Half a qubit: an algebraic fractionalization

• URL: http://arxiv.org/abs/2608.16183v1
• Date: Mon, 17 Aug 2026 06:57:54 GMT
• Title: Half a qubit: an algebraic fractionalization
• Abstract summary: Fractionalizing a quantum two-level system is usually associated with encodings based on pairs of Majorana fermions.<n>We show an alternative algebraic fractionalization by embedding Székely's classical half-coin'' into a non-Hermitian Krein space.
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• License: http://creativecommons.org/licenses/by/4.0/
• Abstract: 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.

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