entangled subspace

This repository is to reproduce results in the paper arxiv-link "Quantifying Subspace Entanglement with Geometric Measures"

🚀 Exciting News! We've launched the numqi package github/numqi, combining all the functionalities of this repository and even more! 🌟 To dive into these features, just install numqi using pip install numqi, and explore the relevant functions within the package. 🛠️

quick start

This project is based on numqi package github-link, which is a package for numerical optimization for quantum information. To install numqi, run the following command (see numqi-doc for detailed installation instruction):

pip install numqi

Then, clone this repository and cd into the directory

git clone [email protected]:numqi/entangled-subspace.git
cd entangled-subspace

1. link
   • github/tensorspace magma based
2. completely entangled subspace (CES)
   • maximal dimension in space $\bigotimes_{i=1}^{k}\mathcal{H}_k$: $d_1d_2\cdots d_k-(d_1+\cdots+d_k)+k-1$ doi-link
3. perfectly entangled subspace

demo_schmidt_rank.py, GenTiles1

dim	loss(1)	loss(2)	time (second)
4	$0.030$	$9.2\times 10^{-15}$	0.82
8	$0.016$	$2.2\times 10^{-12}$	2.5
16	$0.0079$	$2.3\times 10^{-12}$	3.4
32	$0.0038$	$4.3\times 10^{-12}$	20.0
64	$0.0018$	$5.8\times 10^{-12}$	352

draft_seesaw.py/demo_compare_with_seesaw_dicke()

(n,k)	analytical	err(GD)	err(seesaw)	err(PPT)	time(GD)	time(seesaw)	time(PPT)
(5,1)	0.5904	$2.1\times 10^{-13}$	$2.3\times 10^{-13}$	$1.4\times 10^{-10}$	0.103	0.0101	1.73
(5,2)	0.6544	$2.7\times 10^{-14}$	$7.8\times 10^{-13}$	$5.1\times 10^{-11}$	0.087	0.0091	1.64
(5,3)	0.6544	$1.3\times 10^{-13}$	$7.0\times 10^{-13}$	$5.0\times 10^{-11}$	0.084	0.0091	1.59
(6,1)	0.5981	$2.1\times 10^{-13}$	$2.7\times 10^{-13}$	$8.9\times 10^{-10}$	0.129	0.017	25.1
(6,2)	0.6708	$1.7\times 10^{-14}$	$4.4\times 10^{-13}$	$3.4\times 10^{-11}$	0.122	0.018	26.5
(6,3)	0.6875	$1.2\times 10^{-13}$	$6.5\times 10^{-13}$	$2.6\times 10^{-13}$	0.166	0.018	25.7
(7,1)	0.6034	$1.2\times 10^{-14}$	$1.1\times 10^{-12}$	None	0.173	0.019	None
(7,2)	0.6813	$2.2\times 10^{-13}$	$2.4\times 10^{-13}$	None	0.163	0.019	None
(7,3)	0.7062	$4.2\times 10^{-14}$	$1.6\times 10^{-13}$	None	0.150	0.020	None