Four-qubit pure states as fermionic states

TL;DR

This paper investigates the relationship between four-qubit states and fermionic states, revealing that the embedding preserves entanglement structures under certain conditions and providing a complete classification of 4-qubit states.

Contribution

It demonstrates that the orbit mapping from 4-qubit to fermionic states remains injective under SLOCC, extending known results for fewer qubits and solving the SLOCC equivalence problem for 4-qubit states.

Findings

Orbit mapping is injective under SLOCC for 4-qubit states.

Complete classification of pure 4-qubit states under SLOCC.

Embedding preserves entanglement structure for 4-qubits.

Abstract

The embedding of the $n$-qubit space into the $n$-fermion space with $2n$ modes is a widely used method in studying various aspects of these systems. This simple mapping raises a crucial question: does the embedding preserve the entanglement structure? It is known that the answer is affirmative for $n=2$ and $n=3$. That is, under either local unitary (LU) operations or with respect to stochastic local operations and classical communication (SLOCC), there is a one-to-one correspondence between the 2- (or 3)-qubit orbits and the 2- (or 3)-fermion orbits with 4 (or 6) modes. However these results do not generalize as the mapping from the $n$-qubit orbits to the $n$-fermion orbits with $2n$ modes is no longer surjective for $n>3$. Here we consider the case of $n=4$. We show that surprisingly, the orbit mapping from qubits to fermions remains injective under SLOCC, and a similar result holds under LU for generic orbits. As a byproduct, we obtain a complete answer to the problem of SLOCC equivalence of pure 4-qubit states.