Entanglement measure¶
There are various entanglement measures wiki-link with different meanings:
Operational entanglement measures:
- Distillable entanglement
- Entanglement cost
Abstractly defined based on convex roof:
- Concurrence
- Entanglement of formation (EOF)
- Geometric measure of entanglement (GME)
Distance measure:
- Relative entropy of entanglement (REE)
These measures are zero for all SEP states, positive for ENT states, and non-decreasing under local operations and classical communication (LOCC) quantiki-link. In the following tutorial, we will focus on the bipartite case.
import numpy as np
from tqdm.notebook import tqdm
import matplotlib.pyplot as plt
try:
import numqi
except ImportError:
%pip install numqi
import numqi
Abstractly defined based on convex roof¶
Entanglement of formation¶
Given a density matrix $\rho$, its von Neumann entropy is defined as
$$ S\left(\rho\right)=-\mathrm{Tr}\left[\rho\ln\rho\right]. $$
For a bipartite system, the entanglement of formation (EOF) of a pure state $\left|\psi\right\rangle \in\mathcal{H}_{A}\otimes\mathcal{H}_{B}$ is defined as
$$ E_f\left(\left|\psi\right\rangle \right)=S\left(\rho_{A}\right)=S\left(\rho_{B}\right) $$
where $\rho_A,\rho_B$ is its reduced density matrix on subsystem $A$ and $B$ respectively
$$ \rho_{A}=\mathrm{Tr}_{B}\left[\left|\psi\right\rangle \left\langle \psi\right|\right],\rho_{B}=\mathrm{Tr}_{A}\left[\left|\psi\right\rangle \left\langle \psi\right|\right]. $$
For a bipartite density matrix $\rho$, its EOF is defined as
$$ E_f\left(\rho\right)=\min_{\left\{ \vec{p},\vec{\psi}\right\} }\sum_{i}p_{i}E_f\left(\left|\psi_{i}\right\rangle \right) $$
$$ s.t.\begin{cases} \rho=\sum_{i}p_{i}\left|\psi_{i}\right\rangle \left\langle \psi_{i}\right|\\ p_{i}>0\\ \langle\psi_{i}|\psi_{i}\rangle=1 \end{cases} $$
For Werner state, its EOF is given in paper doi-link. Below, we use the variational method to calculate the EOF of Werner state as proposed in paper doi-link
dim = 3
alpha_list = np.linspace(-1, 1, 150)
eof_analytical = numqi.state.get_Werner_eof(dim, alpha_list)
For variational method, we need to perform an optimization for each of target density matrix.
model = numqi.entangle.EntanglementFormationModel(dim, dim, num_term=2*dim*dim)
eof_variational = []
kwargs = dict(num_repeat=1, tol=1e-9, print_every_round=0)
for x in tqdm(alpha_list): #about 1 min
model.set_density_matrix(numqi.state.Werner(dim, x))
theta_optim = numqi.optimize.minimize(model, **kwargs)
eof_variational.append(theta_optim.fun)
eof_variational = np.array(eof_variational)
Above, the hyper-parameters, e.g. num_term=2*dim*dim and num_repeat=1, may not the best choice. You can try to change them to get a better result.
fig,ax = plt.subplots()
ax.plot(alpha_list, eof_analytical, 'x', label='analytical')
ax.plot(alpha_list, eof_variational, label='variational')
ax.legend()
ax.set_xlabel(r'$\alpha$')
ax.set_yscale('log')
ax.set_ylabel('EOF')
ax.set_title(f'Werner({dim})')
fig.tight_layout()
For $\alpha\in[-1,\frac{1}{3}]$, the true EOF (cross in the figure) is zero, so it's not shown in the yscale='log' figure. We can see that the variational results (solid line) is almost the same as the true EOF (cross) in the whole range$.
TASK: try Isotropic state which also has analytical EOF.
Concurrence¶
A closely related quantity is the concurrence, which is defined as
$$ C\left(\left|\psi\right\rangle \right)=\sqrt{2\left(1-\mathrm{Tr}\left[\rho_{A}^{2}\right]\right)} $$
for pure states and
$$ C\left(\rho\right)=\min_{\left\{ \vec{p},\vec{\psi}\right\} }\sum_{i}p_{i}C\left(\left|\psi_{i}\right\rangle \right) $$
$$ s.t.\begin{cases} \rho=\sum_{i}p_{i}\left|\psi_{i}\right\rangle \left\langle \psi_{i}\right|\\ p_{i}>0\\ \langle\psi_{i}|\psi_{i}\rangle=1 \end{cases} $$
for mixed states.
