Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

arXiv:2504.15511 · Physica Scripta

Cayley’s first hyperdeterminant is an entanglement measure

Isaac Dobes, Naihuan Jing

Published Jul 17, 2026• 0 citations
arXivHTMLPDF

Abstract

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.

DOI
10.1088/1402-4896/ae8c7f

Related papers

Multipartite entanglement in spin chains and the hyperdeterminant

Alba Cervera-Lierta, Albert Gasull, José Ignacio Latorre +1 more

Abstract A way to characterize multipartite entanglement in pure states of a spin chain with sites and local dimension is by means of the Cayley hyperdeterminant. The latter quantity is a polynomial constructed with the components of the wave function which is invariant under local unitary transformation. For spin 1/2 chains (i.e. ) with and sites, the hyperdeterminant coincides with the concurrence and the tangle respectively. In this paper we consider spin chains with sites where the hyperdeterminant is a polynomial of degree 24 containing around terms. This huge object can be written in terms of more simple polynomials and of degrees 8 and 12 respectively. Correspondingly we compute , and the hyperdeterminant for eigenstates of the following spin chain Hamiltonians: the transverse Ising model, the XXZ Heisenberg model and the Haldane–Shastry model. Those invariants are also computed for random states, the ground states of random matrix Hamiltonians in the Wigner–Dyson Gaussian ensembles and the quadripartite entangled states defined by Verstraete et al in 2002. Finally, we propose a generalization of the hyperdeterminant to thermal density matrices. We observe how these polynomials are able to capture the phase transitions present in the models studied as well as a subclass of quadripartite entanglement present in the eigenstates.

arXiv:1802.02596•Journal of Physics A Mathematical and Theoretical•Nov 5, 2018•8 citations

Multipartite entanglement and hyperdeterminants

Akimasa Miyake, Miki Wadati

We classify multipartite entanglement in a unified manner, focusing on a duality between the set of separable states and that of entangled states. Hyperdeterminants, derived from the duality, are natural generalizations of entanglement measures, the concurrence, 3-tangle for 2, 3 qubits respectively. Our approach reveals how inequivalent multipartite entangled classes of pure states constitute a partially ordered structure under local actions, significantly different from a totally ordered one in the bipartite case. Moreover, the generic entangled class of the maximal dimension, given by the nonzero hyperdeterminant, does not include the maximally entangled states in Bell's inequalities in general (e.g., in the \(n \!\geq\! 4\) qubits), contrary to the widely known bipartite or 3-qubit cases. It suggests that not only are they never locally interconvertible with the majority of multipartite entangled states, but they would have no grounds for the canonical \(n\)-partite entangled states. Our classification is also useful for that of mixed states.

arXiv:quant-ph/0212146•Quantum Information and Computation•Nov 1, 2002•53 citations

Topological classification of multipartite entangled states by the hyperdeterminant

Akimasa Miyake

We find that the hyperdeterminant $Det A$, related to an entanglement measure (the concurrence, 3-tangle for the 2,3-qubit respectively), is derived from a duality between the entangled states and separable states. In terms of $Det A$ and its singularities, the single copy of multipartite pure entangled states is classified into an onion structure of every closed subset, similarly to the local rank in the bipartite case. This reveals that many inequivalent multipartite entangled states are partially ordered such that entanglement measures like $Det A$ as well as local ranks are needed to distinguish them. In particular, the nonzero $Det A$ distinguishes generic entangled states of the maximal dimension (the outermost class of the onion structure). It suggests that the majority of multipartite entangled states never locally converts to the maximally entangled states in Bell's inequalities even probabilistically in general (e.g., in the $n \\geq 4$-qubit), contrary to the widely known bipartite or 3-qubit cases. The classification is also useful for that of mixed states.

•Jun 18, 2002•1 citation

Classification of multipartite entangled states by multidimensional determinants

Akimasa Miyake

We find that multidimensional determinants ``hyperdeterminants,'' related to entanglement measures (the so-called concurrence, or 3-tangle for two or three qubits, respectively), are derived from a duality between entangled states and separable states. By means of the hyperdeterminant and its singularities, the single copy of multipartite pure entangled states is classified into an onion structure of every closed subset, similar to that by the local rank in the bipartite case. This reveals how inequivalent multipartite entangled classes are partially ordered under local actions. In particular, the generic entangled class of the maximal dimension, distinguished as the nonzero hyperdeterminant, does not include the maximally entangled states in Bell's inequalities in general (e.g., in the $n>~4$ qubits), contrary to the widely known bipartite or three-qubit cases. It suggests that not only are they never locally interconvertible with the majority of multipartite entangled states, but they would have no grounds for the canonical n-partite entangled states. Our classification is also useful for the mixed states.

arXiv:quant-ph/0206111•Physical Review A•Jan 30, 2003•273 citations

Quantify entanglement by concurrence hierarchy

Heng Fan, Keiji Matsumoto, Hiroshi Imai

We define the concurrence hierarchy as d − 1 independent invariants under local unitary transformations in d -level quantum system. The first one is the original concurrence defined by Wootters (1998 Phys. Rev. Lett. 80 2245) and Hill and Wootters (1997 Phys. Rev. Lett. 78 5022) in a two-level quantum system and generalized to the d -level pure quantum state case. We propose to use this concurrence hierarchy as a measurement of entanglement. This measurement does not increase under local quantum operations and classical communication.

arXiv:quant-ph/0204041•Journal of Physics A Mathematical and General•Mar 27, 2003•59 citations

