape(para_num, 1),
np.array(b_list[i]).reshape(1, para_num),
)
for pj in range(para_num):
for pk in range(para_num):
F_list[pj][pk][i] = F_tp[pj][pk]
B_list[pj][pk][i] = B_tp[pj][pk]
bb_list[pj][pk][i] = bb_tp[pj][pk]
F_res = np.zeros([para_num, para_num])
for para_i in range(0, para_num):
for para_j in range(para_i, para_num):
F_ij = np.array(F_list[para_i][para_j]).reshape(p_shape)
arr = p * F_ij
for si in reversed(range(para_num)):
arr = simps(arr, x[si])
F_res[para_i][para_j] = arr
F_res[para_j][para_i] = arr
B_res = np.zeros([para_num, para_num])
bb_res = np.zeros([para_num, para_num])
for para_m in range(para_num):
for para_n in range(para_num):
B_mn = np.array(B_list[para_m][para_n]).reshape(p_shape)
bb_mn = np.array(bb_list[para_m][para_n]).reshape(p_shape)
arr2 = p * B_mn
arr3 = p * bb_mn
for sj in reversed(range(para_num)):
arr2 = simps(arr2, x[sj])
arr3 = simps(arr3, x[sj])
B_res[para_m][para_n] = arr2
bb_res[para_m][para_n] = arr3
res = np.dot(B_res, np.dot(np.linalg.pinv(F_res), B_res)) + bb_res
return res
elif btype == 3:
F_list = [[[0.0 for i in range(len(p_list))]
for j in range(para_num)] for k in range(para_num)]
for i in range(len(p_list)):
F_tp = QFIM(rho_list[i], drho_list[i], LDtype=LDtype, eps=eps)
I_tp = np.zeros((para_num, para_num))
G_tp = np.zeros((para_num, para_num))
for pm in range(para_num):
for pn in range(para_num):
if pm == pn:
G_tp[pm][pn] = dp_list[i][pn] * b_list[i][
pm] / p_list[i] + (1.0 + db_list[i][pm])
else:
G_tp[pm][pn] = dp_list[i][pn] * b_list[i][
pm] / p_list[i]
I_tp[pm][pn] = dp_list[i][pm] * dp_list[i][
pn] / p_list[i]**2
F_tot = np.dot(G_tp, np.dot(np.linalg.pinv(F_tp + I_tp),
G_tp.T))
for pj in range(para_num):
for pk in range(para_num):
F_list[pj][pk][i] = F_tot[pj][pk]
res = np.zeros([para_num, para_num])
for para_i in range(0, para_num):
for para_j in range(para_i, para_num):
F_ij = np.array(F_list[para_i][para_j]).reshape(p_shape)
arr = p * F_ij
for si in reversed(range(para_num)):
arr = simps(arr, x[si])
res[para_i][para_j] = arr
res[para_j][para_i] = arr
return res
else:
raise NameError("NameError: btype should be choosen in {1, 2, 3}.")
def VTB(x, p, dp, rho, drho, M=[], eps=1e-8):
r"""
Calculation of the Bayesian version of Cramer-Rao bound introduced by
Van Trees (VTB). The covariance matrix with a prior distribution $p(\textbf{x})$
is defined as
\begin{align}
\mathrm{cov}(\hat{\textbf{x}},\{\Pi_y\})=\int p(\textbf{x})\sum_y\mathrm{Tr}
(\rho\Pi_y)(\hat{\textbf{x}}-\textbf{x})(\hat{\textbf{x}}-\textbf{x})^{\mathrm{T}}
\mathrm{d}\textbf{x}
\end{align}
where $\textbf{x}=(x_0,x_1,\dots)^{\mathrm{T}}$ are the unknown parameters to be estimated
and the integral $\int\mathrm{d}\textbf{x}:=\iiint\mathrm{d}x_0\mathrm{d}x_1\cdots$.
$\{\Pi_y\}$ is a set of positive operator-valued measure (POVM) and $\rho$ represent
the parameterized density matrix.
\begin{align}
\mathrm{cov}(\hat{\textbf{x}},\{\Pi_y\})\geq \left(\mathcal{I}_{\mathrm{prior}}
+\mathcal{I}_{\mathrm{Bayes}}\right)^{-1},
\end{align}
where $\mathcal{I}_{\mathrm{prior}}=\int p(\textbf{x})\mathcal{I}_{p}\mathrm{d}\textbf{x}$
is the CFIM for $p(\textbf{x})$ and
$\mathcal{I}_{\mathrm{Bayes}}=\int p(\textbf{x})\mathcal{I}\mathrm{d}\textbf{x}$ is the
average CFIM.
Parameters
----------
> **x:** `list`
-- The regimes of the parameters for the integral.
> **p:** `multidimensional array`
-- The prior distribution.
> **dp:** `list`
-- Derivatives of the prior distribution with respect to the unknown parameters
to be estimated. For example, dp[0] is the derivative vector with respect to the first
parameter.
> **rho:** `multidimensional list`
-- Parameterized density matrix.
> **drho:** `multidimensional list`
-- Derivatives of the parameterized density matrix (rho) with respect to the
unknown parameters to be estimated.
> **M:** `list of matrices`
-- A set of positive operator-valued measure (POVM). The default measurement
is a set of rank-one symmetric informationally complete POVM (SIC-POVM).
> **eps:** `float`
-- Machine epsilon.
Returns
----------
**VTB:** `float or matrix`
-- For single parameter estimation (the length of x is equal to one), the
output is a float and for multiparameter estimation (the length of x is
more than one), it returns a matrix.
