Boundary of detection method¶

We are showing various Joint Algebraic Numerical Range (JANR) related to different boundaries obtained by different methods (PPT, CHA, and PureB).

In [1]:

import numpy as np
import matplotlib.pyplot as plt

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

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

Fixed operators¶

Here, we choose two different operators $((\sigma_z+\sigma_0)\sigma_z)/(2\sqrt{2})$ and $(\sigma_y\sigma_x-\sigma_x\sigma_y)/2$.

In [2]:

# about 1 minute
sx = np.array([[0,1],[1,0]])
sy = np.array([[0,-1j],[1j,0]])
sz = np.array([[1,0],[0,-1]])
op0 = np.kron(np.array([[1/np.sqrt(2),0],[0,0]]), sz)
op1 = (np.kron(sy, sx) - np.kron(sx, sy)) / 2
dimA = 2
dimB = 2
num_theta = 100
kext_list = [8,16,32,64,128]

theta_list = np.linspace(0, 2*np.pi, num_theta)
direction = np.stack([np.cos(theta_list), np.sin(theta_list)], axis=1)
beta_ppt = numqi.entangle.get_ppt_numerical_range([op0,op1], direction, (dimA,dimB))
op_nr_ppt = beta_ppt.reshape(-1,1)*direction
model_cha = numqi.entangle.AutodiffCHAREE((dimA, dimB))
ret_cha = model_cha.get_numerical_range(op0, op1, num_theta=num_theta, num_repeat=3)
ret_pureb = []
for kext_i in kext_list:
    model = numqi.entangle.PureBosonicExt(2, 2, kext=kext_i)
    ret_pureb.append(model.get_numerical_range(op0, op1, num_theta=num_theta, num_repeat=3))
ret_pureb = np.stack(ret_pureb)

fig,ax = plt.subplots()
ax.plot(op_nr_ppt[:,0], op_nr_ppt[:,1], 'x', label=f'PPT')
ax.plot(ret_cha[:,0], ret_cha[:,1], linewidth=0.5, label=f'CHA')
for ind0 in range(len(kext_list)):
    ax.plot(ret_pureb[ind0,:,0], ret_pureb[ind0,:,1], linewidth=0.5, label=f'PureB({kext_list[ind0]})')
ax.legend()
ax.set_xlabel('$O_1$')
ax.set_ylabel('$O_2$')
ax.set_xlabel(r'$((\sigma_z+\sigma_0)\sigma_z)/(2\sqrt{2})$')
ax.set_ylabel(r'$(\sigma_y\sigma_x-\sigma_x\sigma_y)/2$')
ax.set_title(f'numerical range {dimA}x{dimB}')
fig.tight_layout()

# about 1 minute sx = np.array([[0,1],[1,0]]) sy = np.array([[0,-1j],[1j,0]]) sz = np.array([[1,0],[0,-1]]) op0 = np.kron(np.array([[1/np.sqrt(2),0],[0,0]]), sz) op1 = (np.kron(sy, sx) - np.kron(sx, sy)) / 2 dimA = 2 dimB = 2 num_theta = 100 kext_list = [8,16,32,64,128] theta_list = np.linspace(0, 2*np.pi, num_theta) direction = np.stack([np.cos(theta_list), np.sin(theta_list)], axis=1) beta_ppt = numqi.entangle.get_ppt_numerical_range([op0,op1], direction, (dimA,dimB)) op_nr_ppt = beta_ppt.reshape(-1,1)*direction model_cha = numqi.entangle.AutodiffCHAREE((dimA, dimB)) ret_cha = model_cha.get_numerical_range(op0, op1, num_theta=num_theta, num_repeat=3) ret_pureb = [] for kext_i in kext_list: model = numqi.entangle.PureBosonicExt(2, 2, kext=kext_i) ret_pureb.append(model.get_numerical_range(op0, op1, num_theta=num_theta, num_repeat=3)) ret_pureb = np.stack(ret_pureb) fig,ax = plt.subplots() ax.plot(op_nr_ppt[:,0], op_nr_ppt[:,1], 'x', label=f'PPT') ax.plot(ret_cha[:,0], ret_cha[:,1], linewidth=0.5, label=f'CHA') for ind0 in range(len(kext_list)): ax.plot(ret_pureb[ind0,:,0], ret_pureb[ind0,:,1], linewidth=0.5, label=f'PureB({kext_list[ind0]})') ax.legend() ax.set_xlabel('$O_1$') ax.set_ylabel('$O_2$') ax.set_xlabel(r'$((\sigma_z+\sigma_0)\sigma_z)/(2\sqrt{2})$') ax.set_ylabel(r'$(\sigma_y\sigma_x-\sigma_x\sigma_y)/2$') ax.set_title(f'numerical range {dimA}x{dimB}') fig.tight_layout()

