Cayley's First Hyperdeterminant is an Entanglement Measure

Previously, it was shown that both the concurrence and n-tangle on $2n$-qubits can be expressed in terms of Cayley's first hyperdeterminant, indicating that Cayley's first hyperdeterminant captures at least some aspects of a quantum state's entanglement. In this paper, we rigorously prove that Cayley's first hyperdeterminant is an entanglement measure on $2n$-qudits, and thus a legitimate generalization of the concurrence and n-tangle. In particular, we show that the modulus of the hyperdeterminant and its square both vanish on separable $2n$-qudits, are LU invariants, and are non-increasing on average under LOCC. Lastly, we then consider their convex roof extensions and show that they may in fact be viewed as entanglement measures on mixed states of $2n$-qudits.