Complexity equals Volume (CV) in Holography

• CV is a holographic principle equating quantum complexity with the volume of an extremal bulk hypersurface anchored on a boundary time slice.
• The prescription uses maximal spatial slices in black-hole settings and subregion entanglement wedges to extract geometric insights from quantum states.
• Extensions of CV to de Sitter metrics, higher-curvature theories, and information-geometric analogues highlight its implications for nonlocal gate models and normalization ambiguities.

Complexity equals Volume (CV) is a holographic prescription that associates quantum complexity with the volume of a codimension-one bulk hypersurface anchored on a specified boundary time slice. In its standard form, the proposal is written as

$\mathcal{C}_V=\frac{V(\Sigma)}{G_N\,\ell},$

or equivalently $V(\Sigma)/(G_N R)$, where $\Sigma$ is an extremal or maximal-volume slice, $G_N$ is Newton’s constant, and the length scale is typically taken to be the AdS radius or another reference scale (Fu et al., 2018, Balushi et al., 2020). The same idea has been extended to subregions, finite-cutoff holography, time-dependent quenches, rotating and charged black holes, de Sitter settings, higher-curvature theories, and even information-geometric analogues of Krylov complexity (Geng, 2019, Astaneh, 2024, Fang et al., 2 Jun 2026, Zhai et al., 2024).

1. Standard prescription and normalizations

In the two-sided black-hole setting, CV assigns the complexity of a thermofield-double-like state on boundary times $(t_L,t_R)$ to the maximal spatial volume of a bulk hypersurface $\Sigma_{t_L,t_R}$ connecting the two asymptotic boundaries (Hashimoto et al., 2021). For static metrics and at moments of time-reflection symmetry, the relevant hypersurface is often the $t=0$ slice, which is extremal and, in the cases treated explicitly, maximal (Fu et al., 2018). In subregion versions, following the entanglement-wedge construction, one replaces the full maximal slice by the codimension-zero region bounded by a boundary subregion and its RT or HRT surface (Geng, 2019, Ageev et al., 2018).

The normalization is not unique. Several papers use the standard $V/(G_N\ell)$ form, with $\ell$ taken to be the AdS radius or kept as an arbitrary scale (Fu et al., 2018, Mahapatra et al., 2018). A refinement proposed for stationary black holes replaces this by

$\mathcal{C}(t)=\frac{V(t)}{A_P\,\tau_f},$

where $V(\Sigma)/(G_N R)$0 and $V(\Sigma)/(G_N R)$1 is the maximal proper time from the horizon to the late-time “final slice”; this was argued to remove the apparent mismatch between large and small black holes and to recover the universal scaling $V(\Sigma)/(G_N R)$2 (Couch et al., 2018). Other contexts require adapted choices: for timelike extremal slices in Schwarzschild–de Sitter, the volume is imaginary and one introduces an imaginary scale $V(\Sigma)/(G_N R)$3 so that $V(\Sigma)/(G_N R)$4 is real and positive (Fang et al., 2 Jun 2026).

A closely related observable is the complexity of formation, defined by subtracting a reference state. In AdS black holes this is often taken as the difference between the wormhole volume and the corresponding vacuum or massless-BTZ contribution, with the subtraction fixing the additive ambiguity and removing UV divergences (Fu et al., 2018, Balushi et al., 2020).

2. Extremal hypersurfaces, conserved quantities, and reconstruction formulas

For static geometries of the form

$V(\Sigma)/(G_N R)$5

the maximal slice can be described in ingoing Eddington–Finkelstein coordinates by embedding functions $V(\Sigma)/(G_N R)$6. Defining

$V(\Sigma)/(G_N R)$7

the volume functional takes the form

$V(\Sigma)/(G_N R)$8

with a conserved quantity $V(\Sigma)/(G_N R)$9 because $\Sigma$0 appears only through $\Sigma$1 (Hashimoto et al., 2021). The turning point $\Sigma$2 of the slice satisfies

$\Sigma$3

and the growth rate simplifies to

$\Sigma$4

with the late-time limit controlled by the interior radius $\Sigma$5 maximizing $\Sigma$6 (Hashimoto et al., 2021).

This structure underlies a reconstruction program: given $\Sigma$7 and the exterior geometry, one can invert Abel-type integral equations to recover the black-hole interior metric function $\Sigma$8, and, together with Hartman–Maldacena entropy growth, also $\Sigma$9 (Hashimoto et al., 2021). In BTZ, the method reconstructs the exact interior relation $G_N$0 (Hashimoto et al., 2021).

A complementary formulation uses the conserved volume current $G_N$1, the future-directed unit normal to a maximal foliation. Since the trace of the extrinsic curvature vanishes on each leaf, $G_N$2, and the flux of $G_N$3 through a hypersurface equals its volume (Couch et al., 2018). Under the assumptions stated there, a boundary foliation determines a bulk maximal foliation without gaps, and the horizon flux of $G_N$4 yields a second-law-like statement for interior CV complexity (Couch et al., 2018).