**Note:**
SIC-POVM is calculated by the Weyl-Heisenberg covariant SIC-POVM fiducial state
which can be downloaded from [here](http://www.physics.umb.edu/Research/QBism/
solutions.html).
"""
para_num = len(x)
p_num = len(p)
if para_num == 1:
#### single parameter scenario ####
if M == []:
M = SIC(len(rho[0]))
else:
if type(M) != list:
raise TypeError("Please make sure M is a list!")
if type(drho[0]) == list:
drho = [drho[i][0] for i in range(p_num)]
if type(dp[0]) == list or type(dp[0]) == np.ndarray:
dp = [dp[i][0] for i in range(p_num)]
F_tp = np.zeros(p_num)
for m in range(p_num):
F_tp[m] = CFIM(rho[m], [drho[m]], M=M, eps=eps)
arr1 = [np.real(dp[i] * dp[i] / p[i]) for i in range(p_num)]
I = simps(arr1, x[0])
arr2 = [np.real(F_tp[j] * p[j]) for j in range(p_num)]
F = simps(arr2, x[0])
return 1.0 / (I + F)
else:
#### multiparameter scenario ####
p_shape = np.shape(p)
p_ext = extract_ele(p, para_num)
dp_ext = extract_ele(dp, para_num)
rho_ext = extract_ele(rho, para_num)
drho_ext = extract_ele(drho, para_num)
p_list, dp_list, rho_list, drho_list = [], [], [], []
for p_ele, dp_ele, rho_ele, drho_ele in zip(p_ext, dp_ext, rho_ext,
drho_ext):
p_list.append(p_ele)
dp_list.append(dp_ele)
rho_list.append(rho_ele)
drho_list.append(drho_ele)
dim = len(rho_list[0])
if M == []:
M = SIC(dim)
else:
if type(M) != list:
raise TypeError("Please make sure M is a list!")
F_list = [[[0.0 for i in range(len(p_list))] for j in range(para_num)]
for k in range(para_num)]
I_list = [[[0.0 for i in range(len(p_list))] for j in range(para_num)]
for k in range(para_num)]
for i in range(len(p_list)):
F_tp = CFIM(rho_list[i], drho_list[i], M=M, eps=eps)
for pj in range(para_num):
for pk in range(para_num):
F_list[pj][pk][i] = F_tp[pj][pk]
I_list[pj][pk][i] = (dp_list[i][pj] * dp_list[i][pk] /
p_list[i]**2)
F_res = np.zeros([para_num, para_num])
I_res = np.zeros([para_num, para_num])
for para_i in range(0, para_num):
for para_j in range(para_i, para_num):
F_ij = np.array(F_list[para_i][para_j]).reshape(p_shape)
I_ij = np.array(I_list[para_i][para_j]).reshape(p_shape)
arr1 = p * F_ij
arr2 = p * I_ij
for si in reversed(range(para_num)):
arr1 = simps(arr1, x[si])
arr2 = simps(arr2, x[si])
F_res[para_i][para_j] = arr1
F_res[para_j][para_i] = arr1
I_res[para_i][para_j] = arr2
I_res[para_j][para_i] = arr2
return np.linalg.pinv(F_res + I_res)
def adaptive_Kraus(x, p, M, rho0, K, dK, W, max_episode, eps, savefile):
para_num = len(x)
dim = np.shape(rho0)[0]
if para_num == 1:
#### singleparameter senario ####
p_num = len(p)
F = []
for hi in range(p_num):
rho_tp = sum([np.dot(Ki, np.dot(rho0, Ki.conj().T)) for Ki in K[hi]])
drho_tp = [
sum(
[
(
np.dot(dKi, np.dot(rho0, Ki.conj().T))
+ np.dot(Ki, np.dot(rho0, dKi.conj().T))
)
for (Ki, dKi) in zip(K[hi], dKj)
]
)
for dKj in dK[hi]
]
F_tp = CFIM(rho_tp, drho_tp, M)
F.append(F_tp)
idx = np.argmax(F)
x_opt = x[0][idx]
print("The optimal parameter is %s" % x_opt)
u = 0.0
if savefile == False:
y, xout = [], []
for ei in range(max_episode):
rho = [np.zeros((dim, dim), dtype=np.complex128) for i in range(p_num)]
for hj in range(p_num):
x_idx = np.argmin(np.abs(x[0] - (x[0][hj] + u)))
rho_tp = sum([np.dot(Ki, np.dot(rho0, Ki.conj().T)) for Ki in K[x_idx]])
rho[hj] = rho_tp
print("The tunable parameter is %s" % u)
res_exp = input("Please enter the experimental result: ")
res_exp = int(res_exp)
pyx = np.zeros(p_num)
for xi in range(p_num):
pyx[xi] = np.real(np.trace(np.dot(rho[xi], M[res_exp])))
arr = [pyx[m] * p[m] for m in range(p_num)]
py = simps(arr, x[0])
p_update = pyx * p / py
p = p_update
p_idx = np.argmax(p)
x_out = x[0][p_idx]
print("The estimator is %s (%d episodes)" % (x_out, ei))
u = x_opt - x_out
if (ei + 1) % 50 == 0:
if (x_out + u) > x[0][-1] and (x_out + u) < x[0][0]:
raise ValueError("please increase the regime of the parameters.")
