Linear entropy of entanglement¶

In [1]:

import numpy as np
from tqdm.notebook import tqdm
import matplotlib.pyplot as plt

try:
    import numqi
except ImportError:
    %pip install numqi
    import numqi

import numpy as np from tqdm.notebook import tqdm import matplotlib.pyplot as plt try: import numqi except ImportError: %pip install numqi import numqi

For pure state $|\psi\rangle$

$$ S_{l}(\rho)=1-\mathrm{Tr}[\rho^{2}] $$

$$ E_{l}\left(|\psi\rangle\right)=S_{l}\left(\mathrm{Tr}_{B}\left[|\psi\rangle\langle\psi|\right]\right) $$

For mixed state $\rho$ (convex roof extension)

$$ \begin{align*} E_{l}(\rho)&=\min_{\left\{ p_{i},|\psi_{i}\rangle\right\} }\sum_{i}p_{i}S_{l}\left(|\psi_{i}\rangle\right)\\ &=1-\max_{\left\{ |\tilde{\psi}_{i}\rangle\right\} }\sum_{i}\frac{\mathrm{Tr}\left[\tilde{\psi}_{i}\tilde{\psi}_{i}^{\dagger}\tilde{\psi}_{i}\tilde{\psi}_{i}^{\dagger}\right]}{\mathrm{Tr}\left[\tilde{\psi}_{i}\tilde{\psi}_{i}^{\dagger}\right]} \end{align*} $$

Below, LEE is calculated for some two-qubit states via semi-definite programming (SDP) method doi-link.

In [2]:

rhoA = numqi.random.rand_density_matrix(2)
rhoB = numqi.random.rand_density_matrix(2)
rho = np.kron(rhoA, rhoB)
print('LEE for a random product state:', numqi.entangle.get_linear_entropy_entanglement_ppt(rho, (2,2)))

rho = numqi.random.rand_separable_dm(2, 2)
print('LEE for a random separable state:', numqi.entangle.get_linear_entropy_entanglement_ppt(rho, (2,2)))

rho = numqi.random.rand_density_matrix(4)
print('LEE for a random density matrix:', numqi.entangle.get_linear_entropy_entanglement_ppt(rho, (2,2)))

psi = numqi.state.maximally_entangled_state(2)
rho = psi.reshape(-1,1) * psi.conj()
print('LEE for a maximally entangled state:', numqi.entangle.get_linear_entropy_entanglement_ppt(rho, (2,2)))

rhoA = numqi.random.rand_density_matrix(2) rhoB = numqi.random.rand_density_matrix(2) rho = np.kron(rhoA, rhoB) print('LEE for a random product state:', numqi.entangle.get_linear_entropy_entanglement_ppt(rho, (2,2))) rho = numqi.random.rand_separable_dm(2, 2) print('LEE for a random separable state:', numqi.entangle.get_linear_entropy_entanglement_ppt(rho, (2,2))) rho = numqi.random.rand_density_matrix(4) print('LEE for a random density matrix:', numqi.entangle.get_linear_entropy_entanglement_ppt(rho, (2,2))) psi = numqi.state.maximally_entangled_state(2) rho = psi.reshape(-1,1) * psi.conj() print('LEE for a maximally entangled state:', numqi.entangle.get_linear_entropy_entanglement_ppt(rho, (2,2)))

LEE for a random product state: 0
LEE for a random separable state: 0

LEE for a random density matrix: 0.0013973684432351252
LEE for a maximally entangled state: 0.4999986936612686

Werner states¶

Below, gradient-based optimization is used to find the LEE of Werner states.

In [3]:

alpha_list = np.linspace(0, 1, 100)
dim = 3

model = numqi.entangle.DensityMatrixLinearEntropyModel([dim,dim], num_ensemble=27, kind='convex')
ret0 = []
for alpha_i in tqdm(alpha_list):
    model.set_density_matrix(numqi.state.Werner(dim, alpha=alpha_i))
    ret0.append(numqi.optimize.minimize(model, num_repeat=3, tol=1e-10, print_every_round=0).fun)
ret0 = np.array(ret0)

model = numqi.entangle.DensityMatrixLinearEntropyModel([dim,dim], num_ensemble=27, kind='concave')
ret1 = []
for alpha_i in tqdm(alpha_list):
    model.set_density_matrix(numqi.state.Werner(dim, alpha=alpha_i))
    ret1.append(-numqi.optimize.minimize(model, num_repeat=3, tol=1e-10, print_every_round=0).fun)
ret1 = np.array(ret1)

