Half a qubit: an algebraic fractionalization

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 Sz'{e}kely'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 Nge2 vectors are null in Krein space. L₁ norm of the half-qubit follows in closed form, lVert prVert₁ = 2sqrt{q}-sqrt{q-p}. 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 eta-metric generate a distinguished commuting mathbb Z_2timesmathbb Z₂ subgroup. Second, we find the eta-metric obstructs any local eta-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 nth root.