xout.append(x_out)
y.append(res_exp)
fp = open('pout.csv','a')
fp.write('\n')
np.savetxt(fp, np.array(p))
fp.close()
fx = open('xout.csv','a')
fx.write('\n')
np.savetxt(fx, np.array(xout))
fx.close()
fy = open('y.csv','a')
fy.write('\n')
np.savetxt(fy, np.array(y))
fy.close()
else:
for ei in range(max_episode):
rho = [np.zeros((dim, dim), dtype=np.complex128) for i in range(p_num)]
for hj in range(p_num):
x_idx = np.argmin(np.abs(x[0] - (x[0][hj] + u)))
rho_tp = sum([np.dot(Ki, np.dot(rho0, Ki.conj().T)) for Ki in K[x_idx]])
rho[hj] = rho_tp
print("The tunable parameter is %s" % u)
res_exp = input("Please enter the experimental result: ")
res_exp = int(res_exp)
pyx = np.zeros(p_num)
for xi in range(p_num):
pyx[xi] = np.real(np.trace(np.dot(rho[xi], M[res_exp])))
arr = [pyx[m] * p[m] for m in range(p_num)]
py = simps(arr, x[0])
p_update = pyx * p / py
p = p_update
p_idx = np.argmax(p)
x_out = x[0][p_idx]
print("The estimator is %s (%d episodes)" % (x_out, ei))
u = x_opt - x_out
if (ei + 1) % 50 == 0:
if (x_out + u) > x[0][-1] and (x_out + u) < x[0][0]:
raise ValueError("please increase the regime of the parameters.")
fp = open('pout.csv','a')
fp.write('\n')
np.savetxt(fp, [np.array(p)])
fp.close()
fx = open('xout.csv','a')
fx.write('\n')
np.savetxt(fx, [x_out])
fx.close()
fy = open('y.csv','a')
fy.write('\n')
np.savetxt(fy, [res_exp])
fy.close()
else:
#### miltiparameter senario ####
p_shape = np.shape(p)
x_list = []
for x_tp in product(*x):
x_list.append([x_tp[i] for i in range(para_num)])
p_ext = extract_ele(p, para_num)
K_ext = extract_ele(K, para_num)
dK_ext = extract_ele(dK, para_num)
p_list, K_list, dK_list = [], [], []
for p_ele, K_ele, dK_ele in zip(p_ext, K_ext, dK_ext):
p_list.append(p_ele)
K_list.append(K_ele)
dK_list.append(dK_ele)
k_num = len(K_list[0])
F = []
for hi in range(len(p_list)):
rho_tp = sum([np.dot(Ki, np.dot(rho0, Ki.conj().T)) for Ki in K_list[hi]])
dK_reshape = [
[dK_list[hi][i][j] for i in range(k_num)]
for j in range(para_num)
]
drho_tp = [
sum(
[
np.dot(dKi, np.dot(rho0, Ki.conj().T))
+ np.dot(Ki, np.dot(rho0, dKi.conj().T))
for (Ki, dKi) in zip(K_list[hi], dKj)
]
)
for dKj in dK_reshape
]
F_tp = CFIM(rho_tp, drho_tp, M)
if np.linalg.det(F_tp) < eps:
F.append(eps)
else:
F.append(1.0 / np.trace(np.dot(W, np.linalg.inv(F_tp))))
F = np.array(F).reshape(p_shape)
idx = np.unravel_index(F.argmax(), F.shape)
x_opt = [x[i][idx[i]] for i in range(para_num)]
print("The optimal parameter is %s" % (x_opt))
u = [0.0 for i in range(para_num)]
if savefile == False:
y, xout = [], []
for ei in range(max_episode):
rho = [
np.zeros((dim, dim), dtype=np.complex128) for i in range(len(p_list))
]
for hj in range(len(p_list)):
idx_list = [
np.argmin(np.abs(x[i] - (x_list[hj][i] + u[i])))
for i in range(para_num)
]
x_idx = int(
sum(
[
idx_list[i] * np.prod(np.array(p_shape[(i + 1) :]))
for i in range(para_num)
]
)
)
rho[hj] = sum(
[np.dot(Ki, np.dot(rho0, Ki.conj().T)) for Ki in K_list[x_idx]]
)
print("The tunable parameter are %s" % (u))
res_exp = input("Please enter the experimental result: ")
res_exp = int(res_exp)
pyx_list = np.zeros(len(p_list))
for xi in range(len(p_list)):
pyx_list[xi] = np.real(np.trace(np.dot(rho[xi], M[res_exp])))
pyx = pyx_list.reshape(p_shape)
arr = p * pyx
for si in reversed(range(para_num)):
arr = simps(arr, x[si])
py = arr
p_update = p * pyx / py
p = p_update
p_idx = np.unravel_index(p.argmax(), p.shape)
x_out = [x[i][p_idx[i]] for i in range(para_num)]
print("The estimator are %s (%d episodes)" % (x_out, ei))
u = np.array(x_opt) - np.array(x_out)
if (ei + 1) % 50 == 0:
for un in range(para_num):
if (x_out[un] + u[un]) > x[un][-1] and (x_out[un] + u[un]) < x[un][
0
]:
raise ValueError(
"please increase the regime of the parameters."
