Timelike Subregion Complexity in Holography

• Timelike subregion complexity is a geometric concept in holography that defines the minimal quantum circuit depth for states associated with timelike intervals.
• The CV and CA approaches compute complexity using extremal bulk volumes and on-shell gravitational actions, revealing unique causal structure and phase transitions.
• This framework underpins tensor network interpretations and connects directly to black hole interior dynamics and emergent time phenomena in quantum gravity.

Timelike subregion complexity is a geometric probe of quantum information in gravitational systems, defined within the holographic correspondence in terms of the “complexity=volume” (CV) or “complexity=action” (CA) conjectures. Unlike traditional spatial (spacelike) subregions, timelike subregion complexity explores Lorentzian or causal intervals on the boundary, encoding the minimal quantum circuit depth required to prepare states corresponding to timelike intervals. This observable reveals distinctive ultraviolet divergences, causal behavior, and phase structure, and provides insight into black-hole interiors, the emergence of time, and the structure of @@@@1@@@@ within AdS/CFT and related settings (Alishahiha, 29 Oct 2025, Prihadi et al., 26 Jan 2026, Aguilar-Gutierrez et al., 2024, Pedraza et al., 2021, Alishahiha et al., 2018).

1. Definitions and Geometric Framework

Timelike subregion complexity is defined by selecting a Lorentzian interval or strip—i.e., a connected timelike region $\mathcal{T}$—on the conformal boundary of an asymptotically AdS spacetime or, more generally, on the boundary of a black-hole or cosmological solution. This region typically takes the form

$$\mathcal{T} = \{\, t \in [\!-\Delta t/2, +\Delta t/2],\, \vec{x}_\perp = 0 \,\}$$

where $t$ measures boundary time and $\vec{x}_\perp$ parameterizes transverse spatial directions.

The central CV prescription computes the complexity $\mathcal{C}_V(\mathcal{T})$ as proportional to the bulk volume of a co-dimension-one extremal hypersurface $\Sigma$ that is anchored on the boundary of $\mathcal{T}$: $$\mathcal{C}_V(\mathcal{T}) = \frac{\mathrm{Vol}(\Sigma)}{G_N L}$$ where $G_N$ is the bulk Newton constant, $L$ is the AdS or curvature scale, and $\mathrm{Vol}(\Sigma)$ is regularized by removing leading divergences as $r \to 0$.

The problem reduces to extremizing a volume functional subject to the Lorentzian (timelike) anchoring, often requiring careful handling of multiple branches (spacelike, timelike, and horizon-penetrating) (Alishahiha, 29 Oct 2025, Prihadi et al., 26 Jan 2026).

2. Lorentzian Flows, Threads, and the Max-Flow Min-Cut Duality

An alternative formulation, generalizing bit threads, replaces the maximization over bulk slices with minimization over divergenceless, future-directed timelike flows $v^\mu$ satisfying

$$\nabla_\mu v^\mu = 0, \quad v^0 > 0, \quad -g_{\mu\nu}v^\mu v^\nu \geq \alpha^2$$

for fixed $\alpha$, typically set by the gravitational coupling. The minimal flux of $v^\mu$ through the boundary subregion equals the maximal volume of a homologous Cauchy slice; this is the Lorentzian analog of the max-flow min-cut theorem: $$\inf_v \Phi_R[v] = \alpha \sup_{\Sigma \sim R} \mathrm{Vol}(\Sigma)$$ Establishing this equivalence requires convex optimization and strong duality arguments. The resulting Lorentzian threads (“gatelines”) provide a tensor-network interpretation: each thread corresponds to a quantum gate required for circuit preparation of the target state, with complexity measuring the minimal number of threads intersecting the maximal slice (Pedraza et al., 2021).

3. Holographic Computation: Volume and Action Approaches

(a) Complexity=Volume (CV) for Timelike Subregions

For a timelike interval in the Poincaré patch of AdS$_{d+2}$,

$$ds^2 = \frac{R^2}{r^2}[ dr^2 + dy^2 - d\xi^2 + \xi^2 dX_{d-1}^2 ],$$

the extremal hypersurface parameterized by $\xi = f(r)$ produces a regularized volume functional with two branches: spacelike ($\xi^2 = r^2 + (T/2)^2$) and timelike ($\xi^2 = r^2 - (T/2)^2$ for $r \geq T/2$). The explicit expression after integrating over $\xi$ features a sum of two terms, corresponding to each branch, regulated by a cutoff $\epsilon$ (Alishahiha, 29 Oct 2025).

The complexity exhibits purely real values, even for Lorentzian embedding, with the universal ultraviolet divergence structure matching the spacelike case: $$\mathcal{C}_T \sim \sum_n \frac{c_n}{\epsilon^{k_n}} + (\text{universal log divergence for even } d)$$ with coefficients $c_n$ depending on $T$, $d$, and geometrical data (Alishahiha, 29 Oct 2025, Prihadi et al., 26 Jan 2026).

(b) Complexity=Action (CA) for Timelike Subregions

The CA approach computes the on-shell gravitational action over a region bounded by both null and timelike surfaces. The full diffeomorphism-invariant action includes the Einstein–Hilbert bulk term, Gibbons–Hawking–York (GHY) boundary terms on timelike slices, joint contributions at intersections, null boundary ambiguity terms, and holographic counterterms.

