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Conjecture on TCS from partial stabilizers under a homogeneous condition

Prove that for Pauli stabilizer states satisfying the specified homogeneous condition—where a spatially extended stabilizer and its restrictions to a subdimensional entanglement subsystem A and its complement share the same bulk form—any subsystem A whose strong symmetries are generated by stabilizers fully supported on A has weak symmetries given by the restrictions to A of stabilizers partially supported on A, and these weak symmetries form the transparent-patch operator algebra (t-patch operators) of the strong symmetries.

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Background

Beyond model-specific examples, the authors formulate a structural conjecture intended to capture when the TCS structure of mixed-state symmetries arises generically in Pauli stabilizer states. The conjecture posits that, under a homogeneous condition relating the bulk form of stabilizers and their parts, the restrictions of partially supported stabilizers on an SES supply precisely the weak symmetries that act as transparent-patch operators for the SES’s strong symmetries.

Establishing this result would provide a general criterion for the emergence of TCS in stabilizer states and clarify the algebraic foundations linking SEE, mixed-state symmetries, and categorical structures.

References

In this section, we propose a conjecture that for Pauli stabilizer states satisfying a 'homogeneous condition', if an SES A has strong symmetries specified by a set of stabilizers {W_Ai} fully supported in A , where i is a label of stabilizer, then stabilizers partially supported in A (such a stabilizer is denoted by W_{AB}{i}, the part supported in A is denoted by w_{A}{i} and the part outside A is denoted by w_{B}{i}) would compose the t-patch operators of the strong symmetry.

— Subdimensional entanglement entropy: from virtual response to mixed-state holography (2510.15766 - Li et al., 17 Oct 2025) in Appendix, Section “A conjecture about the generality of transparent composite symmetries in Pauli stabilizer states”