)
xout.append(x_out)
y.append(res_exp)
fp = open('pout.csv','a')
fp.write('\n')
np.savetxt(fp, np.array(p))
fp.close()
fx = open('xout.csv','a')
fx.write('\n')
np.savetxt(fx, np.array(xout))
fx.close()
fy = open('y.csv','a')
fy.write('\n')
np.savetxt(fy, np.array(y))
fy.close()
else:
for ei in range(max_episode):
rho = [
np.zeros((dim, dim), dtype=np.complex128) for i in range(len(p_list))
]
for hj in range(len(p_list)):
idx_list = [
np.argmin(np.abs(x[i] - (x_list[hj][i] + u[i])))
for i in range(para_num)
]
x_idx = int(
sum(
[
idx_list[i] * np.prod(np.array(p_shape[(i + 1) :]))
for i in range(para_num)
]
)
)
rho[hj] = sum(
[np.dot(Ki, np.dot(rho0, Ki.conj().T)) for Ki in K_list[x_idx]]
)
print("The tunable parameter are %s" % (u))
res_exp = input("Please enter the experimental result: ")
res_exp = int(res_exp)
pyx_list = np.zeros(len(p_list))
for xi in range(len(p_list)):
pyx_list[xi] = np.real(np.trace(np.dot(rho[xi], M[res_exp])))
pyx = pyx_list.reshape(p_shape)
arr = p * pyx
for si in reversed(range(para_num)):
arr = simps(arr, x[si])
py = arr
p_update = p * pyx / py
p = p_update
p_idx = np.unravel_index(p.argmax(), p.shape)
x_out = [x[i][p_idx[i]] for i in range(para_num)]
print("The estimator are %s (%d episodes)" % (x_out, ei))
u = np.array(x_opt) - np.array(x_out)
if (ei + 1) % 50 == 0:
for un in range(para_num):
if (x_out[un] + u[un]) > x[un][-1] and (x_out[un] + u[un]) < x[un][
0
]:
raise ValueError(
"please increase the regime of the parameters."
)
fp = open('pout.csv','a')
fp.write('\n')
np.savetxt(fp, [np.array(p)])
fp.close()
fx = open('xout.csv','a')
fx.write('\n')
np.savetxt(fx, [x_out])
fx.close()
fy = open('y.csv','a')
fy.write('\n')
np.savetxt(fy, [res_exp])
fy.close()
def BQFIM(x, p, rho, drho, LDtype="SLD", eps=1e-8):
r"""
Calculation of the Bayesian quantum Fisher information (BQFI) and the
Bayesian quantum Fisher information matrix (BQFIM) of the form
\begin{align}
\mathcal{F}_{\mathrm{Bayes}}=\int p(\textbf{x})\mathcal{F}\mathrm{d}\textbf{x}
\end{align}
with $\mathcal{F}$ the QFIM of all types and $p(\textbf{x})$ the prior distribution.
Parameters
----------
> **x:** `list`
-- The regimes of the parameters for the integral.
> **p:** `multidimensional array`
-- The prior distribution.
> **rho:** `multidimensional list`
-- Parameterized density matrix.
> **drho:** `multidimensional list`
-- Derivatives of the parameterized density matrix (rho) with respect to the unknown
parameters to be estimated.
> **LDtype:** `string`
-- Types of QFI (QFIM) can be set as the objective function. Options are:
"SLD" (default) -- QFI (QFIM) based on symmetric logarithmic derivative (SLD).
"RLD" -- QFI (QFIM) based on right logarithmic derivative (RLD).
"LLD" -- QFI (QFIM) based on left logarithmic derivative (LLD).
> **eps:** `float`
-- Machine epsilon.
Returns
----------
**BQFI or BQFIM:** `float or matrix`
-- For single parameter estimation (the length of x is equal to one), the output
is BQFI and for multiparameter estimation (the length of x is more than one),
it returns BQFIM.
"""
para_num = len(x)
if para_num == 1:
#### single parameter scenario ####
p_num = len(p)
if type(drho[0]) == list:
drho = [drho[i][0] for i in range(p_num)]
F_tp = np.zeros(p_num)
for m in range(p_num):
F_tp[m] = QFIM(rho[m], [drho[m]], LDtype=LDtype, eps=eps)
arr = [p[i] * F_tp[i] for i in range(p_num)]
return simps(arr, x[0])
else:
#### multiparameter scenario ####
p_shape = np.shape(p)
p_ext = extract_ele(p, para_num)
rho_ext = extract_ele(rho, para_num)
drho_ext = extract_ele(drho, para_num)
p_list, rho_list, drho_list = [], [], []
for p_ele, rho_ele, drho_ele in zip(p_ext, rho_ext, drho_ext):
p_list.append(p_ele)
rho_list.append(rho_ele)
drho_list.append(drho_ele)
F_list = [[[0.0 for i in range(len(p_list))] for j in range(para_num)]
for k in range(para_num)]
for i in range(len(p_list)):
F_tp = QFIM(rho_list[i], drho_list[i], LDtype=LDtype, eps=eps)
for pj in range(para_num):
for pk in range(para_num):
F_list[pj][pk][i] = F_tp[pj][pk]
BQFIM_res = np.zeros([para_num, para_num])
for para_i in range(0, para_num):
for para_j in range(para_i, para_num):
F_ij = np.array(F_list[para_i][para_j]).reshape(p_shape)
arr = p * F_ij
for si in reversed(range(para_num)):
arr = simps(arr, x[si])
BQFIM_res[para_i][para_j] = arr
BQFIM_res[para_j][para_i] = arr
return BQFIM_res
def Mopt(self, W=[]):
r"""
Measurement optimization for the optimal x.
Parameters
----------
> **W:** `matrix`
-- Weight matrix.
