⚛️ high-energy theory

Timelike Holographic Complexity

Motivated by the pseudo-entropy program, this paper extends the holographic Complexity-equal-Volume framework to timelike subregions in AdS and black brane geometries, demonstrating that the resulting timelike complexity remains purely real with universal UV divergences, thereby distinguishing its geometric nature from the complex-valued pseudo-entropy.

Original authors: Mohsen Alishahiha

Published 2026-02-06

📖 6 min read🧠 Deep dive

CC BY 4.0

Original authors: Mohsen Alishahiha

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

The Big Picture: Measuring "How Hard" It Is to Change the Future

Imagine you have a complex machine (the universe) and you want to know how much "work" or "effort" it takes to turn the machine from one setting (State A) into another setting (State B). In the world of quantum physics, this "effort" is called Complexity.

Usually, scientists measure this complexity by looking at a snapshot of the machine at a single moment in time (like taking a photo). This paper asks a different question: What if we look at a movie clip instead of a photo? What if we measure the effort required to evolve a system through time?

The author, Mohsen Alishahiha, explores this idea using a famous theory called Holography. Think of Holography as a cosmic translation tool: it says that a 3D universe with gravity (like a black hole) can be perfectly described by a 2D surface without gravity (like a computer screen). The paper tries to translate the concept of "time-evolution complexity" from the 2D screen into the 3D universe.

The Main Discovery: Real vs. Imaginary Numbers

In recent years, physicists discovered a way to measure "entanglement" (how connected two things are) across time. They call this Pseudo-Entropy. When they calculated this, they got complex numbers (numbers with a "real" part and an "imaginary" part, like $3 + 4i$ ). In physics, "imaginary" numbers often signal that something tricky is happening with time and causality.

The Surprise:
Alishahiha calculated the Complexity for these same time-based regions. He expected it to also be a "complex" number (with an imaginary part).
It wasn't.
The result was purely real.

The Analogy:
Imagine you are measuring the distance between two cities.

Pseudo-Entropy is like measuring the distance while also counting the number of time-travel paradoxes involved. Because time travel is weird, your ruler gives you a "weird" number (complex).
Timelike Complexity is like measuring the actual physical road distance you would drive. Even though the road goes through time, the distance you drive is a solid, real number.

The paper argues that while "entanglement" across time gets messy and imaginary, "complexity" (the effort to change the state) remains a solid, geometric, real-world quantity.

How They Did It: The "Volume" Recipe

The paper uses a specific recipe called Complexity = Volume (CV).

The Idea: The amount of "complexity" in a region is equal to the volume of the space inside the holographic universe that is "locked" behind that region.
The Twist: Usually, you measure the volume behind a static wall (a spatial slice). Here, the author measures the volume behind a "time wall" (a slice that moves through time).

He looked at two scenarios:

1. The Empty Universe (Pure AdS)

He calculated the volume of space enclosed by a time-based region in a universe with no black holes.

Result: The volume was perfectly real.
Why it matters: Even though the math involved "imaginary" branches of surfaces (like a ghost path that exists in the math but not in reality), when you add up the real volume, the imaginary parts cancel out perfectly. It's like having a bank account where you have a deposit and a withdrawal of the exact same imaginary amount; they cancel, leaving you with a real balance.

2. The Black Hole Universe (AdS Black Branes)

Next, he added a black hole to the mix. Black holes are tricky because they have an "event horizon" (a point of no return).

The Question: Can the "time wall" of our measurement dip inside the black hole?
The Result: Yes, it can. The surface can cross the horizon and go inside.
The Surprise: Even when the surface goes deep inside the black hole, the calculated complexity stays real. It does not turn into a complex number like pseudo-entropy does.

The "Depth Limit" of the Black Hole

One of the most interesting findings is a limit on how deep the complexity can probe inside a black hole.

The Analogy:
Imagine the inside of a black hole is a deep ocean. You are a diver (the complexity measurement) trying to go deeper and deeper.

In standard physics, you might think you can dive forever.
Alishahiha found that there is a maximum depth ( $r_{max}$ ) you can reach.
If you try to dive deeper than this limit, the "diving equipment" (the math describing the surface) breaks down. There is no valid path for the surface to exist there.

This limit depends on the "dimensions" of the universe (how many directions space has). In higher dimensions, the "ocean" gets steeper, and you can't dive as deep. The paper calls this a "Complexity Horizon." It's not a wall of light (like the event horizon); it's a wall of possibility. Beyond this point, the concept of "complexity for a finite time" simply doesn't exist.

The Growth Pattern: How Complexity Changes Over Time

The paper also looked at how this complexity grows as time passes:

Early Times: When the time interval is short, the complexity grows slowly (like a power law). It doesn't "feel" the black hole yet.
Late Times: As time goes on, the complexity starts growing linearly (a straight line). This is a very important feature. In chaotic quantum systems (like a messy room or a complex computer), complexity is expected to grow at a steady, linear rate for a long time before it stops.
- The paper shows that this linear growth happens even inside the black hole, up until the "Complexity Horizon" is reached.

Summary of the Takeaway

Complexity is Real: Unlike other time-based quantum measurements that produce "imaginary" numbers, this new way of measuring complexity (Timelike Holographic Complexity) always produces a real, physical number. This suggests it is a robust, geometric property of the universe.
It Probes the Inside: This complexity measure can "see" inside black holes, unlike some other measures that get stuck at the horizon.
There is a Limit: You cannot probe the entire black hole interior with this method. There is a hard geometric limit ( $r_{max}$ ) determined by the size and shape of the universe.
It Matches Chaos: The way this complexity grows over time (slowly at first, then steadily linear) matches what we expect from chaotic quantum systems, confirming that this is a valid way to measure quantum complexity.