TODO: add torch.nn.Module for this task
Geometric measure of entanglement¶
For a given pure state $\vert \psi \rangle$, the geometric measure of entanglement (GME) is defined as the geometric distance between the given state and fully product states
$$ E_G(\vert \psi \rangle)=1-\max_{\vert \varphi_{\text{prod}}\rangle}\vert \langle \varphi_{\text{prod}} \vert \psi \rangle\vert^2. $$
With the convex roof construction, we can extend this measure to the mixed state case as follows,
$$ E_G\left(\rho\right)=\min_{\left\{ \vec{p},\vec{\psi}\right\} }\sum_{i}p_{i}E_G\left(\left|\psi_{i}\right\rangle \right) $$
$$ s.t.\begin{cases} \rho=\sum_{i}p_{i}\left|\psi_{i}\right\rangle \left\langle \psi_{i}\right|,\\ p_{i}>0,\\ \langle\psi_{i}|\psi_{i}\rangle=1. \end{cases} $$
If we want to quantify different levels of entanglement, we can generalize the geometric measure of entanglement to the geometric measure of $r$-bounded rank
$$ E_r(\vert \psi \rangle)=1-\max_{\vert\varphi\rangle \in \sigma_{r-1}}\vert \langle \varphi \vert \psi \rangle\vert^2. $$
Here, $\sigma_{r-1}$ is the states of rank at most $r-1$ (in the bipartite setting, it means Schmidt rank). Obviously, it can be reduced into the form of GME when $r=2$. And similarly, the definition can be also extended into mixed state case,
$$ E_r\left(\rho\right)=\min_{\left\{ \vec{p},\vec{\psi}\right\} }\sum_{i}p_{i}E_r\left(\left|\psi_{i}\right\rangle \right). $$
Connection between quantum state and subspace¶
For the entanglement measures defined by convex roof, i.e.,
$$ E\left(\rho\right)=\min_{\left\{ \vec{p},\vec{\psi}\right\} }\sum_{i}p_{i}E\left(\left|\psi_{i}\right\rangle \right). $$
There is a useful relation for all these entanglement measures
$$ E\left(\rho\right) \geq E(\text{supp}(\rho)),$$
where $\text{supp}(\rho)$ is the support space of the state $\rho$. The entanglement measure for a quantum subspace $\mathcal{S}$ is defined as follows,
$$ E(\mathcal{S})= \min_{\vert \psi \rangle \in \mathcal{S}} E(\vert \psi \rangle). $$
Distance measure¶
Relative entropy of entanglement¶
Relative entropy of entanglement (REE) is defined as the follows
$$ E_R\left( \rho \right) =\min_{\sigma \in \mathrm{SEP}} S\left( \rho ||\sigma \right) $$
where $S$ is the quantum relative entropy (von Neumann entropy) defined as
$$ S\left( \rho ||\sigma \right) =\mathrm{Tr}\left[ \rho \ln \rho -\rho \ln \sigma \right] $$
The relative entropy $S$ is jointly convex in its arguments, and non-negative. It is zero if and only if $\rho=\sigma$. In the following, we will discuss Werner state and Isotropic state which has analytical solution for REE.
Werner state:
Werner state is defined as wiki-link over $\mathcal{H}_d\otimes \mathcal{H}_d$
$$ \rho_{W,d}\left(\alpha\right)=\frac{1}{d^{2}-d\alpha}I-\frac{\alpha}{d^{2}-d\alpha}\sum_{ij}\left|ij\right\rangle \left\langle ji\right|,\quad\alpha\in\left[-1,1\right] $$
(You might see different definitions for Werner state, you can check doc-link for the conversion among different definitions)
Werner state is separable for $\alpha\in\left[-1,\frac{1}{d}\right]$, and entangled for $\alpha\in\left(\frac{1}{d},1\right]$. The REE of Werner state is given in paper "Entanglement measures under symmetry" doi-link.
$$ E_R\left(\rho_{W,d}(\alpha)\right)=\begin{cases} 0 & \alpha\in\left[-1,\frac{1}{d}\right]\\ S\left(\rho_{W,d}(\alpha)\lvert\rho_{W,d}\left(\frac{1}{d}\right)\right) & \alpha\in\left[\frac{1}{d},1\right] \end{cases} $$
Below, we can calculate the REE via various methods including the PPT criterion (lower bound), CHA criterion (upper bound) (doc-link), and compare with the analytical solution.
dim = 3
alpha_list = np.linspace(0, 1, 100, endpoint=False) # alpha=1 is unstable for matrix logarithm
dm_target_list = [numqi.state.Werner(dim, x) for x in alpha_list]
PPT criterion is to solve the following semidefinite program (SDP).