"""
if W == []:
W = np.identity(self.para_num)
else:
W = W
if self.dynamic_type == "dynamics":
if self.para_num == 1:
F = []
for i in range(len(self.H)):
dynamics = Lindblad(
self.tspan,
self.rho0,
self.H[i],
self.dH[i],
decay=self.decay,
Hc=self.Hc,
ctrl=self.ctrl,
)
rho_tp, drho_tp = dynamics.expm()
rho, drho = rho_tp[-1], drho_tp[-1]
F_tp = QFIM(rho, drho)
F.append(F_tp)
idx = np.argmax(F)
H_res, dH_res = self.H[idx], self.dH[idx]
else:
p_ext = extract_ele(self.p, self.para_num)
H_ext = extract_ele(self.H, self.para_num)
dH_ext = extract_ele(self.dH, self.para_num)
p_list, H_list, dH_list = [], [], []
for p_ele, H_ele, dH_ele in zip(p_ext, H_ext, dH_ext):
p_list.append(p_ele)
H_list.append(H_ele)
dH_list.append(dH_ele)
F = []
for i in range(len(p_list)):
dynamics = Lindblad(
self.tspan,
self.rho0,
self.H_list[i],
self.dH_list[i],
decay=self.decay,
Hc=self.Hc,
ctrl=self.ctrl,
)
rho_tp, drho_tp = dynamics.expm()
rho, drho = rho_tp[-1], drho_tp[-1]
F_tp = QFIM(rho, drho)
if np.linalg.det(F_tp) < self.eps:
F.append(self.eps)
else:
F.append(1.0 / np.trace(np.dot(W, np.linalg.inv(F_tp))))
idx = np.argmax(F)
H_res, dH_res = self.H_list[idx], self.dH_list[idx]
m = MeasurementOpt(mtype="projection", minput=[], method="DE")
m.dynamics(
self.tspan,
self.rho0,
H_res,
dH_res,
Hc=self.Hc,
ctrl=self.ctrl,
decay=self.decay,
)
m.CFIM(W=W)
elif self.dynamic_type == "Kraus":
if self.para_num == 1:
F = []
for hi in range(len(self.K)):
rho_tp = sum(
[np.dot(Ki, np.dot(self.rho0, Ki.conj().T)) for Ki in self.K[hi]]
)
drho_tp = sum(
[
np.dot(dKi, np.dot(self.rho0, Ki.conj().T))
+ np.dot(Ki, np.dot(self.rho0, dKi.conj().T))
for (Ki, dKi) in zip(self.K[hi], self.dK[hi])
]
)
F_tp = QFIM(rho_tp, drho_tp)
F.append(F_tp)
idx = np.argmax(F)
K_res, dK_res = self.K[idx], self.dK[idx]
else:
p_shape = np.shape(self.p)
p_ext = extract_ele(self.p, self.para_num)
K_ext = extract_ele(self.K, self.para_num)
dK_ext = extract_ele(self.dK, self.para_num)
p_list, K_list, dK_list = [], [], []
for K_ele, dK_ele in zip(K_ext, dK_ext):
p_list.append(p_ele)
K_list.append(K_ele)
dK_list.append(dK_ele)
F = []
for hi in range(len(p_list)):
rho_tp = sum(
[np.dot(Ki, np.dot(self.rho0, Ki.conj().T)) for Ki in K_list[hi]]
)
dK_reshape = [
[dK_list[hi][i][j] for i in range(self.k_num)]
for j in range(self.para_num)
]
drho_tp = [
sum(
[
np.dot(dKi, np.dot(self.rho0, Ki.conj().T))
+ np.dot(Ki, np.dot(self.rho0, dKi.conj().T))
for (Ki, dKi) in zip(K_list[hi], dKj)
]
)
for dKj in dK_reshape
]
F_tp = QFIM(rho_tp, drho_tp)
if np.linalg.det(F_tp) < self.eps:
F.append(self.eps)
else:
F.append(1.0 / np.trace(np.dot(W, np.linalg.inv(F_tp))))
F = np.array(F).reshape(p_shape)
idx = np.where(np.array(F) == np.max(np.array(F)))
K_res, dK_res = self.K_list[idx], self.dK_list[idx]
m = MeasurementOpt(mtype="projection", minput=[], method="DE")
m.Kraus(self.rho0, K_res, dK_res, decay=self.decay)
m.CFIM(W=W)
else:
raise ValueError(
"{!r} is not a valid value for type of dynamics, supported values are 'dynamics' and 'Kraus'.".format(
self.dynamic_type
)
)
def Bayes(x, p, rho, y, M=[], estimator="mean", savefile=False):
"""
Bayesian estimation. The prior distribution is updated via the posterior
distribution obtained by the Bayes’ rule and the estimated value of parameters
are updated via the expectation value of the distribution or maximum a
posteriori probability (MAP).
Parameters
----------
> **x:** `list`
-- The regimes of the parameters for the integral.
> **p:** `multidimensional array`
-- The prior distribution.
> **rho:** `multidimensional list`
-- Parameterized density matrix.
> **y:** `array`
-- The experimental results obtained in practice.
> **M:** `list of matrices`
-- A set of positive operator-valued measure (POVM). The default measurement
is a set of rank-one symmetric informationally complete POVM (SIC-POVM).
> **estimator:** `string`
-- Estimators for the bayesian estimation. Options are:
"mean" -- The expectation value of the distribution.
"MAP" -- Maximum a posteriori probability.
> **savefile:** `bool`
-- Whether or not to save all the posterior distributions.
If set `True` then two files "pout.npy" and "xout.npy" will be generated including
the posterior distributions and the estimated values in the iterations. If set
`False` the posterior distribution in the final iteration and the estimated values
in all iterations will be saved in "pout.npy" and "xout.npy".
Returns
----------
**pout and xout:** `array and float`
-- The posterior distribution and the estimated values in the final iteration.
**Note:**
SIC-POVM is calculated by the Weyl-Heisenberg covariant SIC-POVM fiducial state
which can be downloaded from [here](http://www.physics.umb.edu/Research/QBism/
solutions.html).
