Quantum Decimation in Hilbert Space: Coarse-Graining without Structure

TL;DR

This paper introduces a PCA-based method for coarse-graining quantum states in finite-dimensional Hilbert spaces without relying on traditional structural assumptions, preserving essential entanglement features with fewer degrees of freedom.

Contribution

It proposes a novel, structure-independent coarse-graining technique using PCA that maintains key entanglement properties of quantum states.

Findings

Enables efficient description of quantum state collections

Retains most global entanglement features

Operates without predefined Hilbert space structure

Abstract

We present a technique to coarse-grain quantum states in a finite-dimensional Hilbert space. Our method is distinguished from other approaches by not relying on structures such as a preferred factorization of Hilbert space or a preferred set of operators (local or otherwise) in an associated algebra. Rather, we use the data corresponding to a given set of states, either specified independently or constructed from a single state evolving in time. Our technique is based on principle component analysis (PCA), and the resulting coarse-grained quantum states live in a lower dimensional Hilbert space whose basis is defined using the underlying (isometric embedding) transformation of the set of fine-grained states we wish to coarse-grain. Physically, the transformation can be interpreted to be an "entanglement coarse-graining" scheme that retains most of the global, useful entanglement structure of each state, while needing fewer degrees of freedom for its reconstruction. This scheme could be useful for efficiently describing collections of states whose number is much smaller than the dimension of Hilbert space, or a single state evolving over time.