3. Exact AdS$G_N$5 results, topology, and nonlocality

The sharpest exact result for CV is the AdS$G_N$6 wormhole formula derived at a moment of time symmetry. For orientable multi-boundary wormholes with genus $G_N$7 and $G_N$8 asymptotic regions, the time-symmetric slice is a hyperbolic surface $G_N$9 with Euler character

$(t_L,t_R)$0

Using the Brown–Henneaux relation $(t_L,t_R)$1, Gauss–Bonnet on the compact interior $(t_L,t_R)$2, and subtraction against $(t_L,t_R)$3 copies of the $(t_L,t_R)$4 BTZ black hole, one finds

$(t_L,t_R)$5

or explicitly

$(t_L,t_R)$6

(Fu et al., 2018). The derivation uses

$(t_L,t_R)$7

so the result depends only on topology and not on any continuous moduli (Fu et al., 2018).

Because the exterior BTZ contributions cancel in the relative complexity, $(t_L,t_R)$8 is independent of the temperatures $(t_L,t_R)$9 of the outer regions and of the Fenchel–Nielsen moduli of the interior geometry (Fu et al., 2018). This exact independence led to a strong conclusion: any circuit model reproducing CV in this regime cannot be based on strictly local gates, since locally thermofield-double-like entanglement between thermal-sized patches carries the same CV cost regardless of separation (Fu et al., 2018).

The same topology can also modify full time dependence. In the simplest Lorentzian three-boundary AdS$\Sigma_{t_L,t_R}$0 wormhole, CV was computed at all times and found to grow nonlinearly and saturate at late times, in contrast to the familiar linear growth of the eternal BTZ geometry (Zolfi, 2023). In the symmetric pair-of-pants case, the complexity of formation approaches a finite constant $\Sigma_{t_L,t_R}$1 in the large-$\Sigma_{t_L,t_R}$2 limit (Zolfi, 2023).

4. Time dependence, late-time growth, and thermodynamic interpretations

For two-sided AdS black holes, CV typically exhibits monotone growth in time and a constant late-time slope. In the Einstein–dilaton family generated by $\Sigma_{t_L,t_R}$3, the growth rate approaches a constant from below, and numerically satisfies

$\Sigma_{t_L,t_R}$4

with saturation only at sufficiently high temperature (Mahapatra et al., 2018). The same paper emphasizes that, unlike CA in that model, CV behaves qualitatively as in earlier AdS black-hole studies: no overshoot of the late-time limit and no Lloyd-bound-type pathology was reported (Mahapatra et al., 2018).

For rotating AdS black holes, the complexity of formation appears to be controlled by thermodynamic volume rather than entropy. In equal-spin Myers–Perry–AdS black holes, the proposed large-black-hole scaling is

$\Sigma_{t_L,t_R}$5

for both CV and CA, with $\Sigma_{t_L,t_R}$6 the thermodynamic volume and $\Sigma_{t_L,t_R}$7 (Balushi et al., 2020). In slowly accelerating Kerr–AdS spacetimes with conical deficits, the paper computes CV only for the complexity of formation at $\Sigma_{t_L,t_R}$8, finding that it increases with the average and differential deficits near the static limit but decreases with them near extremality (Zhang et al., 2022).

CV has also been adapted to de Sitter holography. In Schwarzschild–de Sitter, timelike extremal slices anchored either on a stretched horizon or on $\Sigma_{t_L,t_R}$9 produce linear late-time growth in both static-patch and dS/CFT schemes, with identical asymptotic slope

$t=0$0

where $t=0$1 is the maximum of the effective potential $t=0$2 (Fang et al., 2 Jun 2026). In global de Sitter foliations of AdS, by contrast, the maximal volume scales with the spatial volume of the boundary slice, so at fixed cutoff $t=0$3; it is even in $t=0$4, minimized at $t=0$5, and grows exponentially at large $t=0$6 rather than exhibiting any finite-time divergence (Parihar et al., 23 Apr 2026).

5. Subregion CV, deformations, and phase-sensitive phenomenology

In the subregion version of CV, the relevant object is the codimension-zero region of the entanglement wedge on a fixed time slice. In AdS$t=0$7/CFT$t=0$8, this was used to motivate a field-theoretic interpretation via $t=0$9 flow: the deformation acts as a reversible circuit, the layer density

$V/(G_N\ell)$0

maps through $V/(G_N\ell)$1 to the radial measure $V/(G_N\ell)$2, and

$V/(G_N\ell)$3

reproduces the wedge volume underlying subregion CV (Geng, 2019).

Boundary conditions can change the UV structure. In AdS$V/(G_N\ell)$4/BCFT$V/(G_N\ell)$5, the global CV complexity on the $V/(G_N\ell)$6 slice is

$V/(G_N\ell)$7

so the boundary introduces a subleading logarithmic divergence with coefficient $V/(G_N\ell)$8 (Braccia et al., 2019). The same work shows that subregion CV can jump discontinuously across RT phase transitions; for an interval at critical distance $V/(G_N\ell)$9, the jump is

$\ell$0

(Braccia et al., 2019).