"""
para_num = len(x)
max_episode = len(y)
if para_num == 1:
#### single parameter scenario ####
if M == []:
M = SIC(len(rho[0]))
else:
if type(M) != list:
raise TypeError("Please make sure M is a list!")
if savefile == False:
x_out = []
if estimator == "mean":
for mi in range(max_episode):
res_exp = int(y[mi])
pyx = np.zeros(len(x[0]))
for xi in range(len(x[0])):
p_tp = np.real(np.trace(np.dot(rho[xi], M[res_exp])))
pyx[xi] = p_tp
arr = [pyx[m] * p[m] for m in range(len(x[0]))]
py = simps(arr, x[0])
p_update = pyx * p / py
p = p_update
mean = simps([p[m] * x[0][m] for m in range(len(x[0]))],
x[0])
x_out.append(mean)
elif estimator == "MAP":
for mi in range(max_episode):
res_exp = int(y[mi])
pyx = np.zeros(len(x[0]))
for xi in range(len(x[0])):
p_tp = np.real(np.trace(np.dot(rho[xi], M[res_exp])))
pyx[xi] = p_tp
arr = [pyx[m] * p[m] for m in range(len(x[0]))]
py = simps(arr, x[0])
p_update = pyx * p / py
p = p_update
indx = np.where(p == max(p))[0][0]
x_out.append(x[0][indx])
else:
raise ValueError(
"{!r} is not a valid value for estimator, supported values are 'mean' and 'MAP'."
.format(estimator))
np.save("pout", p)
np.save("xout", x_out)
return p, x_out[-1]
else:
p_out, x_out = [], []
if estimator == "mean":
for mi in range(max_episode):
res_exp = int(y[mi])
pyx = np.zeros(len(x[0]))
for xi in range(len(x[0])):
p_tp = np.real(np.trace(np.dot(rho[xi], M[res_exp])))
pyx[xi] = p_tp
arr = [pyx[m] * p[m] for m in range(len(x[0]))]
py = simps(arr, x[0])
p_update = pyx * p / py
p = p_update
mean = simps([p[m] * x[0][m] for m in range(len(x[0]))],
x[0])
p_out.append(p)
x_out.append(mean)
elif estimator == "MAP":
for mi in range(max_episode):
res_exp = int(y[mi])
pyx = np.zeros(len(x[0]))
for xi in range(len(x[0])):
p_tp = np.real(np.trace(np.dot(rho[xi], M[res_exp])))
pyx[xi] = p_tp
arr = [pyx[m] * p[m] for m in range(len(x[0]))]
py = simps(arr, x[0])
p_update = pyx * p / py
p = p_update
indx = np.where(p == max(p))[0][0]
p_out.append(p)
x_out.append(x[0][indx])
else:
raise ValueError(
"{!r} is not a valid value for estimator, supported values are 'mean' and 'MAP'."
.format(estimator))
np.save("pout", p_out)
np.save("xout", x_out)
return p, x_out[-1]
else:
#### multiparameter scenario ####
p_shape = np.shape(p)
p_ext = extract_ele(p, para_num)
rho_ext = extract_ele(rho, para_num)
p_list, rho_list = [], []
for p_ele, rho_ele in zip(p_ext, rho_ext):
p_list.append(p_ele)
rho_list.append(rho_ele)
dim = len(rho_list[0])
if M == []:
M = SIC(dim)
else:
if type(M) != list:
raise TypeError("Please make sure M is a list!")
if savefile == False:
x_out = []
if estimator == "mean":
for mi in range(max_episode):
res_exp = int(y[mi])
pyx_list = np.zeros(len(p_list))
for xi in range(len(p_list)):
p_tp = np.real(
np.trace(np.dot(rho_list[xi], M[res_exp])))
pyx_list[xi] = p_tp
pyx = pyx_list.reshape(p_shape)
arr = p * pyx
for si in reversed(range(para_num)):
arr = simps(arr, x[si])
py = arr
p_update = p * pyx / py
p = p_update
mean = integ(x, p)
x_out.append(mean)
elif estimator == "MAP":
for mi in range(max_episode):
res_exp = int(y[mi])
pyx_list = np.zeros(len(p_list))
for xi in range(len(p_list)):
p_tp = np.real(
np.trace(np.dot(rho_list[xi], M[res_exp])))
pyx_list[xi] = p_tp
pyx = pyx_list.reshape(p_shape)
arr = p * pyx
for si in reversed(range(para_num)):
arr = simps(arr, x[si])
py = arr
p_update = p * pyx / py
p = p_update
indx = np.where(np.array(p) == np.max(np.array(p)))
x_out.append([x[i][indx[i][0]] for i in range(para_num)])
else:
raise ValueError(
"{!r} is not a valid value for estimator, supported values are 'mean' and 'MAP'."
.format(estimator))
np.save("Lout", p)
np.save("xout", x_out)
return p, x_out[-1]
else:
p_out, x_out = [], []
if estimator == "mean":
for mi in range(max_episode):
res_exp = int(y[mi])
pyx_list = np.zeros(len(p_list))
for xi in range(len(p_list)):
p_tp = np.real(
np.trace(np.dot(rho_list[xi], M[res_exp])))
pyx_list[xi] = p_tp
pyx = pyx_list.reshape(p_shape)
arr = p * pyx
for si in reversed(range(para_num)):
arr = simps(arr, x[si])
py = arr
p_update = p * pyx / py
p = p_update
mean = integ(x, p)
p_out.append(p)
x_out.append(mean)
elif estimator == "MAP":
for mi in range(max_episode):
res_exp = int(y[mi])
pyx_list = np.zeros(len(p_list))
for xi in range(len(p_list)):
p_tp = np.real(
np.trace(np.dot(rho_list[xi], M[res_exp])))
pyx_list[xi] = p_tp
pyx = pyx_list.reshape(p_shape)
arr = p * pyx
for si in reversed(range(para_num)):
arr = simps(arr, x[si])
py = arr
p_update = p * pyx / py
p = p_update
indx = np.where(np.array(p) == np.max(np.array(p)))
p_out.append(p)
x_out.append([x[i][indx[i][0]] for i in range(para_num)])
else:
raise ValueError(
"{!r} is not a valid value for estimator, supported values are 'mean' and 'MAP'."
