Description
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≥2 vectors are null in Krein space. L1 norm of the half-qubit follows in closed form. Its L1 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 ℤ2×ℤ2 subgroup. Second, we find the η-metric obstructs any local η-self-adjoint partner of the parity, so a half-qubit carries a ℤ2 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 nth root.