Uniqueness of Complementary Recovery in Holographic Error-Correcting Codes

• URL: http://arxiv.org/abs/2509.19299v1
• Date: Tue, 23 Sep 2025 17:59:16 GMT
• Title: Uniqueness of Complementary Recovery in Holographic Error-Correcting Codes
• Authors: Julia Jones, Jason Pollack,
• Abstract summary: Holographic codes are error-correcting codes with extra geometric structure ensured by a complementary recovery'' property.<n>We show that the failure of uniqueness is due to a failure to enforce error correction against erasure of $mathcalHbar A$.
• Score: 0.0
• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• Abstract: Holographic codes are a type of error-correcting code with extra geometric structure ensured by a ``complementary recovery'' property: given a division of the physical Hilbert space $\mathcal{H}$ into $\mathcal{H}_A$ and $\mathcal{H}_{\bar A}$, and an algebra of physical operators $\mathcal{M}\subseteq (\mathcal{L}(\mathcal{H}_A)\otimes I_{\mathcal{H}_{\bar A}})$, the logical operators in $\mathcal{L}(\mathcal{H}_L)\simeq \mathcal{L}(P\mathcal{H})$ which can be created by acting in $\mathcal{M}$ are identical to the logical operators whose expectation values cannot be altered by acting in the commutant $\mathcal{M}^\prime$, and vice versa. In arXiv:2110.14691, a uniqueness theorem was stated: the only possible tuple of (code, bipartition, algebra) which can exhibit complementary recovery is the maximal one $\mathcal{M}=P(\mathcal{L}(\mathcal{H}_A)\otimes I_{\mathcal{H}_{\bar A}})P$. We point out a counterexample to this result, using a ``non-adjacent'' bipartition of a four-qubit code proposed in arXiv:2110.14691. We show that the failure of uniqueness is due to a failure to enforce error correction against erasure of $\mathcal{H}_{\bar A}$, which requires enforcing the algebraic Knill-Laflamme condition $[P E_i^\dagger E_j P,\mathcal{M}]=0$ for each pair of error operators. When we add the additional requirement that $\mathcal{M}$ be correctable with respect to this channel, uniqueness is restored, and we re-prove the theorem of arXiv:2110.14691 with this added assumption. We present the list of bipartitions of the ``atomic'' holographic codes in arXiv:2110.14691 in which the correctability assumption can be violated.

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