.format(estimator))
np.save("pout", p_out)
np.save("xout", x_out)
return p, x_out[-1]
def BCB(x, p, rho, W=[], eps=1e-8):
"""
Calculation of the Bayesian cost bound with a quadratic cost function.
Parameters
----------
> **x:** `list`
-- The regimes of the parameters for the integral.
> **p:** `multidimensional array`
-- The prior distribution.
> **rho:** `multidimensional list`
-- Parameterized density matrix.
> **W:** `array`
-- Weight matrix.
> **eps:** `float`
-- Machine epsilon.
Returns
----------
**BCB:** `float`
-- The value of the minimum Bayesian cost.
"""
para_num = len(x)
if para_num == 1:
# single-parameter scenario
dim = len(rho[0])
p_num = len(x[0])
value = [p[i] * x[0][i]**2 for i in range(p_num)]
delta2_x = simps(value, x[0])
rho_avg = np.zeros((dim, dim), dtype=np.complex128)
rho_pri = np.zeros((dim, dim), dtype=np.complex128)
for di in range(dim):
for dj in range(dim):
rho_avg_arr = [p[m] * rho[m][di][dj] for m in range(p_num)]
rho_pri_arr = [
p[n] * x[0][n] * rho[n][di][dj] for n in range(p_num)
]
rho_avg[di][dj] = simps(rho_avg_arr, x[0])
rho_pri[di][dj] = simps(rho_pri_arr, x[0])
Lambda = Lambda_avg(rho_avg, [rho_pri], eps=eps)
minBC = delta2_x - np.real(
np.trace(np.dot(np.dot(rho_avg, Lambda[0]), Lambda[0])))
return minBC
else:
# multi-parameter scenario
p_shape = np.shape(p)
p_ext = extract_ele(p, para_num)
rho_ext = extract_ele(rho, para_num)
p_list, rho_list = [], []
for p_ele, rho_ele in zip(p_ext, rho_ext):
p_list.append(p_ele)
rho_list.append(rho_ele)
dim = len(rho_list[0])
p_num = len(p_list)
x_pro = product(*x)
x_list = []
for x_ele in x_pro:
x_list.append([x_ele[i] for i in range(para_num)])
if W == []:
W = np.identity(para_num)
value = [0.0 for i in range(p_num)]
for i in range(p_num):
x_tp = np.array(x_list[i])
xCx = np.dot(x_tp.reshape(1, -1), np.dot(W, x_tp.reshape(-1,
1)))[0][0]
value[i] = p_list[i] * xCx
delta2_x = np.array(value).reshape(p_shape)
for si in reversed(range(para_num)):
delta2_x = simps(delta2_x, x[si])
rho_avg = np.zeros((dim, dim), dtype=np.complex128)
rho_pri = [
np.zeros((dim, dim), dtype=np.complex128) for i in range(para_num)
]
for di in range(dim):
for dj in range(dim):
rho_avg_arr = [
p_list[m] * rho_list[m][di][dj] for m in range(p_num)
]
rho_avg_tp = np.array(rho_avg_arr).reshape(p_shape)
for si in reversed(range(para_num)):
rho_avg_tp = simps(rho_avg_tp, x[si])
rho_avg[di][dj] = rho_avg_tp
for para_i in range(para_num):
rho_pri_arr = [
p_list[n] * x_list[n][para_i] * rho_list[n][di][dj]
for n in range(p_num)
]
rho_pri_tp = np.array(rho_pri_arr).reshape(p_shape)
for si in reversed(range(para_num)):
rho_pri_tp = simps(rho_pri_tp, x[si])
rho_pri[para_i][di][dj] = rho_pri_tp
Lambda = Lambda_avg(rho_avg, rho_pri, eps=eps)
Mat = np.zeros((para_num, para_num), dtype=np.complex128)
for para_m in range(para_num):
for para_n in range(para_num):
Mat += W[para_m][para_n] * np.dot(Lambda[para_m],
Lambda[para_n])
minBC = delta2_x - np.real(np.trace(np.dot(rho_avg, Mat)))
return minBC
def BayesCost(x, p, xest, rho, M, W=[], eps=1e-8):
"""
Calculation of the average Bayesian cost with a quadratic cost function.
Parameters
----------
> **x:** `list`
-- The regimes of the parameters for the integral.
> **p:** `multidimensional array`
-- The prior distribution.
> **xest:** `list`
-- The estimators.
> **rho:** `multidimensional list`
-- Parameterized density matrix.
> **M:** `array`
-- A set of POVM.
> **W:** `array`
-- Weight matrix.
> **eps:** `float`
-- Machine epsilon.
Returns
----------
**The average Bayesian cost:** `float`
-- The average Bayesian cost.
