Lorentzian threads and nonlocal computation in holography

TL;DR

The paper identifies a fundamental limitation of the single-flavor Lorentzian-thread formulation of CV complexity in capturing subregion complexity and its inequalities. It then develops a multi-flavor measure-based program that assigns separate thread families to each boundary subregion, producing both subregion and full-state complexities; in bipartite and multipartite cases this leads to soft and tight superadditivity and reveals a binding contribution via the hole. To maintain micro-level consistency with the complexity cone, the authors perform a change of basis to a set of generalized Lorentzian hyperthreads, which act as intrinsically nonlocal gates enabling nonlocal computations in the CFT while preserving macroscopic complexity relations. This framework suggests that nonlocal gates are an unavoidable ingredient for faithfully encoding holographic complexity and motivates further work connecting these generalized gates to explicit tensor-network models and other holographic prescriptions.

Abstract

Recent advances in holography and quantum gravity have shown that CFTs with classical gravity duals can implement nonlocal quantum computation protocols that appear local from the bulk perspective. We examine the extent to which current prescriptions for holographic complexity support this claim, focusing on the Complexity=Volume (CV) proposal. The reformulation of CV in terms of Lorentzian threads suggests that bulk computations are performed with local gates. However, we find that the original formalism is insufficient when it comes to analyzing the complexity of subsystems and their inequalities. Specifically, standard Lorentzian threads cannot account for the negativity of `mutual complexity' and its higher-partite generalizations. To address this deficiency, we modify the Lorentzian threads program by introducing multiple flavors of threads. Our analysis reveals that an optimal solution for this new program implies the existence of additional types of gates that enable nonlocal computations in the dual CFT. We give a tentative interpretation of the multiflavor program in terms of Lorentzian `hyperthreads,' in analogy with the Riemannian case.

Figures (7)

• Figure 1: Boundary region $A$, boundary Cauchy slice $\sigma(A)$, and bulk Cauchy slice $\Sigma(A)$. The WdW patch associated with this Cauchy slice, depicted in blue, is a bulk causal diamond $D(A)$ anchored to $\sigma(A)$.
• Figure 2: Discretization of the bulk slice $\Sigma(A)$ using a physical tensor network. On the right, two threads of different thicknesses are illustrated. The number of degrees of freedom these threads act upon represents the level of $k$-locality in the bulk quantum circuit.
• Figure 3: Illustration of a bipartite system: the boundary Cauchy slice $\sigma(A)$ is divided into two regions, $\sigma_1(A)$ and $\sigma_2(A)$, which share a common HRT surface, $\gamma$. The surfaces $\Sigma_1(A)$ and $\Sigma_2(A)$ are bounded by $\sigma_1(A) \cup \gamma$ and $\sigma_2(A) \cup \gamma$, respectively, with their corresponding domains of dependence, $D_1(A)$ and $D_2(A)$, depicted in red and green.
• Figure 4: Threads in the manifold $\mathcal{M}$ with measures $\mu_{1}$ or $\mu_{2}$, depicted in orange and blue, respectively. In this scenario, the complexity of $\sigma_1(A)$ is determined by the number of orange threads traversing $D_1(A)$, while the complexity of $\sigma_2(A)$ corresponds to the number of blue threads passing through $D_2(A)$. The complexity of $\sigma(A)$ is calculated as half of the total number of threads present, regardless of their color.
• Figure 5: Left: elementary threads of the multi-flavor program. Right: A Lorentzian hyperthread obtained by combining two of these elementary threads. The resulting weights, +1, +1 and +1, signaling the contributions to subregion complexities and the complexity of the full state, are analogous to those in Riemannian hyperthreads. However, in the Lorentzian case, they lead to violations of micro-superadditivity.

Theorems & Definitions (6)

• Theorem 1
• Theorem 2
• Theorem 2
• proof
• Theorem 2
• proof