New Identity for Cayley's First Hyperdeterminant with Applications to Symmetric Tensors and Entanglement

Isaac Dobes

In this article, a new formula for computing Cayley's first hyperdeterminant in terms of the Levi-Civita symbol is given. It is then shown that this formula can be used to compute the hyperdeterminant of symmetric tensors in polynomial time with respect to their order (assuming fixed side length). Applications to quantifying the entanglement of states of bosonic quantum systems are then discussed. Additionally, in order to obtain the fast calculation of the hyperdeterminant on symmetric tensors, generalized elimination and duplication matrices are defined and their explicit formulas are derived.

quant-pharXiv:2512.03093•Dec 2, 2025
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.
Abstract A way to characterize multipartite entanglement in pure states of a spin chain with sites and local dimension is by means of the Cayley hyperdeterminant. The latter quantity is a polynomial constructed with the components of the wave function which is invariant under local unitary transformation. For spin 1/2 chains (i.e. ) with and sites, the hyperdeterminant coincides with the concurrence and the tangle respectively. In this paper we consider spin chains with sites where the hyperdeterminant is a polynomial of degree 24 containing around terms. This huge object can be written in terms of more simple polynomials and of degrees 8 and 12 respectively. Correspondingly we compute , and the hyperdeterminant for eigenstates of the following spin chain Hamiltonians: the transverse Ising model, the XXZ Heisenberg model and the Haldane–Shastry model. Those invariants are also computed for random states, the ground states of random matrix Hamiltonians in the Wigner–Dyson Gaussian ensembles and the quadripartite entangled states defined by Verstraete et al in 2002. Finally, we propose a generalization of the hyperdeterminant to thermal density matrices. We observe how these polynomials are able to capture the phase transitions present in the models studied as well as a subclass of quadripartite entanglement present in the eigenstates.
We classify multipartite entanglement in a unified manner, focusing on a duality between the set of separable states and that of entangled states. Hyperdeterminants, derived from the duality, are natural generalizations of entanglement measures, the concurrence, 3-tangle for 2, 3 qubits respectively. Our approach reveals how inequivalent multipartite entangled classes of pure states constitute a partially ordered structure under local actions, significantly different from a totally ordered one in the bipartite case. Moreover, the generic entangled class of the maximal dimension, given by the nonzero hyperdeterminant, does not include the maximally entangled states in Bell's inequalities in general (e.g., in the n ⁣≥ ⁣4n \!\geq\! 4n≥4 qubits), contrary to the widely known bipartite or 3-qubit cases. It suggests that not only are they never locally interconvertible with the majority of multipartite entangled states, but they would have no grounds for the canonical nnn-partite entangled states. Our classification is also useful for that of mixed states.
We find that the hyperdeterminant DetADet ADetA, related to an entanglement measure (the concurrence, 3-tangle for the 2,3-qubit respectively), is derived from a duality between the entangled states and separable states. In terms of DetADet ADetA and its singularities, the single copy of multipartite pure entangled states is classified into an onion structure of every closed subset, similarly to the local rank in the bipartite case. This reveals that many inequivalent multipartite entangled states are partially ordered such that entanglement measures like DetADet ADetA as well as local ranks are needed to distinguish them. In particular, the nonzero DetADet ADetA distinguishes generic entangled states of the maximal dimension (the outermost class of the onion structure). It suggests that the majority of multipartite entangled states never locally converts to the maximally entangled states in Bell's inequalities even probabilistically in general (e.g., in the ngeq4n \\geq 4ngeq4-qubit), contrary to the widely known bipartite or 3-qubit cases. The classification is also useful for that of mixed states.
We find that multidimensional determinants ``hyperdeterminants,'' related to entanglement measures (the so-called concurrence, or 3-tangle for two or three qubits, respectively), are derived from a duality between entangled states and separable states. By means of the hyperdeterminant and its singularities, the single copy of multipartite pure entangled states is classified into an onion structure of every closed subset, similar to that by the local rank in the bipartite case. This reveals how inequivalent multipartite entangled classes are partially ordered under local actions. In particular, the generic entangled class of the maximal dimension, distinguished as the nonzero hyperdeterminant, does not include the maximally entangled states in Bell's inequalities in general (e.g., in the n> 4n>~4n> 4 qubits), contrary to the widely known bipartite or three-qubit cases. It suggests that not only are they never locally interconvertible with the majority of multipartite entangled states, but they would have no grounds for the canonical n-partite entangled states. Our classification is also useful for the mixed states.
We define the concurrence hierarchy as d − 1 independent invariants under local unitary transformations in d -level quantum system. The first one is the original concurrence defined by Wootters (1998 Phys. Rev. Lett. 80 2245) and Hill and Wootters (1997 Phys. Rev. Lett. 78 5022) in a two-level quantum system and generalized to the d -level pure quantum state case. We propose to use this concurrence hierarchy as a measurement of entanglement. This measurement does not increase under local quantum operations and classical communication.
In this article, a new formula for computing Cayley's first hyperdeterminant in terms of the Levi-Civita symbol is given. It is then shown that this formula can be used to compute the hyperdeterminant of symmetric tensors in polynomial time with respect to their order (assuming fixed side length). Applications to quantifying the entanglement of states of bosonic quantum systems are then discussed. Additionally, in order to obtain the fast calculation of the hyperdeterminant on symmetric tensors, generalized elimination and duplication matrices are defined and their explicit formulas are derived.