On superqubits
Steven Duplij, Raimund Vogl
TL;DR
This work extends quantum information into a supersymmetric framework by formulating superqubits within $\mathbb{Z}_{2}$-graded, Grassmann-valued spaces. It develops a comprehensive mathematical infrastructure, including $\Lambda_{N}(\mathbb{C})$ Grassmann algebras, graded tensor products, and supermatrices with $\mathsf{sT}$, $\operatorname{str}$, and $\operatorname{Ber}$, alongside the entanglement surrogate sdTr. The authors define even and odd superqubits, derive density matrices and inner products in the graded setting, and introduce generalized entanglement measures such as even/odd superconcurrence and supertangle, with LOCC/SLOCC group structures like $uOSp(2|1)$. These contributions offer a formal pathway to a superquantum theory that interpolates between standard quantum mechanics and more exotic nonlocal models, with potential applications in condensed-matter systems and beyond.
Abstract
We first reconsider the mathematical background of superqubit theory and describe important peculiarities of superspaces and supermatrices which are usually out of attention. Then we study states in super Hilbert spaces using super-bra/super-ket formalism in details. The qubit (qudit) and superqubit (superqudit) are defined as linear spans in the corresponding Hilbert subspaces. A new kind of superqubit carring the odd parity is introduced. The multi-superqubit states are studied, and the superconcurrence which distinguishes separable states is proposed.