Embedding qubits into fermionic Fock space: Peculiarities of the four-qubit case

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 $\text{Spin}(16,\mathbb{C})$ 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 $E_8$ is playing in these considerations.

Figure 1

Single occupancy embedding of the $n$-qubit Hilbert space ($2^n$ basis vectors) inside ${\mathcal{F}}_{+}$ (for $n=2k$ boxes and $N=2n$ single particle states) or ${\mathcal{F}}_{-}$ (for $n=2k+1$ boxes and $N=2n$ single particle states).

Figure 2

Double occupancy embedding of the $n$-qubit Hilbert space ($2^n$ basis vectors) inside ${\mathcal{F}}_{+}$ ($n$ boxes and $N=2n$ single particle states).

Figure 3

Single occupancy embedding of the two-qubit Hilbert space inside ${\mathcal{F}}_{+}$.

Figure 4

Double occupancy embedding of the two-qubit Hilbert space inside ${\mathcal{F}}_{+}$.

Figure 5

Mixed occupancy embedding of the two-qubit Hilbert space inside ${\mathcal{F}}_{-}$.