On Infinite Tensor Networks, Complementary Recovery and Type II Factors

TL;DR

The paper develops an operator-algebraic framework for the continuum limit of layered tensor networks using inductive limits, with a focus on holographic quantum error-correcting codes that exhibit complementary recovery. It shows that HaPPY-type networks yield boundary subregion algebras that are hyperfinite type II factors in the infinite-layer limit, with the RT surface contributing a $II_1$ piece and bulk/ancilla sectors contributing a type I part, resulting in an overall type $II_{ obreaksp{\infty}}$ algebra. The approach extends to Majorana-dimer and dimerized networks, where similar decompositions lead to $II$ factors, while MERA networks lacking complementary recovery may exhibit type III algebras in suitable limits. These results establish a concrete link between entanglement geometry in tensor networks and operator-algebra classifications, offering new tools for holography, quantum gravity, and quantum information theory.

Abstract

We initiate a study of local operator algebras at the boundary of infinite tensor networks, using the mathematical theory of inductive limits. In particular, we consider tensor networks in which each layer acts as a quantum code with complementary recovery, a property that features prominently in the bulk-to-boundary maps intrinsic to holographic quantum error-correcting codes. In this case, we decompose the limiting Hilbert space and the algebras of observables in a way that keeps track of the entanglement in the network. As a specific example, we describe this inductive limit for the holographic HaPPY code model and relate its algebraic and error-correction features. We find that the local algebras in this model are given by the hyperfinite type II$_\infty$ factor. Next, we discuss other networks that build upon this framework and comment on a connection between type II factors and stabilizer circuits. We conclude with a discussion of MERA networks in which complementary recovery is broken. We argue that this breaking possibly permits a limiting type III von Neumann algebra, making them more suitable ansätze for approximating subregions of quantum field theories.

Figures (16)

• Figure 1: (a) In the Araki-Woods construction of a type II von Neumann algebra $\mathcal{A}$, one constructs an infinite series of maximally entangled pairs of qubits (EPR pairs), one side of which constitutes a subsystem $A$ (with complement $A^c$) on which operators in $\mathcal{A}$ act. (b) A layered tensor-network code forms an isometric map between bulk qubits (red) and boundary qubits (white). Layers can be added iteratively until both the number of bulk and boundary qubits become infinite. The tensor network contraction (black connecting lines) itself acts as a projection onto EPR pairs. (c) For a holographic tensor-network code with complementary recovery, a bipartition of the boundary qubits induces a clean bipartition of the bulk qubits along a Ryu-Takayanagi surface $\gamma_A$. Adding more layers increases EPR-like entanglement across $\gamma_A$, again ultimately leading to a type II von Neumann algebra for operators acting on $A$ in the limit of infinitely many layers, provided that boundary states contain only finite bulk entanglement.
• Figure 2: Turning a holographic tensor network into an encoding circuit. (a) We take a small HaPPY code with four contracted perfect tensors and consider a boundary bipartition into $A$ and $A^c$. From each region, two logical qubits (red dots) can be recovered, forming the "bulk regions" $a$ and $a^c$, separated by a cut $\gamma_A$ through the tensor network. (b) Using the property that the six-leg perfect tensor acts as a unitary $U$ from any three legs to the remaining three, we can reorganize the tensor network into a circuit from the logical qubits in $a$ and $a^c$ to the physical qubits in $A$ and $A^c$. In this circuit, some of the tensor contractions become insertions of maximally entangled pairs into the circuit. Three of such pairs cross between $A$ and $A^c$, leading to an entanglement entropy $S_A = \log 3 + S_a$. (c) The generic holographic encoding circuit in terms of two unitaries $U_A$ and $U_{A^c}$ (or equivalently, isometries $V_A$ and $V_{A^c}$), with resource states $|\chi_a\rangle$ and $|\chi_{a^c}\rangle$ contributing only to entanglement within each subregion and $|\chi_{\gamma}\rangle$ contributing to the entanglement between $A$ and $A^c$. For HaPPY codes, these resource states are copies of maximally entangled pairs.
• Figure 3: Subregion algebra reconstruction in the HaPPY model. (a) A boundary bipartition into $A$ and $A^c$ of the full $\{5,4\}$ HaPPY code. The Ryu-Takayanagi cut $\gamma_A$ separates the bulk into two wedges $a$ and $a^c$, logical qubits (red dots) in which are reconstructable (only) on $A$ and $A^c$ (white dots), respectively. (b) Mapping the full boundary subregion algebra $\mathcal{A}_A$ back into the bulk: Removing $a^c$ and bonds corresponding to (one choice of) ancillas $\ket{\chi_a}$ turns the remaining tensors into a unitary circuit (following Fig. \ref{['fig:happy_to_unitary']}). $\mathcal{A}_A$ is unitarily mapped to the bulk algebra $\mathcal{A}_a$ (red), the wedge ancilla algebra $\mathcal{A}_{\chi_a}$ (black), and the Ryu-Takayanagi algebra $\mathcal{A}_{\gamma_A}$ (gray). (c) With the ancilla bonds removed, operator-pushing a logical operator (here $\bar{X}$ acting on one bulk qubit) follows a unique flow towards the boundary, resulting in a unique boundary representation of the logical operator.
• Figure 4: Subregion algebra mapping with one layer of the HaPPY model. (a) A single vertex inflation layer of the "opened-up" HaPPY code of Fig. \ref{['fig:happy_to_unitary_algebras']}, acting as a unitary map from the subregion algebra $\mathcal{A}_A^\Lambda$ at layer $\Lambda$ and the algebras of the degrees of freedom of the new layer, the bulk algebra $\mathcal{A}_{\delta a}^\Lambda$ (red), wedge ancilla algebra $\mathcal{A}_{\chi_{\delta a}}^\Lambda$ (black), and Ryu-Takayanagi algebra $\mathcal{A}_{\delta \gamma_A}^\Lambda$ (gray) to the subregion algebra $\mathcal{A}_A^{\Lambda+1}$ on the next layer. (b) The generic form of a layer of the HaPPY code with ancillas, written as a circuit diagram with the two unitary subregion maps $U_A^{\Lambda,\Lambda+1}$ (highlighted in (a)) and $U_{A^c}^{\Lambda,\Lambda+1}$.
• Figure 5: Commutative diagram summarizing the structure required for an inductive limit of codes. The sequence of logical Hilbert spaces and their isometries is shown on the top diagram, while the sequence of algebras and their operator pushing maps is shown on the bottom diagram. We ask that the arrows of the same color on the commutative diagram satisfy the compatibility conditions \ref{['eq:pushing_layer_commutativity']} (for the red ones), and \ref{['eq:pushing_layer_commutativity2']} (for the blue and green ones). A similar diagram to the bottom one must also hold for commutant algebras and maps.

Theorems & Definitions (6)

• Theorem 1
• Theorem 2
• Theorem 3
• Theorem 4
• Lemma 1
• Corollary 1