Cayley’s first hyperdeterminant is an entanglement measure

Abstract Previously, it was shown that both the concurrence and n -tangle on 2 n -qubit pure quantum states can be expressed in terms of Cayley’s first hyperdeterminant Dür et al (2000 Phys. Rev. A 62 062314), indicating that Cayley’s first hyperdeterminant, denoted hdet , captures some aspects of a state’s 2 n -way entanglement. In this paper, we rigorously prove that on both pure and mixed states, | hdet | 2 / d is identically zero on separable states, is an local unitary invariant, and is non-increasing on average under local operations and classical communications, thus demonstrating that | hdet | d / 2 is a physically meaningful and legitimate entanglement measure. Moreover, we discuss a few key examples to illustrate the particular type of entanglement Cayley’s first hyperdeterminant is detecting: genuine full d -level GHZ-type entanglement across all 2 n parties. Combined, this establishes Cayley’s first hyperdeterminant (or | hdet | 2 / d to be precise), as a physically significant generalization of the G -concurrence and the n -tangle to 2 n -qudit states.