"""
para_num = len(x)
if para_num == 1:
# single-parameter scenario
if M == []:
M = SIC(len(rho[0]))
else:
if type(M) != list:
raise TypeError("Please make sure M is a list!")
p_num = len(x[0])
value = [
p[i] * sum([
np.trace(np.dot(rho[i], M[mi])) * (x[0][i] - xest[mi][0])**2
for mi in range(len(M))
]) for i in range(p_num)
]
C = simps(value, x[0])
return np.real(C)
else:
# multi-parameter scenario
p_shape = np.shape(p)
p_ext = extract_ele(p, para_num)
rho_ext = extract_ele(rho, para_num)
p_list, rho_list = [], []
for p_ele, rho_ele in zip(p_ext, rho_ext):
p_list.append(p_ele)
rho_list.append(rho_ele)
x_pro = product(*x)
x_list = []
for x_ele in x_pro:
x_list.append([x_ele[i] for i in range(para_num)])
dim = len(rho_list[0])
p_num = len(p_list)
if W == []:
W = np.identity(para_num)
if M == []:
M = SIC(dim)
else:
if type(M) != list:
raise TypeError("Please make sure M is a list!")
value = [0.0 for i in range(p_num)]
for i in range(p_num):
x_tp = np.array(x_list[i])
xCx = 0.0
for mi in range(len(M)):
xCx += np.trace(np.dot(rho_list[i], M[mi])) * np.dot(
(x_tp - xest[mi]).reshape(1, -1),
np.dot(W, (x_tp - xest[mi]).reshape(-1, 1)))[0][0]
value[i] = p_list[i] * xCx
C = np.array(value).reshape(p_shape)
for si in reversed(range(para_num)):
C = simps(C, x[si])
return np.real(C)
def MLE(x, rho, y, M=[], savefile=False):
"""
Bayesian estimation. The estimated value of parameters obtained via the
maximum likelihood estimation (MLE).
Parameters
----------
> **x:** `list`
-- The regimes of the parameters for the integral.
> **rho:** `multidimensional list`
-- Parameterized density matrix.
> **y:** `array`
-- The experimental results obtained in practice.
> **M:** `list of matrices`
-- A set of positive operator-valued measure (POVM). The default measurement
is a set of rank-one symmetric informationally complete POVM (SIC-POVM).
> **savefile:** `bool`
-- Whether or not to save all the likelihood functions.
If set `True` then two files "Lout.npy" and "xout.npy" will be generated including
the likelihood functions and the estimated values in the iterations. If set
`False` the likelihood function in the final iteration and the estimated values
in all iterations will be saved in "Lout.npy" and "xout.npy".
Returns
----------
**Lout and xout:** `array and float`
-- The likelihood function and the estimated values in the final iteration.
**Note:**
SIC-POVM is calculated by the Weyl-Heisenberg covariant SIC-POVM fiducial state
which can be downloaded from [here](http://www.physics.umb.edu/Research/QBism/
solutions.html).
"""
para_num = len(x)
max_episode = len(y)
if para_num == 1:
#### single parameter scenario ####
if M == []:
M = SIC(len(rho[0]))
else:
if type(M) != list:
raise TypeError("Please make sure M is a list!")
if savefile == False:
x_out = []
L_out = np.ones(len(x[0]))
for mi in range(max_episode):
res_exp = int(y[mi])
for xi in range(len(x[0])):
p_tp = np.real(np.trace(np.dot(rho[xi], M[res_exp])))
L_out[xi] = L_out[xi] * p_tp
indx = np.where(L_out == max(L_out))[0][0]
x_out.append(x[0][indx])
np.save("Lout", L_out)
np.save("xout", x_out)
return L_out, x_out[-1]
else:
L_out, x_out = [], []
L_tp = np.ones(len(x[0]))
for mi in range(max_episode):
res_exp = int(y[mi])
for xi in range(len(x[0])):
p_tp = np.real(np.trace(np.dot(rho[xi], M[res_exp])))
L_tp[xi] = L_tp[xi] * p_tp
indx = np.where(L_tp == max(L_tp))[0][0]
L_out.append(L_tp)
x_out.append(x[0][indx])
np.save("Lout", L_out)
np.save("xout", x_out)
return L_tp, x_out[-1]
else:
#### multiparameter scenario ####
p_shape = []
for i in range(para_num):
p_shape.append(len(x[i]))
rho_ext = extract_ele(rho, para_num)
rho_list = []
for rho_ele in rho_ext:
rho_list.append(rho_ele)
dim = len(rho_list[0])
if M == []:
M = SIC(dim)
else:
if type(M) != list:
raise TypeError("Please make sure M is a list!")
if savefile == False:
x_out = []
L_list = np.ones(len(rho_list))
for mi in range(max_episode):
res_exp = int(y[mi])
for xi in range(len(rho_list)):
p_tp = np.real(np.trace(np.dot(rho_list[xi], M[res_exp])))
L_list[xi] = L_list[xi] * p_tp
L_out = L_list.reshape(p_shape)
indx = np.where(L_out == np.max(L_out))
x_out.append([x[i][indx[i][0]] for i in range(para_num)])
np.save("Lout", L_out)
np.save("xout", x_out)
return L_out, x_out[-1]
else:
L_out, x_out = [], []
L_list = np.ones(len(rho_list))
for mi in range(max_episode):
res_exp = int(y[mi])
for xi in range(len(rho_list)):
p_tp = np.real(np.trace(np.dot(rho_list[xi], M[res_exp])))
L_list[xi] = L_list[xi] * p_tp
L_tp = L_list.reshape(p_shape)
indx = np.where(L_tp == np.max(L_tp))
L_out.append(L_tp)
x_out.append([x[i][indx[i][0]] for i in range(para_num)])
np.save("Lout", L_out)
np.save("xout", x_out)
return L_tp, x_out[-1]