Set $$V^{(t)}_i  =
    \begin{cases}
    \tilde{V}_i    &\text{with probability }   \min \{ 1,A(V^{(t-1)}, \tilde{V}) \} ,
    \\
     V^{(t-1)}_i        & \text{otherwise}  ,
    \end{cases}$$ where $A(V^{(t-1)}, \tilde{V})$ is the acceptance ratio given below.

Let us now give precisely $h$, $R$ and $A$. We define $ h(\cdot|Y^{(t-1)}_i)$ as the probability distribution of $U = Y^{(t-1)}_i \exp(y)$ where $y \sim \mathcal{U}(-0.5,0.5) $. Following [@robert2013monte] the acceptance ratios are then given by: $$\begin{aligned}
\log(R( \tilde{Y}, Y^{(t-1)}))
&= \lambda \ell\left(\sum_{i=1}^d \tilde{\gamma_i}
 V_i V_i^{\dagger} ,\mathcal{D}\right)
- \lambda \ell\left(\sum_{i=1}^d \gamma^{(t-1)}_i
V_i V_i^{\dagger} ,\mathcal{D}\right)
\\
& + \sum_{i=1}^d((\alpha-1)\log(\tilde{Y}_i) -\tilde{Y}_i )
-
\sum_{i=1}^d((\alpha-1)\log(Y_i^{(t-1)}) -Y_i^{(t-1)})
\\
& + \sum_{i=1}^d
\tilde{Y}_i - \sum_{i=1}^d Y_i^{(t-1)}\end{aligned}$$ and $$\begin{aligned}
\log (A(V^{(t-1)}, \tilde{V})) 
= 
\lambda \ell\left(\sum_{i=1}^d \gamma_i
 \tilde{V}_i \tilde{V}_i ^{\dagger} ,\mathcal{D}\right)
- \lambda \ell\left(\sum_{i=1}^d \gamma_i
V^{(t-1)}_i (V^{(t-1)}_i)^{\dagger} ,\mathcal{D}\right)\end{aligned}$$ where $\ell(\cdot,\mathcal{D})$ stands for $\ell^{dens}(\cdot,\mathcal{D})$ or $\ell^{prob}(\nu,\mathcal{D})$ depending on the estimator we are computing.

Experiments
-----------

We study the numerical performance of the prob-estimators with $ \lambda = m/2 $, i.e. $\tilde{\rho}_{m/2}^{prob}$ and the dens-estimator with $ \lambda =  \frac{N}{4} $, i.e. $\tilde{\rho}_{ N/4}^{dens}$ on the following settings, all with $ n=2,3,4$ ($d=4,8,16 $):

-   a pure state density (rank-$ 1$) $\rho = \psi \psi^{\dagger}$ with $ \psi \in \mathbb{C}^{d\times 1} $,

-   a rank-$2 $ density matrix that $ 