Embedding qubits into fermionic Fock space, peculiarities of the four-qubit case

Abstract

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 physically 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 $Spin(16,\mathbb{C})$ invariant polynomials related to the structure of possible measures of entanglement for fermionic systems with 8 modes. Here we find an algebraically independent generating set of the generalized SLOCC invariants and calculate their restriction to the dense orbit. We point out the special role the largest exceptional group $E_8$ is playing in these considerations.

arXiv comment35 pages, 5 figures

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• Entanglement of four-qubit systems: a geometric atlas with polynomial compass II (the tame world)
• The magic three-qubit Veldkamp line: A finite geometric underpinning for form theories of gravity and black hole entropy
• Quantum marginals from pure doubly excited states
• Skew-Symmetric Tensor Decomposition
• Weighted Graph Theory Representation of Quantum Information Inspired by Lie Algebras
• Classification of multipartite systems featuring only $|W⟩$ and $|GHZ⟩$ genuine entangled states
• Hyperdeterminants from the $E_8$ Discriminant
• Sequential generation of Polynomial Invariants and N-body non-local correlations
• A fermionic code related to the exceptional group $E_8$
• Multi-qubits and Polyvalent Singularity in Type II Supestring Theory
• Identifiability and singular locus of secant varieties to spinor varieties
• The Coupled Cluster Method and Entanglement