Compression and reduction of N∗1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$N*1$$\end{document} states by unitary matrices

In recent experiments, the compression from qutrit to qubit is realized by the autoencoder. Inspired by the idea of dimensionality reduction, we apply the rotation transformation to compress the states. Starting from Lie algebra, we construct a 3*3 unitary matrix acting on 3*1 state and realize the rotation transformation of the states and then achieve compression of 3*1 state. Each rotation of a state is a compression, and each compression-only needs to adjust two parameters. According to the compression of 3*1 and 4*1 states by unitary matrices, we further discuss the compression law of N*1 states by unitary matrices. In the process of compression, we can adjust the form of the unitary matrix according to the system condition to change the compression position. In this paper, we focus on the compression law along the diagonal from top to bottom. We redesigned the autoencoder and added the waveplate combination to reduce the parameters without affecting the results and achieve the purpose of state compression.