Random operators¶

In [3]:

# about 1 minute
hf_trace0 = lambda x: x - np.eye(x.shape[0])*np.trace(x)/x.shape[0]
dimA = 3
dimB = 3
kext = 32
op0 = hf_trace0(numqi.random.rand_hermitian_matrix(dimA*dimB))
op1 = hf_trace0(numqi.random.rand_hermitian_matrix(dimA*dimB))
num_theta = 100

theta_list = np.linspace(0, 2*np.pi, num_theta)
direction = np.stack([np.cos(theta_list), np.sin(theta_list)], axis=1)
beta_ppt = numqi.entangle.get_ppt_numerical_range([op0, op1], direction, (dimA,dimB))
op_nr_ppt = beta_ppt.reshape(-1,1)*direction
model_cha = numqi.entangle.AutodiffCHAREE((dimA, dimB))
ret_cha = model_cha.get_numerical_range(op0, op1, num_theta=num_theta, num_repeat=3)
model_pureb = numqi.entangle.PureBosonicExt(dimA, dimB, kext=kext)
ret_pureb = model_pureb.get_numerical_range(op0, op1, num_theta=num_theta, num_repeat=3)

fig,ax = plt.subplots()
ax.plot(op_nr_ppt[:,0], op_nr_ppt[:,1], 'x', label=f'PPT')
ax.plot(ret_cha[:,0], ret_cha[:,1], linewidth=0.5, label=f'CHA')
ax.plot(ret_pureb[:,0], ret_pureb[:,1], linewidth=0.5, label=f'PureB({kext})')
ax.legend()
ax.grid()
ax.set_xlabel('random $O_1$')
ax.set_ylabel('random $O_2$')
ax.set_title(f'numerical range {dimA}x{dimB}')
fig.tight_layout()

# about 1 minute hf_trace0 = lambda x: x - np.eye(x.shape[0])*np.trace(x)/x.shape[0] dimA = 3 dimB = 3 kext = 32 op0 = hf_trace0(numqi.random.rand_hermitian_matrix(dimA*dimB)) op1 = hf_trace0(numqi.random.rand_hermitian_matrix(dimA*dimB)) num_theta = 100 theta_list = np.linspace(0, 2*np.pi, num_theta) direction = np.stack([np.cos(theta_list), np.sin(theta_list)], axis=1) beta_ppt = numqi.entangle.get_ppt_numerical_range([op0, op1], direction, (dimA,dimB)) op_nr_ppt = beta_ppt.reshape(-1,1)*direction model_cha = numqi.entangle.AutodiffCHAREE((dimA, dimB)) ret_cha = model_cha.get_numerical_range(op0, op1, num_theta=num_theta, num_repeat=3) model_pureb = numqi.entangle.PureBosonicExt(dimA, dimB, kext=kext) ret_pureb = model_pureb.get_numerical_range(op0, op1, num_theta=num_theta, num_repeat=3) fig,ax = plt.subplots() ax.plot(op_nr_ppt[:,0], op_nr_ppt[:,1], 'x', label=f'PPT') ax.plot(ret_cha[:,0], ret_cha[:,1], linewidth=0.5, label=f'CHA') ax.plot(ret_pureb[:,0], ret_pureb[:,1], linewidth=0.5, label=f'PureB({kext})') ax.legend() ax.grid() ax.set_xlabel('random $O_1$') ax.set_ylabel('random $O_2$') ax.set_title(f'numerical range {dimA}x{dimB}') fig.tight_layout()