Twenty years ago, Cayley's hyperdeterminant, the degree four invariant of the polynomial ring \mathbb{C}[\mathbb{C}^2\otimes\mathbb{C}^2\otimes \mathbb{C}^2]^{{\operatorname{SL}_2(\mathbb{C})}^{\times 3}} , was popularized in modern physics as it separates genuine entanglement classes in the three-qubit Hilbert space and is connected to entropy formulas for special solutions of black holes. In this note, we compute the analogous invariant on the fermionic Fock space for N=8 , i.e., spin particles with four different locations, and show how this invariant projects to other well-known invariants in quantum information. We also give combinatorial interpretations of these formulas.
Abstract In this paper, we represent n-qubits as hypermatrices and consider various applications to quantum entanglement. In particular, we use the higher-order singular value decomposition of hypermatrices to prove that the π-transpose is an LU invariant. Additionally, through our construction we show that the matrix representation of the combinatorial hyperdeterminant of 2n-qubits can be expressed as a product of the second Pauli matrix, allowing us to derive a formula for the combinatorial hyperdeterminant of 2n-qubits in terms of the n-tangle.
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.
Let
Xn=∑i=1stiAi⊗n, where
A1,…,As∈Md(C) and
t1,…,ts∈C are fixed, while
n grows. Direct computation of determinants or Schatten norms of
Xn is exponential in
n. For a single tensor power these quantities are elementary, and even the determinant of a generic two-term combination admits a reduction to polynomially many scalar factors; however, no analogous elementary reduction is available for three or more terms.
We give an exact representation-theoretic method which, for fixed
d and
s, computes
∥Xn∥p,
0<p<∞, and determinants in polynomial time in
n. Schur--Weyl duality yields a simultaneous block decomposition, while Jacobi--Trudi identities in the Grothendieck ring replace Schur modules by signed combinations of tensor products of symmetric powers.
For
d=3, each irreducible contribution reduces to the difference of two explicitly computable symmetric-power terms, leading to an open-source implementation. In a single-thread CPU benchmark, a genuine three-term
3×3 trace-norm problem with
n=18 is evaluated in about
47 seconds, whereas just storing the unreduced matrix would require approximately
2.4×1018 bytes. Direct and reduced computations agree to relative error below
3.4×10−15 throughout their common range
n≤9.