Polynomial invariants of degree 4 for even-$n$ qubits and their applications in entanglement classification

TL;DR

This paper introduces a straightforward method to construct degree-4 polynomial invariants for even-numbered qubits, enabling entanglement classification and the development of entanglement measures based on these invariants.

Contribution

The paper provides explicit expressions for degree-4 polynomial invariants for even-$n$ qubits and demonstrates their use in classifying entangled states and constructing entanglement measures.

Findings

Polynomial invariants are invariant under SLOCC operations.

Absolute values of invariants serve as entanglement monotones.

The invariants facilitate entanglement classification for even-$n$ qubits.

Abstract

We develop a simple method for constructing polynomial invariants of degree 4 for even-$n$ qubits and give explicit expressions for these polynomial invariants. We demonstrate the invariance of the polynomials under stochastic local operations and classical communication and exemplify the use of the invariance in classifying entangled states. The absolute values of these polynomial invariants are entanglement monotones, thereby allowing entanglement measures to be built. Finally, we discuss the properties of these entanglement measures.