$$ \min_{\sigma}\;S\left(\rho_{W,d}\left(\alpha\right)\|\sigma\right) $$
$$ s.t.\;\begin{cases} \rho\succeq0\\ \mathrm{Tr}\left[\rho\right]=1\\ \rho^{\Gamma}\succeq0 \end{cases} $$
It can be approximately evaluated by semidefinite programming (SDP) according to the paper "Semidefinite approximations of the matrix logarithm" doi-link, arxiv-link, github/cvxquad.
ree_analytical = np.array([numqi.state.get_Werner_ree(dim, x) for x in alpha_list])
ree_ppt = numqi.entangle.get_ppt_ree(dm_target_list, dim, dim) #about 30 seconds
Convex hull approximation (CHA) is to solve the following non-convex optimization problem via gradient descent method. The key idea is to formulize the problem as a non-convex optimization problem, and use gradient descent method to find the optimal solution. The gradient of the objective function is solved by gradient back-propagation method from Pytorch framework. We will dive into more details in the following tutorial (doc-link).
model = numqi.entangle.AutodiffCHAREE((dim,dim), distance_kind='ree')
ree_cha = [] #about 11 seconds
for dm_target_i in tqdm(dm_target_list):
model.set_dm_target(dm_target_i)
tmp0 = numqi.optimize.minimize(model, theta0='uniform', tol=1e-12, num_repeat=1, print_every_round=0).fun
ree_cha.append(tmp0)
ree_cha = np.array(ree_cha)
fig, ax = plt.subplots()
ax.plot(alpha_list, ree_analytical, "x", label="analytical")
ax.plot(alpha_list, ree_ppt, label="PPT")
ax.plot(alpha_list, ree_cha, label="CHA")
ax.set_xlim(min(alpha_list), max(alpha_list))
ax.set_ylim(1e-13, 1)
ax.set_yscale('log')
ax.set_xlabel(r"$\beta$")
ax.set_ylabel("REE")
ax.set_title(f'Werner({dim})')
ax.legend()
fig.tight_layout()
Be careful, there is NOT a phase transition in the figure. the sharp shift is only due to the logarithmic scale used in the y-axis. We can see that the upper bound (CHA) and lower bound (PPT) are very close to the analytical solution.
Isotropic state:
Let's repeat the same procedure for Isotropic state. Isotropic state is defined as quantiki-link over $\mathcal{H}_d\otimes \mathcal{H}_d$
$$ \rho_{I,d}^{\prime}\left(\alpha\right)=\frac{1-\alpha}{d^{2}}I+\frac{\alpha}{d}\sum_{i,j}\left|ii\right\rangle \left\langle jj\right|,\quad\alpha\in\left[-\frac{1}{d^{2}-1},1\right] $$
(You might see different definitions for Werner state, you can check doc-link for the conversion among different definitions)
Isotropic state is separable for $\alpha\in\left[-\frac{1}{d^2-1},\frac{1}{d+1}\right]$, and entangled for $\alpha\in\left(\frac{1}{d+1},1\right]$. The REE of Werner state is given in paper "Bound on distillable entanglement" doi-link.
$$ E_R\left(\rho_{I,d}(\alpha)\right)=\begin{cases} 0 & \alpha\in\left[-\frac{1}{d^{2}-1},\frac{1}{d+1}\right]\\ S\left(\rho_{I,d}(\alpha)\lvert\rho_{I,d}\left(\frac{1}{d+1}\right)\right) & \alpha\in\left[\frac{1}{d+1},1\right] \end{cases} $$
dim = 3
alpha_list = np.linspace(0, 1, 100, endpoint=False) # alpha=1 is unstable for matrix logarithm
dm_target_list = [numqi.state.Isotropic(dim, x) for x in alpha_list]
ree_analytical = np.array([numqi.state.get_Isotropic_ree(dim, x) for x in alpha_list])
ree_ppt = numqi.entangle.get_ppt_ree(dm_target_list, dim, dim) #about 30 seconds
model = numqi.entangle.AutodiffCHAREE((dim, dim), distance_kind='ree')
ree_cha = [] #about 11 seconds
for dm_target_i in tqdm(dm_target_list):
model.set_dm_target(dm_target_i)
tmp0 = numqi.optimize.minimize(model, theta0='uniform', tol=1e-12, num_repeat=1, print_every_round=0).fun
ree_cha.append(tmp0)
ree_cha = np.array(ree_cha)
fig, ax = plt.subplots()
ax.plot(alpha_list, ree_analytical, "x", label="analytical")
ax.plot(alpha_list, ree_ppt, label="PPT")
ax.plot(alpha_list, ree_cha, label="CHA")
ax.set_xlim(min(alpha_list), max(alpha_list))
ax.set_ylim(1e-13, 1)
ax.set_yscale('log')
ax.set_xlabel(r"$\beta$")
ax.set_ylabel("REE")
ax.legend()
ax.set_title(f"Isotropic({dim})")
fig.tight_layout()