alpha_list = np.linspace(0, 1, 100) dim = 3 model = numqi.entangle.DensityMatrixLinearEntropyModel([dim,dim], num_ensemble=27, kind='convex') ret0 = [] for alpha_i in tqdm(alpha_list): model.set_density_matrix(numqi.state.Werner(dim, alpha=alpha_i)) ret0.append(numqi.optimize.minimize(model, num_repeat=3, tol=1e-10, print_every_round=0).fun) ret0 = np.array(ret0) model = numqi.entangle.DensityMatrixLinearEntropyModel([dim,dim], num_ensemble=27, kind='concave') ret1 = [] for alpha_i in tqdm(alpha_list): model.set_density_matrix(numqi.state.Werner(dim, alpha=alpha_i)) ret1.append(-numqi.optimize.minimize(model, num_repeat=3, tol=1e-10, print_every_round=0).fun) ret1 = np.array(ret1)

In [4]:

fig,ax = plt.subplots()
ax.axvline(1/dim, color='r')
ax.plot(alpha_list, ret0, label='convex (LEE)')
ax.plot(alpha_list, ret1, label='concave')
ax.legend()
# ax.set_yscale('log')
ax.set_xlabel('alpha')
ax.set_ylabel('linear entropy')
ax.set_title(f'Werner({dim})')
fig.tight_layout()

fig,ax = plt.subplots() ax.axvline(1/dim, color='r') ax.plot(alpha_list, ret0, label='convex (LEE)') ax.plot(alpha_list, ret1, label='concave') ax.legend() # ax.set_yscale('log') ax.set_xlabel('alpha') ax.set_ylabel('linear entropy') ax.set_title(f'Werner({dim})') fig.tight_layout()

Horodecki states¶

In [5]:

rho = numqi.state.get_bes3x3_Horodecki1997(0.23)
plist = np.linspace(0.92, 1, 30)
plist_ppt = plist[::3] #to save time

# about 3 minutes
tmp0 = np.stack([numqi.utils.hf_interpolate_dm(rho,alpha=p) for p in plist_ppt])
ret_ppt = numqi.entangle.get_linear_entropy_entanglement_ppt(tmp0, (3,3), use_tqdm=True)

# about 1 minute
ret_gd = []
model = numqi.entangle.DensityMatrixLinearEntropyModel([3,3], num_ensemble=27, kind='convex')
for p in tqdm(plist):
    model.set_density_matrix(numqi.utils.hf_interpolate_dm(rho, alpha=p))
    ret_gd.append(numqi.optimize.minimize(model, num_repeat=3, tol=1e-10, print_every_round=0).fun)
ret_gd = np.array(ret_gd)

rho = numqi.state.get_bes3x3_Horodecki1997(0.23) plist = np.linspace(0.92, 1, 30) plist_ppt = plist[::3] #to save time # about 3 minutes tmp0 = np.stack([numqi.utils.hf_interpolate_dm(rho,alpha=p) for p in plist_ppt]) ret_ppt = numqi.entangle.get_linear_entropy_entanglement_ppt(tmp0, (3,3), use_tqdm=True) # about 1 minute ret_gd = [] model = numqi.entangle.DensityMatrixLinearEntropyModel([3,3], num_ensemble=27, kind='convex') for p in tqdm(plist): model.set_density_matrix(numqi.utils.hf_interpolate_dm(rho, alpha=p)) ret_gd.append(numqi.optimize.minimize(model, num_repeat=3, tol=1e-10, print_every_round=0).fun) ret_gd = np.array(ret_gd)

In [6]:

fig,ax = plt.subplots()
ax.plot(plist_ppt, ret_ppt, 'x', label='PPT')
ax.plot(plist, ret_gd, label='gradient descent')
ax.legend()
ax.set_xlabel('p')
ax.set_ylabel('linear entropy')
ax.set_yscale('log')
ax.set_title('Horodecki1997-2qutrit(0.23)')
fig.tight_layout()

fig,ax = plt.subplots() ax.plot(plist_ppt, ret_ppt, 'x', label='PPT') ax.plot(plist, ret_gd, label='gradient descent') ax.legend() ax.set_xlabel('p') ax.set_ylabel('linear entropy') ax.set_yscale('log') ax.set_title('Horodecki1997-2qutrit(0.23)') fig.tight_layout()