QPIP

Instruments and methods to solve the quantum photocounting inverse problem QPIP module consists of some parts:

• epscon: Pyomo model and epsilon-constrained algorithm to recover photon-number statistics from photocounting statistics in the presence of noise The approach based on minimization of differential entropy of photocounting statistics and its estimation and maximization of photon-number statistics estimation entropy qpip.epscon can be used to recover any photon-number statistics with any quantum efficiency of the detector. One can use both binomial and subbinomial models of photodetection. See also the related material for additional details.
• stat: Some examples of photon-number and photocounting statistics corresponding to quantum light fields
• denoise: Pyomo model and epsilon-constrained algorithm to recover noise statistics of laser source or signal statistics against the background of coherent noise (for example in collinear SPDC analyzing). Here we use a method to determine photon-number (or photocounting) statistics of excess noise in laser radiation from measured photocounting statistics. The method based on the multi-objective optimization approach is applied to blind deconvolution problem to determine excess noise distribution from the convolution of this one and Poissonian photon-number distribution of laser radiation. See the related material for additional details.

How to use qpip.epscon?

Import necessary modules:

import numpy as np
from fpdet import normalize, P2Q, fidelity

from qpip.stat import ppoisson
from qpip.epscon import InvPBaseModel, invpopt

Make photocounting statistics of the sample of finite size in presence of the noise

N = 25 # length of the photon-number statistics
M = 25 # length of the photocounting statistics
N0 = int(1E6) # number of photocounting events
qe = 0.3 # quantum efficiency
ERR = 1e-2 # relative error for photocounting statistics

P = ppoisson(7, N) # photon-number distribution
Q = P2Q(P, qe, M) # photocounting distribution
QND = np.random.choice(range(M), size=N0, p=Q.astype(float)) # random photocounting events
QN = np.histogram(QND, bins=range(M + 1), density=True)[0] 
QN = np.abs(QN*(1 + np.random.uniform(-ERR, ERR, size=len(QN))))
QN = normalize(QN) # photocounting statistics for the sample of N0 size

Find optimal estimation

invpmodel = InvPBaseModel(QN, qe, N) # make optimization model
res = invpopt(invpmodel, eps_tol=1e-5) # optimize it!
print(res) # print result (OptimizeResult)
print(fidelity(res.x, P))

How to use qpip.denoise?

Import necessary modules:

import numpy as np
from fpdet import normalize, P2Q, fidelity, p_convolve

from qpip.stat import ppoisson, pthermal
from qpip.denoise import DenoisePBaseModel, denoiseopt

Make noised statistics of the sample of finite size in presence of the noise

N = 25 # length of the laser distribution
M = 10 # length of the noise distribution
N0 = int(1E6) # number of events
ERR = 1e-2 # relative error for photocounting statistics

pnoise = pthermal(1, M)
P = normalize(p_convolve(ppoisson(6, N), pnoise)) # noised distribution
PND = np.random.choice(range(N), size=N0, p=P.astype(float)) # random events
PN = np.histogram(PND, bins=range(N + 1), density=True)[0] 
PN = np.abs(PN*(1 + np.random.uniform(-ERR, ERR, size=N)))
PN = normalize(PN) # statistics for the sample of N0 size

Find optimal estimation

dnpmodel = DenoisePBaseModel(PN, M) # make optimization model
res = denoiseopt(dnpmodel, g2_lbound=2) # optimize it!
print(res) # print result (OptimizeResult)
print(fidelity(res.x, pnoise))

Requirements

See requirements.txt