Geometric Measure of Coherence¶

In [1]:

import numpy as np
from tqdm.auto import tqdm
import matplotlib.pyplot as plt
import torch

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

np_rng = np.random.default_rng()

import numpy as np from tqdm.auto import tqdm import matplotlib.pyplot as plt import torch try: import numqi except ImportError: %pip install numqi import numqi np_rng = np.random.default_rng()

Pure state¶

$$ C_{g}\left(|\psi\rangle\right)=1-\max_{i}\left|\langle i|\psi\rangle\right|^{2} $$

In [2]:

psi = np.array([1,0,0,0], dtype=np.float64)
print('computational basis: ', numqi.coherence.get_geometric_coherence_pure(psi))
# TODO for some random pure state

psi = numqi.random.rand_haar_state(4)
print('random pure state: ', numqi.coherence.get_geometric_coherence_pure(psi))

psi = np.array([1,1,1,1], dtype=np.float64)/2
print('maximally coherent state: ', numqi.coherence.get_geometric_coherence_pure(psi))

psi = np.array([1,0,0,0], dtype=np.float64) print('computational basis: ', numqi.coherence.get_geometric_coherence_pure(psi)) # TODO for some random pure state psi = numqi.random.rand_haar_state(4) print('random pure state: ', numqi.coherence.get_geometric_coherence_pure(psi)) psi = np.array([1,1,1,1], dtype=np.float64)/2 print('maximally coherent state: ', numqi.coherence.get_geometric_coherence_pure(psi))

computational basis:  0
random pure state:  0.6738805592986645
maximally coherent state:  0.75

Mixed state¶

$$ \begin{align*} C_{g}(\rho)&=\min_{\left\{ p_{\alpha},|\psi_{\alpha}\rangle\right\} }\sum_{\alpha}p_{\alpha}C_{g}\left(|\psi_{\alpha}\rangle\right)\\ &=1-\max_{\left\{ \tilde{\rho}_{\alpha}\right\} }\sum_{\alpha}\max_{i}\tilde{\rho}_{\alpha,ii} \end{align*} $$

$$ \max_{i}\tilde{\rho}_{\alpha,ii}=\lim_{T\to0^{+}} T \log\left(\sum_{i}\mathrm{exp}\left(\tilde{\rho}_{\alpha,ii}/T\right)\right) $$

log-sum-exp is the "real" SoftMax wiki-link. The naive "$\max_i$" is hard to converge (easily trapped in local minimum), while log-sum-exp works much better.

In [3]:

dim = 32 #when dim=64, more num_repeat is needed
psi = numqi.state.maximally_coherent_state(dim)
dm_target = psi.reshape(-1,1) * psi.conj()

alpha_list = np.linspace(0, 1, 50)

model = numqi.coherence.GeometricCoherenceModel(dim, num_term=4*dim, temperature=0.3)
kwargs = dict(theta0='uniform', num_repeat=3, tol=1e-10, print_every_round=0)
gc_list = []
for alpha_i in tqdm(alpha_list):
    model.set_density_matrix(numqi.utils.hf_interpolate_dm(dm_target, alpha=alpha_i))
    theta_optim = numqi.optimize.minimize(model, **kwargs).fun
    with torch.no_grad():
        gc_list.append(model(use_temperature=False).item())
gc_list = np.array(gc_list)

gc_analytical = numqi.state.get_maximally_coherent_state_gmc(dim, alpha=alpha_list)

fig,ax = plt.subplots()
ax.plot(alpha_list, gc_list, label='manifold-opt')
ax.plot(alpha_list, gc_analytical, 'x', label='analytical')
ax.legend()
ax.set_xlabel('alpha')
ax.set_ylabel('geometric coherence')
ax.set_title(f'dim={dim}')
fig.tight_layout()

dim = 32 #when dim=64, more num_repeat is needed psi = numqi.state.maximally_coherent_state(dim) dm_target = psi.reshape(-1,1) * psi.conj() alpha_list = np.linspace(0, 1, 50) model = numqi.coherence.GeometricCoherenceModel(dim, num_term=4*dim, temperature=0.3) kwargs = dict(theta0='uniform', num_repeat=3, tol=1e-10, print_every_round=0) gc_list = [] for alpha_i in tqdm(alpha_list): model.set_density_matrix(numqi.utils.hf_interpolate_dm(dm_target, alpha=alpha_i)) theta_optim = numqi.optimize.minimize(model, **kwargs).fun with torch.no_grad(): gc_list.append(model(use_temperature=False).item()) gc_list = np.array(gc_list) gc_analytical = numqi.state.get_maximally_coherent_state_gmc(dim, alpha=alpha_list) fig,ax = plt.subplots() ax.plot(alpha_list, gc_list, label='manifold-opt') ax.plot(alpha_list, gc_analytical, 'x', label='analytical') ax.legend() ax.set_xlabel('alpha') ax.set_ylabel('geometric coherence') ax.set_title(f'dim={dim}') fig.tight_layout()