In short, the paper builds a new, sturdy ruler for measuring the "effort" of time evolution in the universe. It turns out that even when looking through the lens of a black hole, this ruler gives us a clear, real, and finite answer.

Technical Summary: Timelike Holographic Complexity

Problem Statement
Motivated by the "pseudo-entropy" program, which extends entanglement entropy to transitions between non-identical quantum states ( $|\psi_i\rangle \to |\psi_f\rangle$ ) and timelike subregions, this work investigates the extension of holographic quantum complexity to timelike domains. While pseudo-entropy is generically complex-valued due to the non-Hermitian nature of the transition matrix, the paper addresses whether holographic complexity—specifically within the "Complexity=Volume" (CV) framework—admits a consistent timelike generalization. A central question is whether the resulting quantity retains a real, geometric character or acquires imaginary contributions similar to pseudo-entropy, particularly when extremal surfaces penetrate black hole horizons.

Methodology
The author employs the subregion CV conjecture, defining holographic complexity as proportional to the bulk volume enclosed by extremal surfaces anchored to a boundary subregion. The analysis proceeds in two stages:

Pure AdS Spacetime: The study focuses on timelike hyperbolic subregions in $AdS_{d+2}$ . The author utilizes explicit solutions for extremal surfaces derived in prior work [8, 9], which consist of two branches: a spacelike branch and a timelike branch. The bulk volume is computed as the region enclosed between these two branches, bounded by the timelike boundary interval of duration $T$ .
AdS Black Brane Geometries: The analysis is extended to thermal states dual to AdS black branes. The author solves the extremization problem for timelike embeddings $t(r)$ $t (r)$ in the presence of an event horizon. This yields two classes of solutions based on the location of the turning point $r_0$ $r_{0}$ :
- Exterior solutions: The turning point lies outside the horizon ( $r_0 < r_h$ ).
- Interior solutions: The turning point lies inside the horizon ( $r_0 > r_h$ ).
  The complexity is calculated by integrating the bulk volume enclosed by these surfaces, carefully handling UV divergences (via cutoff $\epsilon$ ) and IR divergences near the horizon.

Key Contributions and Results

Reality of Timelike Complexity: A primary finding is that timelike subregion complexity, unlike pseudo-entropy, is strictly real. In pure AdS, the bulk volume is computed over a real Lorentzian domain ( $r \ge T/2$ ). Although the timelike branch of the extremal surface contributes an imaginary term to entanglement entropy (via analytic continuation across the light cone), the volume integral for complexity yields a purely real result. This holds even when the extremal surface penetrates the black hole interior in thermal states.
Divergence Structure: The UV divergence structure of timelike complexity mirrors that of spacelike subregion complexity, featuring power-law divergences associated with the boundary volume and universal logarithmic terms in even dimensions. However, the finite terms differ; notably, for odd-dimensional AdS backgrounds, the universal constant term present in the spacelike case is absent in the timelike case.
Dynamics in Black Brane Backgrounds:
- Early Times (Deep Exterior): When the turning point is far outside the horizon ( $r_0 \ll r_h$ ), the finite complexity scales as $C_{finite} \sim T^{1-d}$ , behaving similarly to pure AdS.
- Late Times (Near Horizon): As the boundary time $T$ increases, the turning point approaches the horizon. The complexity exhibits linear growth, $C_{finite} \sim T$ , with a slope proportional to the black brane's ADM energy ( $E$ ). This linear growth persists even when the surface penetrates the horizon.
- Interior Penetration Limit: The paper identifies a dimension-dependent upper bound on how far timelike extremal surfaces can penetrate the black hole interior. A valid turning point exists only for $r_0 < r_{max}(d)$ , where:
  $r_{max}(d) = r_h \left( \frac{2d}{d-1} \right)^{\frac{1}{d+1}} \quad (d > 1)$
  Beyond this "complexity horizon," the effective potential for the extremal surface equation develops a double root, and no real solutions exist for finite boundary time intervals. For $d=1$ (BTZ), no such geometric cutoff exists, though redshift effects still bound the growth.
Distinction from Entanglement Measures: The paper highlights that while pseudo-entropy acquires imaginary parts when surfaces cross horizons (reflecting phase information of state transitions), timelike complexity remains real. This suggests complexity is a geometric observable measuring the "depth" of a causal process rather than an analytically continued entropic quantity.

Significance and Claims
The paper claims that timelike subregion complexity provides a robust, well-defined holographic observable for studying quantum processes in thermal states. Its significance lies in several aspects:

Geometric Interpretation: The reality of the result reinforces the interpretation of complexity as a geometric volume in Lorentzian bulk, distinct from the complex-valued nature of pseudo-entropy.
Complexity Horizon: The existence of a maximal penetration depth $r_{max}(d)$ introduces a new geometric constraint on the black hole interior. This "complexity horizon" implies that the interior is stratified into layers of decreasing informational accessibility for a boundary observer with finite temporal resolution.
Dynamical Features: The emergence of linear growth at late times ( $C \sim T$ ) aligns with the expected behavior of quantum complexity in chaotic systems, providing evidence that the dual boundary quantity is indeed a notion of quantum complexity.
Complementarity: The work complements entanglement-based probes (like pseudo-entropy) by offering a measure that avoids analytic continuation subtleties and imaginary contributions, while still probing causal structures and interior regions inaccessible to standard spatial subregion complexity.

The author concludes that this framework offers a sharp geometric tool for probing black hole interiors and thermalization dynamics, with the dimension-dependent bound $r_{max}(d)$ serving as a concrete target for any consistent boundary definition of timelike complexity.