A link between Quantum Entanglement, Secant varieties and Sphericity

TL;DR

This paper explores the deep connections between algebraic geometry, representation theory, and quantum information, revealing how geometric properties influence quantum entanglement classification and state approximation limitations.

Contribution

It establishes a link between sphericity in group actions and the existence of exceptional states in quantum entanglement, providing new insights into state classification.

Findings

Sphericity implies states of a given rank cannot be approximated by lower-rank states.

Non-sphericity indicates the presence of exceptional states in quantum systems.

Identifies specific exceptional states related to secant varieties and entanglement classes.

Abstract

In this paper, we shed light on relations between three concepts studied in representations theory, algebraic geometry and quantum information theory. First - spherical actions of reductive groups on projective spaces. Second - secant varieties of homogeneous projective varieties, and the related notions of rank and border rank. Third - quantum entanglement. Our main result concerns the relation between the problem of the state reconstruction from its reduced one-particle density matrices and the minimal number of separable summands in its decomposition. More precisely, we show that sphericity implies that states of a given rank cannot be approximated by states of a lower rank. We call states for which such approximation is possible exceptional states. For three, important from quantum entanglement perspective cases of distinguishable, fermionic and bosonic particles, we also show that non-sphericity implies the existence of exceptional states. Remarkably, the exceptional states belong to non-bipartite entanglement classes. In particular, we show that the $W$-type states and their appropriate modifications are exceptional states stemming from the second secant variety for three cases above. We point out that the existence of the exceptional states is a physical obstruction for deciding the local unitary equivalence of states by means of the one-particle reduced density matrices. Finally, for a number of systems of distinguishable particles with known orbit structure we list all exceptional states and discuss their possible importance in entanglement theory.