Cayley's First Hyperdeterminant is an Entanglement Measure

TL;DR

This work addresses quantifying entanglement for multipartite $2n$-qudit states where standard measures are incomplete, by leveraging Cayley's first hyperdeterminant $\mathrm{hdet}$. It proves that the modulus $|\mathrm{hdet}(\widehat{\psi})|$ and its square are valid entanglement monotones on $2n$-qudits, with LU invariance, vanishing on separable states, and non-increasing on average under LOCC, extended to mixed states via convex roofs. The results recover concurrence and the $n$-tangle in the $2n$-qubit limit, and establish $|\mathrm{hdet}|$ as a genuine generalization to high-dimensional systems, offering a new fundamental entanglement measure. The paper also suggests directions for practical computation and potential closed-form expressions for mixed-state entanglement in terms of spectra or hypermatrix analogues, broadening the scope of entanglement quantification in quantum information.

Abstract

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.

Theorems & Definitions (10)

• Proposition 1
• Proposition 2
• Proposition 3
• proof
• Theorem 1
• proof
• Theorem 2
• proof
• Lemma 1
• proof