Quantum minimal surfaces from quantum error correction

TL;DR

This work extends holographic entanglement wedge ideas beyond state-independent codes by introducing state-specific bulk reconstruction and a quantum minimal surface prescription applicable to arbitrary quantum codes. It defines a robust area functional A_B via the Choi–Jamiołkowski state and proves an exact equivalence between complementary state-specific product unitary reconstruction and the holographic entropy formula, with a corresponding one-shot minimality property. The framework accommodates approximate and non-isometric codes, thereby addressing more realistic holographic scenarios (e.g., black hole interiors and the Page curve) and enriching the notion of emergent bulk geometry from boundary entanglement. It also surveys subtleties arising from algebras with centers and discusses extensions toward quantum extremality and future connections to gravitational dynamics.

Abstract

We show that complementary state-specific reconstruction of logical (bulk) operators is equivalent to the existence of a quantum minimal surface prescription for physical (boundary) entropies. This significantly generalizes both sides of an equivalence previously shown by Harlow; in particular, we do not require the entanglement wedge to be the same for all states in the code space. In developing this theorem, we construct an emergent bulk geometry for general quantum codes, defining "areas" associated to arbitrary logical subsystems, and argue that this definition is "functionally unique." We also formalize a definition of bulk reconstruction that we call "state-specific product unitary" reconstruction. This definition captures the quantum error correction (QEC) properties present in holographic codes and has potential independent interest as a very broad generalization of QEC; it includes most traditional versions of QEC as special cases. Our results extend to approximate codes, and even to the "non-isometric codes" that seem to describe the interior of a black hole at late times.

Figures (5)

• Figure 1: An AdS/CFT setup without complementary state-independent recovery. Between the two candidate extremal surfaces lies a black hole with horizon area much greater than the difference in area between the two surfaces, $A_\mathrm{BH} \gg A_{\gamma_2} - A_{\gamma_1}$. Neither $B$ nor $\overline{B}$ can reconstruct operators in $b'$ in a state-independent way.
• Figure 2: Two illustrations of the quantum codes used throughout this paper. (a) A linear map $V: \otimes_i \mathcal{H}_{b_i} \to \mathcal{H}_B \otimes \mathcal{H}_{\overline{B}}$. (b) This linear map encompasses situations in holography. Each input leg corresponds to a bulk point or subregion, while the output legs correspond to boundary subregions. The map $V$ is implicit.
• Figure 3: The Choi-Jamiolkowski state $\ket{\mathrm{CJ}}$. Angled lines represent maximally entangled states, with one half of the maximally entangled state input into $V$.
• Figure 4: An AdS/CFT setup in which a unitary acting within an entanglement wedge changes the location of the quantum minimal surface. On the left, we have two black holes in an entangled pure state. Because of the large amount of entanglement, $\gamma_1$ is quantum minimal and therefore both black holes are inside the entanglement wedge of $B$. On the right, we have the situation after acting on both black holes with a unitary $U_{bb'}$ that maps the original state to a factorized pure state. Now $\gamma_2$ is quantum minimal. This unitary has changed the entropies $S(B)$ and $S(\overline{B})$ and therefore cannot be represented by a unitary on $B$.
• Figure 5: An example tensor network. The blue circles each represent tensors, four-partite states on the union of the small red circle and three black legs touching that blue circle. Each tensor can be viewed as a map from a state on the red circle to a state on the black legs. A solid black line between tensors represents postselection of the state on one leg from each tensor onto the state $\ket{\mathrm{MAX}}$. The entire tensor network forms a map from states on the union of the small red circles to the union of the boundary legs, which are those legs crossing the outer circle.

Theorems & Definitions (93)

• Definition 2.1
• Remark 2.2
• Remark 2.3
• Definition 2.4
• Remark 2.5
• Remark 2.6
• Definition 2.7
• Theorem 2.8
• proof
• Theorem 2.9