Algebraic characterization of the physical subspace

Characterize algebraically whether there exists a class of operators that leaves the total-number sector N≤1 invariant for the fused half-qubit construction, thereby identifying the physical subspace and its null-state structure without postulating the truncation.

Background

The fused construction selects the total-number sectors N=0 and N=1 as the physical two-dimensional code space, while sectors with N≥2 are null with respect to the restricted Krein form. The paper establishes this quotient structure but does not derive the physical subspace from an operator algebra. The unresolved question is whether operators preserving the N≤1 sector can be identified, in analogy with the role of gauge-invariant observables and null states in Gupta–Bleuler quantization.

References

A second question is whether the physical subspace can be characterized algebraically rather than postulated---that is, whether there exists a class of operators leaving the $N\le1$ sector invariant, which would place the present truncation on the same footing as the Gupta--Bleuler quantization of the electromagnetic field, where the null states form a subspace preserved by every gauge-invariant observable.

— Half a qubit: an algebraic fractionalization  (2608.16183 - Chang, 17 Aug 2026) in Section Conclusion