摘要
Fractionalizing a quantum two-level system is usually associated with encodings based on pairs of Majorana fermions---an operational fractionalization. We show an alternative algebraic fractionalization by embedding Székely's classical ``half-coin'' into a non-Hermitian Krein space. The coefficients of$(q+pz)^{1/2}$define a signed sequence and a normalized, non-Hermitian biorthogonal operator describing a biorthogonal half-qubit. We prove that two such objects fuse into an arbitrary pure qubit through the signed Vandermonde convolution that the collective$N\ge2$vectors are null in Krein space.$L_1$norm of the half-qubit follows in closed form,$\lVert p\rVert_1 = 2\sqrt{q}-\sqrt{q-p}$ . Its$L_1$norm increases monotonically with the bias and attains its supremum$\sqrt{2}$precisely at the unbiased point$p=q=1/2$ . Interestingly, we identify two structural results as follows. First, number parity and the$η$ -metric generate a distinguished commuting$\mathbb Z_2\times\mathbb Z_2$subgroup. Second, we find the$η$ -metric obstructs any local$η$ -self-adjoint partner of the parity, so a half-qubit carries a$\mathbb{Z}_2$observable but no local$SU(2)$ . The full Pauli algebra emerges only upon fusion. We then show that the construction survives truncation of the Fock basis: the fused qubit is exact at every cutoff, and the Vandermonde cancellation is visible in sign-weighted photon-number statistics, and can be tested using existing cavity and trapped-ion state-synthesis methods. Finally, we generalize this algebraic fractionalization to a$1/n$ -qubit, which can be achieved by replacing the square root with an$n$ th root.