Four-qubit pure states as fermionic states

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.

arXiv comment8 pages, 2 figures

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• Entanglement of four-qubit systems: a geometric atlas with polynomial compass II (the tame world)
• Entanglement classification of three fermions with up to nine single-particle states
• The Quadrifocal Variety
• Fine-Structure Classification of Multiqubit Entanglement by Algebraic Geometry
• Teleporting quantum information encoded in fermionic modes
• Coffman-Kundu-Wootters inequality for fermions
• A finite geometric toy model of space-time as an error correcting code
• Classification of four qubit states and their stabilisers under SLOCC operations
• Hyperdeterminants from the $E_8$ Discriminant
• A fermionic code related to the exceptional group $E_8$
• Classifying Entanglement by Algebraic Geometry