Recoverable Information and Emergent Conservation Laws in Fracton Stabilizer Codes

TL;DR

This work defines recoverable information $\mu$ for stabilizer Hamiltonians as a Hamiltonian-dependent, physically interpretable counterpart to topological entanglement entropy, capturing nonlocal constraints at entanglement cuts. It presents three equivalent methods to compute $\mu$ and demonstrates their equivalence in several stabilizer models, including cluster models, toric codes, X-cube, and Haah's code. The authors show that $\mu$ equals the dimension of a nonlocal surface stabilizer group $G_{\text{NLSS}}$, linking recoverable information to Gauss-law-type constraints and emergent $Z_2$ conservation laws for point-like excitations in fracton phases. This framework provides a unified lens to understand topological information in higher dimensions and fracton systems, with implications for understanding excitation dynamics and potential error-correcting interpretations.

Abstract

We introduce a new quantity, that we term recoverable information, defined for stabilizer Hamiltonians. For such models, the recoverable information provides a measure of the topological information, as well as a physical interpretation, which is complementary to topological entanglement entropy. We discuss three different ways to calculate the recoverable information, and prove their equivalence. To demonstrate its utility, we compute recoverable information for fracton models using all three methods where appropriate. From the recoverable information, we deduce the existence of emergent $Z_2$ Gauss-law type constraints, which in turn imply emergent $Z_2$ conservation laws for point-like quasiparticle excitations of an underlying topologically ordered phase.

Figures (8)

• Figure 1: $G_c^{X}$ and $G_c^{Z}$ terms in Haah's code. Each site has two spins. $X$ and $Z$ denote the corresponding Pauli operators. $I$ represents the identity operator.
• Figure 2: Depiction of the boundary of Alice and Bob's system. The red (blue) edges represent a good cut stabilizer for Alice (Bob). The dotted edges are not actually present but represent the completion of the cut stabilizer.
• Figure 3: Visualization of the 'quasi-local NLSS' at the corner of a square entanglement cut. For simplicity, the black circles represent $Z$-type operators and the white circles represent the $X$-type operators.
• Figure 4: Depiction of the constraint in the $d=3$ toric code which is formed by the product (sum in $W[\mathcal{S}]$) of all cut local cube constraints, but is itself not a cut constraint.
• Figure 5: Construction of an appropriate basis for the cut stabilizer group in the $d=3$ toric code. (a) Depiction of the straddling surface ribbons a minimal basis must avoid generating in $G_\partial$. (b) Basis construction using the checkerboard method as described in the text. Here, the thick lines indicate the intersection of a cut plaquette stabilizer in the basis with $A$.

Theorems & Definitions (6)

• proof
• Lemma 1
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• Lemma 2
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