In time-dependent states, CV exhibits distinct behavior in full and subregion settings. After a local quench in AdS$\ell$1/CFT$\ell$2, the global CV deviation obeys the early-time expansion

$\ell$3

then continues to grow unboundedly, while the subregion CV of an interval shows early quadratic growth, an intermediate nearly linear rise, and a late return to equilibrium (Ageev et al., 2018). The paper interprets this non-monotonic subregion behavior as “effective-complexity-like,” since it can decrease while entanglement entropy and integrated entanglement density continue to increase (Ageev et al., 2018).

Holographic superconductors provide another phase-sensitive test. For Einstein–Maxwell–scalar models, the superconducting phase was found numerically to have smaller complexity of formation than the unstable normal phase below $\ell$4, and at low temperature the thermal contribution scales as $\ell$5, with $\ell$6 for $\ell$7, independently of $\ell$8 (Yang et al., 2019). In subregion CV for a $\ell$9-dimensional holographic superconductor, the strip complexity has a single UV divergence, grows linearly with large strip width, and tracks phase transitions; in the first-order case it develops an “S” curve, while in the second-order case it shows a kink at $\mathcal{C}(t)=\frac{V(t)}{A_P\,\tau_f},$0 (Chakraborty, 2019).

At finite cutoff, the $\mathcal{C}(t)=\frac{V(t)}{A_P\,\tau_f},$1-deformed theory yields a particularly geometric correction. The difference between the deformed and undeformed CV complexities is

$\mathcal{C}(t)=\frac{V(t)}{A_P\,\tau_f},$2

where

$\mathcal{C}(t)=\frac{V(t)}{A_P\,\tau_f},$3

is the Willmore, or bending, energy of the time-constant slice $\mathcal{C}(t)=\frac{V(t)}{A_P\,\tau_f},$4 (Astaneh, 2024).

6. Generalizations, ambiguities, and ongoing problems

Several works replace the bare volume by higher-curvature or more general geometric functionals. In hyperscaling-violating black branes, generalized volume-complexity takes

$\mathcal{C}(t)=\frac{V(t)}{A_P\,\tau_f},$5

with $\mathcal{C}(t)=\frac{V(t)}{A_P\,\tau_f},$6 in the main example, and retains linear late-time growth whenever the associated effective potential has an interior maximum (Omidi, 2022). In doubly holographic island setups, the near-brane expansion motivates a generalized CV functional involving Wald-like curvature terms and extrinsic-curvature corrections, reducing in Einstein gravity to a generalized island volume on the brane (Hernandez et al., 2020).

Plain CV can also be too restrictive as an interior probe. For multi-horizon Bardeen–AdS black holes, standard CV with $\mathcal{C}(t)=\frac{V(t)}{A_P\,\tau_f},$7 probes only the outermost interior band $\mathcal{C}(t)=\frac{V(t)}{A_P\,\tau_f},$8, whereas the generalized “complexity equals anything” constructions can be tuned to probe all regions with $\mathcal{C}(t)=\frac{V(t)}{A_P\,\tau_f},$9 and distinguish Cauchy horizons from singularities (Jiang et al., 12 Jun 2025). This suggests that CV, in its simplest form, captures only part of the interior structure in multi-horizon geometries.

Another unresolved issue is dimensional uplift. In magnetized holographic plasmas, the DK model yields $V(\Sigma)/(G_N R)$00 and the full ten-dimensional volume is just $V(\Sigma)/(G_N R)$01 times the five-dimensional one, but in the AP model the uplifted five-dimensional slice does not solve the full ten-dimensional extremality equations, so $V(\Sigma)/(G_N R)$02 for $V(\Sigma)/(G_N R)$03 (Ávila et al., 2023). The same work identifies “magnetic simplification,” meaning that a sufficiently strong magnetic field can reduce the complexity relative to the $V(\Sigma)/(G_N R)$04 thermal state in models with a nontrivial scalar profile (Ávila et al., 2023).

Finally, CV has inspired non-holographic analogues. In an information-geometric generalization, the Fubini–Study volume of a two-parameter manifold of states obeys

$V(\Sigma)/(G_N R)$05

for both closed and “open” two-mode squeezed systems, providing a generalized “CV” relation for Krylov complexity (Zhai et al., 2024). This suggests that the volume–complexity relation may extend beyond spacetime wormholes into state-space geometry, although this is a distinct proposal rather than a direct consequence of bulk gravity (Zhai et al., 2024).

Taken together, these developments present CV as a broad but non-unique framework: exact in special geometries, robust in many dynamical black-hole settings, sharply sensitive to topology and phase structure, yet still marked by normalization ambiguities, locality tensions, higher-dimensional uplift issues, and competition from generalized volume functionals (Fu et al., 2018, Couch et al., 2018, Hernandez et al., 2020).