State-dependent geometries from magic-enriched quantum codes

Abstract

Quantum error-correcting codes provide a powerful framework for emergent spacetime, yet existing holographic code models describe only quantum fields on a fixed background: in exact erasure-correcting codes, the entropic area term is state independent and cannot capture gravitational backreaction. We argue that this limitation is intrinsic to exact subsystem recovery and that incorporating backreaction instead requires approximate quantum error correction. We introduce a Ryu-Takayanagi-like entropy decomposition for approximate subsystem erasure-correcting codes, defining bulk matter entropy via optimal recovery and a complementary proto-area entropy as the difference between boundary entropy and recoverable bulk entropy. For a broad class of skewed quantum codes obtained by small nonlocal perturbations of exact codes, the proto-area increases monotonically with bulk entropy, closely aligning with the behavior of quantum extremal surfaces. We identify the origin of this response as a form of tripartite non-local magic in the Choi state of the encoding map, which vanishes in stabilizer codes and controls the leading matter-geometry coupling in approximate subsystem erasure-correcting codes.

Figures (8)

• Figure 1: Schematic of the encode–recover process. A logical input $\sigma^{(L)}_{a\bar{a}}\equiv \sigma^{(L)}$ is supplied to the code, where $\sigma_{a}^{(L)}=\mathop{\mathrm{Tr}}\nolimits_{\bar{a}}(\sigma^{(L)}_{a\bar{a}})$ and $\sigma_{\bar{a}}^{(L)}=\mathop{\mathrm{Tr}}\nolimits_{a}(\sigma^{(L)}_{a\bar{a}})$ denote the logical marginals on $a$ and $\bar{a}$, respectively. The isometry $V$ encodes $\sigma^{(L)}_{a\bar{a}}$ into boundary degrees of freedom, producing the collective encoded state $\tilde{\rho}_{A\bar{A}}$. We group boundary output legs as $A \equiv A_1\cup A_2$ and $\bar{A} \equiv \bar{A}_1\cup\bar{A}_2$. Intermediate encoded marginals are obtained by partial traces, $\tilde{\rho}_{A}=\mathop{\mathrm{Tr}}\nolimits_{\bar{A}}(\tilde{\rho}_{A\bar{A}})$ and $\tilde{\rho}_{\bar{A}}=\mathop{\mathrm{Tr}}\nolimits_{A}(\tilde{\rho}_{A\bar{A}})$. Local recovery unitaries $R_A$ and $R_{\bar{A}}$ act independently on $A$ and $\bar{A}$, producing the recovered components $\sigma^{(R)}_{A_1},\,\sigma^{(R)}_{A_2},\,\sigma^{(R)}_{\bar{A}_1},\,\sigma^{(R)}_{\bar{A}_2}$, which together form the final recovered boundary state $\sigma^{(R)}_{A\bar{A}}$.
• Figure 2: HaPPY code network shown as a hyperbolic tiling of perfect pentagon tensors. Each pentagon corresponds to a perfect tensor ($[[5,1,3]]$ code), with the black dot inside indicating the logical input leg. These logical legs point inward and should be understood as inputs. The red dots on the boundary represent the physical qubits where the encoded state appears.
• Figure 3: Left: an exact subsystem erasure-correcting code can be written such that an entangling state is first generated by $U_\chi$ on $A_2,\bar{A}_2$ followed by local encoding $U_A\otimes U_{\bar{A}}$, which can not produce matter-geometry correlation. Right: A controlled-$\chi$ unitary is needed to generate different amounts of entanglement based on the logical information on $A_1, \bar{A}_1$, so as to produce states of the form in \ref{['eqn:statedep']}. For example, when $U_\chi = CX$, then it is a multi-controlled gate, which is non-Clifford.
• Figure 4: Left: AdS picture for a mixed bulk state supported in the entanglement wedge $\mathrm{EW}(A)$. The shaded region denotes bulk degrees of freedom in $\mathrm{EW}(A)$, and the red dotted line to the external system $r$ indicates that these bulk degrees of freedom are mixed because they are entangled with $r$. The complementary wedge $\mathrm{EW}(\bar{A})$ is taken to contain no relevant bulk degrees of freedom in this setup. We assume the bulk information in $\mathrm{EW}(A)$ is approximately recoverable from the boundary region $A$. Right: Circuit representation of the same setting. The logical state $\sigma^{(L)}$ is a mixed state on $\mathcal{H}_L$, encoded by the isometry $V$ into $\mathcal{H}_A \otimes \mathcal{H}_{\bar{A}}$. Independent recovery maps $R_A$ and $R_{\bar{A}}$ act on the boundary regions, producing the recovered reduced states $\sigma^{(R)}_{A_1}$. We focus on the recovery of bulk information from $A$, treating the resulting map as an effective channel from $\mathcal{H}_L$ to $\mathcal{H}_{A_1}$, with the joint state $\sigma^{(R)}_{A\bar{A}}$ encoding the correlations with the complementary region.
• Figure 5: Top: The proto-area surface in a holographic stabilizer codes is unchanged by the encoded logical information, similar to the RT surface in holography where the bulk background geometry is fixed. Bottom: a magic-enriched code where the encoding circuits are skewed away from the exact encoding maps now permits the area of the surface to become state-dependent, an effect observed in QES and systems with gravitational backreaction in holography.

Theorems & Definitions (29)

• Definition 3.1
• Definition 3.2
• Theorem 4.1
• proof
• Theorem 4.2
• proof
• Corollary 4.1
• Lemma 4.1
• Theorem 4.3
• proof