A salient feature is that the timelike boundary produces a contribution proportional to the classical free energy, $F\,\tilde\tau$, where $\tilde\tau$ is the duration of the interval. The final complexity formula decomposes into a linear-in-$\tilde\tau$ free-energy term plus a “joint” contribution from the null-null intersections: $$\mathcal{C}_{\text{timelike}} = \frac{F}{\pi}\,\tilde\tau + \frac{V_d L^d}{8\pi^2 G_N}\;\frac{\log f(r_p)}{r_p^d}$$ (Alishahiha et al., 2018). Subadditivity and mutual complexity inequalities apply, analogous to mutual information in entanglement theory.

4. Behavior in Black Hole and De Sitter Geometries

AdS Black Branes and Timelike Anchoring

In AdS black brane backgrounds,

$$ds^2 = \frac{R^2}{r^2}[ -f(r)dt^2 + dr^2/f(r) + dx^2 + dY_{d-1}^2 ], \quad f(r) = 1 - (r/r_h)^{d+1},$$

timelike subregion complexity exhibits branch structure:

• Trivial branch: constant-time hypersurfaces.
• Horizon-penetrating branch: surfaces that enter the black hole interior, with turning points $r_0 > r_h$.

Phase transitions can occur as the time interval $T$ increases: for small $T$, trivial branches dominate; for large $T$, horizon-penetrating configurations may minimize the CV functional (Alishahiha, 29 Oct 2025, Prihadi et al., 26 Jan 2026).

The ultraviolet structure remains universal, but the finite part is sensitive to the black hole interior. In the BTZ case ($d=2$), the UV-finite component of the renormalized volume arises entirely from the region inside the horizon, making timelike subregion complexity a sensitive probe of the black hole interior geometry (Prihadi et al., 26 Jan 2026).

Schwarzschild–de Sitter Space and Stretched Horizons

In static patch dS coordinates,

$$ds^2 = -f(r)dt^2 + dr^2/f(r) + r^2 d\Omega_{d-1}^2, \quad f(r) = 1 - 2\mu/r^{d-2} - r^2/L^2,$$

timelike subregion complexity arises when codimension-one extremal surfaces are anchored on stretched horizons, which are timelike surfaces near the cosmological or black hole horizons.

Depending on the anchoring regime (both stretches in cosmological, black hole, or mixed), various complexity proposals (CV, CV2.0, CA, CAny-on-CMC) exhibit distinct time dependencies:

• Hyperfast (divergent) growth for cosmological-only anchoring.
• Linear growth at late times for black hole anchoring.
• Time-independence in mixed anchoring. Constraints on extremal surfaces are fixed by the boundary Dirichlet data provided by the timelike stretches (Aguilar-Gutierrez et al., 2024).

5. UV Structure, Reality Properties, and Distinctions from Entropy

Unlike pseudo-entropy, which develops an imaginary component due to Lorentzian embedding (complex area from crossing light cones), timelike subregion complexity remains strictly real for any dimension and geometry considered (Alishahiha, 29 Oct 2025, Prihadi et al., 26 Jan 2026). This real-valuedness persists for both the total and renormalized (finite) parts.

The universal ultraviolet divergences are structurally identical to those found in spacelike complexity:

• Power-law divergences in $\epsilon$, with $1/\epsilon^d$ (area law), $1/\epsilon^{d-2}$, etc.
• Universal $\log\epsilon$ divergences for even $d$. All finite corrections are determined by the causal and geometric structure of the underlying black hole or AdS configuration.

Distinctively, phase transitions and switchings between branches signal behavior not amenable to Wick rotation from the spacelike problem; the phase structure is genuinely Lorentzian (Alishahiha, 29 Oct 2025).

6. Extensions, Tensor-Network Interpretation, and Bulk Dynamics

Timelike subregion complexity admits a tensor-network circuit interpretation: Lorentzian flows or threads (“gatelines”) map to the minimal number of unitary gates needed to build the corresponding target state from a reference (often vacuum) state. Complexity inequalities (e.g., monotonicity, conditional complexity bounds, and three-region inequalities) hold due to the nesting property of maximal bulk slices and associated flows (Pedraza et al., 2021).

Perturbations of the AdS background induce a shift in the thread form $u_\eta$, governed by the bulk linearized symplectic form. The closedness of this form is equivalent to the linearized Einstein equations, leading to a “first law of holographic complexity”: infinitesimal changes in complexity correspond to the symplectic two-form in boundary theory, which is equivalent to the bulk volume perturbation (CV duality) and to gravitational equations of motion (Pedraza et al., 2021).

A refined ensemble-average definition of complexity, integrating over all Cauchy slices (weighted by the path-integral action), unifies CV and CA proposals and is closely related to path-integral complexity and Hartle–Hawking extremization.

In summary, timelike subregion complexity provides a robust, geometric, and causally sensitive probe of holographic states, with strict real-valuedness, structural analogies to circuit depth, and distinctive connections to black hole interiors, phase transitions, and emergent time in quantum gravity (Pedraza et al., 2021, Alishahiha, 29 Oct 2025, Prihadi et al., 26 Jan 2026, Aguilar-Gutierrez et al., 2024, Alishahiha et al., 2018).