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arXiv:2301.10660 · Annales de l’Institut Henri Poincaré D Combinatorics Physics and their Interactions

A hyperdeterminant on fermionic Fock space

Frédéric Holweck, Luke Oeding

Published Jan 30, 2024• 0 citations
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Abstract

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.

DOI
10.4171/aihpd/187

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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.
In quantum information theory, it is well known that the tripartite entanglement of three qubits is described by the group [SL(2,C)]3[SL(2,C){]}^{3}[SL(2,C)]3 and that the entanglement measure is given by Cayley's hyperdeterminant. This has provided an analogy with certain N=2N=2N=2 supersymmetric black holes in string theory, whose entropy is also given by the hyperdeterminant. In this paper, we extend the analogy to N=8N=8N=8. We propose that a particular tripartite entanglement of seven qubits, encoded in the Fano plane, is described by the exceptional group E7(C){E}_{7}(C)E7​(C) and that the entanglement measure is given by Cartan's quartic E7{E}_{7}E7​ invariant.
We describe explicitly the algebra of polynomial functions on the Hilbert space of four-qubit states that are invariant under the group SL(2,C)4\mathrm{SL}(2,C{)}^{4}SL(2,C)4 of stochastic local quantum operations assisted by classical communication. From this description, we obtain a closed formula for the hyperdeterminant in terms of low degree invariants.
We give a fermionic Fock space description of embedded entangled qubits. Within this framework the problem of classification of pure state entanglement boils down to the problem of classifying spinors. The usual notion of separable states turns out to be just a special case of the one of pure spinors. By using the notion of single, double and mixed occupancy representation with intertwiners relating them a natural physical interpretation of embedded qubits is found. As an application of these ideas one can make a physical sound meaning of some of the direct sum structures showing up in the context of the so-called black-hole/qubit correspondence. We discuss how the usual invariants for qubits serving as measures of entanglement can be obtained from invariants for spinors in an elegant manner. In particular a detailed case study for recovering the invariants for four-qubits within a spinorial framework is presented. We also observe that reality conditions on complex spinors defining Majorana spinors for embedded qubits boil down to self-conjugate states under the Wootters spin flip operation. Finally we conduct a study on the explicit structure of Spin(16,C)\text{Spin}(16,\mathbb{C})Spin(16,C) invariant polynomials related to the structure of possible measures of entanglement for fermionic systems with eight modes. Here we find an algebraically independent generating set of the generalized stochastic local operations and classical communication invariants and calculate their restriction to the dense orbit. We point out the special role the largest exceptional group E8{E}_{8}E8​ is playing in these considerations.
Using a generalization of Cayley's hyperdeterminant as a new measure of tripartite fermionic entanglement we obtain the SLOCC classification of three-fermion systems with six ``single-particle'' states. A special subclass of such three-fermion systems is shown to have the same properties as the well-known three-qubit ones. Our results can be presented in a unified way using Freudenthal triple systems based on cubic Jordan algebras. For systems with an arbitrary number of fermions and single-particle states we propose to use the Pl\"ucker relations as a sufficient and necessary condition of separability.
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.
The geometry of four-qubit entanglement is investigated. We replace some of the polynomial invariants for four qubits introduced recently by new ones of direct geometrical meaning. It is shown that these invariants describe four points, six lines and four planes in complex projective space CP 3 . For the generic entanglement class of stochastic local operations and classical communication they take a very simple form related to the elementary symmetric polynomials in four complex variables. Moreover, suitable powers of their magnitudes are entanglement monotones that fit nicely into the geometric set of n -qubit ones related to Grassmannians of l -planes found recently. We also show that in terms of these invariants the hyperdeterminant of order 24 in the four-qubit amplitudes takes a more instructive form than the previously published expressions available in the literature. Finally, in order to understand two-, three- and four-qubit entanglement in geometric terms we propose a unified setting based on CP 3 furnished with a fixed quadric.