dens-estimator is easier to compute and while it is less accurate than the prob-estimator, it still shows better results than the direct inversion method.

-   The thresholding estimator of [@butucea2015spectral] works well for rank-1 states but seems to bring too much bias for other states.

Besides the square error, the eigenvalues of the estimates are also important when reconstructing density matrices. In Figure \[eigenvalues\_simulated\], the dens-estimator returns with eigenvalues similar to the true eigenvalues of the true density matrix, while the prob-estimator seems not to shrink enough.

![Eigenvalues of estimates for an “approximate rank-2" density with $ d=2^3, m = 200 $.[]{data-label="eigenvalues_simulated"}](eigenvalues_n3.png)

[| l | c | c | c | c |]{} & $m = 20$ & $m = 200$ & $m = 1000$ & $m = 2000$\
\
Inversion & 175 (4e-4) & 14.8 (2e-5) & 2.71 (8e-6) & 1.55 (5e-6)\
Thresholding & 93.5 (3e-4) & **12.6** (3e-5) &**.596** (2e-6) & **.412** (2e-6)\
prob & 86.3 (6e-4) & 22.4 (2e-4) & 10.5 (6e-5) & 5.13 (2e-5)\
dens & **51.5** (2e-4) & 21.7 (7e-5) & 13.1 (3e-5) & 13.2 (2e-5)\
\
Inversion & 16.8 (8e-4) & 15.9 (3e-4) & 15.9 (1e-4) & 15.8 (7e-5)\
Thresholding & 14.9 (3e-4) & 