# -*- coding: utf-8 -*-

"""

Created on Fri Jul 20 11:33:37 2018

File to construct displaced T-states (t_1=t_2, t_3, a_z all nonzero) with LHS models, considering deterministic maps on Alice's side.

This script performs the optimization over u, v in Eq. (24), which are used to create Fig. 3(a).

Optimization results coincide with Eq. (25) of the text.

@author: Trav Baker, travis.baker@griffithuni.edu.au

"""

import numpy as np

import matplotlib.pyplot as plt

from scipy.integrate import dblquad

from scipy.optimize import brentq

from scipy.optimize import minimize_scalar

import time

import qutip as qt

# First, pick the size of Alice's Bloch vector.

a_z = 0.1 #=sin(u)*sin(v) in terms of map parameters.

def integrand(phi,z,u,theta):

z**2*(np.cos(theta)/(np.cos(u)*np.cos(v)))**2)

def objective(u,theta): #v is defined by x (i.e. u) and a_z.

I,err=dblquad(integrand,-1,1,lambda phi: 0,lambda phi:2*np.pi, args=(u,theta))

r = 2*np.pi/I

return -r

def bowles_objective(z,theta,a_z,tau_3,sign=-1):

return sign*((a_z*z)**2 + 2*np.abs(tau_3)*np.sqrt(np.tan(theta)**2*(1-z**2) + z**2))

def bowles_t3_root(tau_3,theta,a_z):

res = minimize_scalar(bowles_objective, method='Bounded', bounds=(-1,1), args=(theta,a_z,tau_3))

return -res.fun - 1

#Solve Eq. (7):

bow_1=[]

bow_3=[]

start=time.time()

nbow=100 #resolution of angular points for Eq. (7)

for theta in np.linspace(0,np.pi/2,nbow)[1:-1]:

bow_res,out = brentq(bowles_t3_root,0,1,args=(theta,a_z),xtol=1e-12, maxiter=100, full_output=True)

if out.converged == True:

bow_3.append(bow_res)

bow_1.append(bow_res*np.tan(theta))

#Compute Eq. (22)

ntheta = 100 #number of points to construct curve

theta_vec = np.linspace(0,np.pi/2,ntheta)

tau_1=[]

tau_3=[]

tau_1.append(0)

tau_3.append(1-a_z)

u_opt=[]

for theta in theta_vec[1:-1]:

res = minimize_scalar(objective, method='Bounded', bounds=(np.arcsin(a_z),np.pi/2), args=(theta))

I, err = dblquad(integrand,-1,1,lambda phi: 0,lambda phi:2*np.pi, args=(res.x,theta))

u_opt.append(res.x)

tau_3.append(np.cos(theta)*2*np.pi/I)

tau_1.append(np.sin(theta)*2*np.pi/I)

print('Finished angle %s of %s' %(np.where(theta_vec == theta)[0][0],ntheta-2))

#Quantum state boundary

def least_eig(r,theta,a_z):

return min(eigs)

npoints=200 #resolution.

qangle=np.linspace(0,np.pi/2,npoints) #angle from -y axis

x_border=[]

y_border=[]

for i in qangle:

root=brentq(least_eig,0,2,args=(i,a_z))

x_border.append(-root*np.sin(i)/np.sqrt(2))

y_border.append(-root*np.cos(i))

print(r'%.2f%% of Q-boundary points' %(2*i/np.pi*npoints))

#Entangled state boundary

def least_ent_eig(r,theta,a_z):

return min(eigs)

npoints=100

qangle=np.linspace(0,np.pi/2,npoints) #angle from -y axis

xent_border=[]

yent_border=[]

for i in qangle:

root=brentq(least_ent_eig,0,2,args=(i,a_z))

xent_border.append(-root*np.sin(i)/np.sqrt(2))

yent_border.append(-root*np.cos(i))

print(r'%.2f%% of Entangled-boundary points' %(2*i/np.pi*npoints))

end=time.time()

minutes=(end-start)/60

print('%f minutes elapsed' %minutes)

bow_1[-1]=bow_1[-2]

bow_3[-1]=0

#save data to a file in the present

np.savez('a_z%s_2DT_%sintegrals_data.npz' %(a_z,ntheta), tau_1, tau_3, bow_1, bow_3, x_border, y_border, xent_border, yent_border)

data = np.load('a_z%s_2DT_%sintegrals_data.npz' %(a_z,ntheta))

tau_1, tau_3, bow_1, bow_3, x_border, y_border, xent_border, yent_border = data['arr_0'], data['arr_1'], data['arr_2'], data['arr_3'], data['arr_4'], data['arr_5'], data['arr_6'], data['arr_7']

#Analytic curve from Eq. (25), based on the ansatz u=v:

alphas=np.linspace(0,np.pi/2,1000)

def radius(alpha,a):

return np.sqrt(2*(1-a))/np.sin(alpha)*(np.sqrt(1+c) + (2*np.sqrt(c))**(-1)*np.log((np.sqrt(1+c)+np.sqrt(c))/(np.sqrt(1+c)-np.sqrt(c))))**(-1)

t1prime=[]

t3prime=[]

for alpha in alphas[1:-129]:

t1prime.append(radius(alpha,a_z)*np.sin(alpha)/np.sqrt(2))

t3prime.append(radius(alpha,a_z)*np.cos(alpha))

# Plot all data

fig, ax = plt.subplots(1,1,figsize=(5,4))

plt.rcParams['font.size'] = 12

ax.plot(-np.array(tau_1)[1:164],-np.array(tau_3)[1:164],'.',label=r'Optimized Data')

ax.set_xlim([-1,0])

ax.set_ylim([-1,0])

ax.set_xlabel(r'$t_1^\prime = t_2^\prime$', fontsize=14)

ax.set_ylabel(r'$t_3^\prime$', fontsize=14)

ax.plot(-np.array(t1prime),-np.array(t3prime),'-', label=r'${\rm Eq. (25)}$')

ax.plot(-np.array(bow_1)[:992],-np.array(bow_3)[:992],'-', label=r'${\rm Eq. (7)}$')

# Borders of quantum states and entanglement

ax.plot(x_border, y_border,'k-')

ax.fill(np.append(x_border,[-1,-1,0]), np.append(y_border,[0,-1,-1]), 'k', alpha=1)

ax.fill(np.append(xent_border[:-1],[-1,-1,0]), np.append(yent_border[:-1],[0,-1,-1]), 'k', alpha=0.1)

ax.legend(loc=2,framealpha=1)

ax.set_xticklabels([r'$%.1f$'%i for i in ([i for i in np.linspace(-1,0,6)])])

ax.set_yticklabels([r'$%.1f$'%i for i in ([i for i in np.linspace(-1,0,6)])])

ax.text(-1,0.02,'$(a)$', fontsize=14)

plt.subplots_adjust(top=0.88,

bottom=0.125,

left=0.145,

right=0.9,

hspace=0.2,

wspace=0.2)