Complexity of Formation: Theory and Applications

• Complexity of formation is the additional cost required to prepare structured states from simpler reference configurations, notably in holography.
• Holographic methods like complexity=action and complexity=volume quantitatively compare black hole and vacuum geometries to isolate entanglement costs.
• Applications span fields from astrophysics to astrochemistry, illustrating how reference subtraction highlights the emergence of complex structures.

Searching arXiv for papers on "complexity of formation" and closely related usage across holography and adjacent contexts. “Complexity of formation” most commonly denotes a relative complexity: the additional cost of preparing a nontrivial state from a simpler reference state. In the holographic literature, which provides the most systematic use of the term in this set of sources, it is the extra complexity required to prepare the thermofield double state dual to an eternal AdS black hole relative to preparing two disconnected vacuum states, usually written as

$$\Delta \mathcal{C}=\mathcal{C}(\mathrm{TFD})-2\,\mathcal{C}(\mathrm{vac}) \, .$$

This quantity is finite after vacuum subtraction and has been studied in the complexity=action (CA), complexity=volume (CV), and generalized “complexity equals anything” frameworks (Chapman et al., 2016). In other arXiv usages represented here, related expressions quantify the creation of structure from a maximally uniform configuration, or the emergence of molecular complexity in astrophysical environments, but these are not definitionally identical to the holographic observable (Barbour et al., 2024).

1. Conceptual scope and reference-state dependence

In holography, the reference configuration is fixed by the vacuum AdS geometry, and the target configuration is typically the eternal black hole dual to a thermofield double state. The resulting observable is therefore vacuum-subtracted and explicitly comparative rather than absolute. This subtraction is central: it removes UV-divergent contributions common to the black hole and vacuum geometries and isolates the extra complexity associated with entanglement and thermal structure (Akhavan et al., 2019).

A central feature of the subject is that the term does not denote a universal invariant across all branches of physics represented here. In relational $N$-body theory, for example, the analogous quantity is the difference between the shape complexity of a configuration and the minimum attained by the maximally uniform configuration, called the “shape age,”

$$a(s)=C_{\rm shape}^s-C_{\rm shape}^{\rm Alpha} \, ,$$

so the reference state is again simple, but the object being measured is geometrical structure rather than circuit complexity (Barbour et al., 2024). In astrochemistry, the issue is the formation of molecules of increasing complexity under low temperature, low density, radiative, and cosmic-ray conditions; the reference notion is chemical simplicity rather than quantum-state preparation (Becker, 2013).

2. Canonical holographic definitions

The standard holographic definitions are organized by the bulk quantity identified with boundary complexity. In the CA proposal, complexity is proportional to the gravitational action on the Wheeler–DeWitt patch,

$\mathcal{C}=\frac{I_{\text{WDW}}}{\pi \hbar} \, ,$

and the complexity of formation is obtained by subtracting twice the vacuum value from the black-hole value (Akhavan et al., 2019). In the CV proposal, complexity is proportional to the maximal bulk codimension-one volume anchored on the boundary time slice, and the corresponding complexity of formation is the regularized difference between the black-hole and vacuum maximal volumes (Chapman et al., 2016).

These constructions agree on the qualitative role of $\Delta \mathcal{C}$ as the cost of forming the entangled thermofield double state, but they need not agree on coefficients, subleading structure, or even sign in more general settings. For $d>2$ in the CA analysis of eternal AdS black holes, the leading high-temperature result is linear in the entropy,

$$\Delta \mathcal{C}=k_d S,\qquad k_d=\frac{d-2}{d\pi}\cot\!\left(\frac{\pi}{d}\right),$$

whereas for $d=2$ the complexity of formation is a temperature-independent constant (Chapman et al., 2016). The CV proposal exhibits the same qualitative high-temperature linearity in $S$ for static cases, but with a different dimension-dependent coefficient (Chapman et al., 2016).

The formalism also extends beyond the original CA and CV prescriptions. In the generalized covariant-volume framework, complexity is defined by extremizing a codimension-one functional weighted by a scalar $F$ built from the geometry. The corresponding generalized complexity of formation is again the difference between the black-hole and AdS vacuum values at zero boundary time (Zhang et al., 2024).

3. Entropy scaling and thermodynamic-volume scaling

The early static-black-hole results suggested that entropy controls the complexity of formation. For boundary dimensions $N$0, the CA computation found $N$1 at high temperature, while in $N$2 the result is constant rather than extensive (Chapman et al., 2016). This established the first systematic dimensional dependence of the observable.

Subsequent work on rotating black holes showed that this entropy law is not fundamental in general. For large rotating AdS black holes, both CA and CV analyses indicate that the relevant thermodynamic quantity is the thermodynamic volume $N$3, not the entropy $N$4, with the principal scaling

$N$5

for large black holes (Balushi et al., 2020). The distinction becomes nontrivial precisely when $N$6 and $N$7 are independent, as in rotating Myers–Perry–AdS solutions; in the static limit they reduce to the same scaling behavior, which explains why earlier results appeared entropy-controlled (Balushi et al., 2020).

A related lower bound follows from the reverse isoperimetric inequality. Assuming the relevant inequality for thermodynamic volume, the large-black-hole complexity of formation obeys

$N$8

so entropy survives as a lower bound even when it no longer determines the leading scaling (Balushi et al., 2020). The generalized volume-complexity analysis reaches the same conclusion for broad classes of regular scalar functionals $N$9: for large near-extremal rotating Myers–Perry–AdS black holes, the generalized complexity of formation maintains the same $$a(s)=C_{\rm shape}^s-C_{\rm shape}^{\rm Alpha} \, ,$$0 scaling as the original holographic proposals, whereas static charged black holes recover the entropy law when the generalized functional remains finite in the vacuum limit and at spatial infinity (Zhang et al., 2024).

4. Rotation, extremality, and the role of counterterms

Angular momentum produces the sharpest departures from the static picture. For odd-dimensional equal-spinning rotating black holes, the complexity of formation exhibits a mixed structure: near extremality there is a logarithmic term controlled by the entropy, while for large black holes at fixed temperature the dominant contribution scales with the thermodynamic volume,

$$a(s)=C_{\rm shape}^s-C_{\rm shape}^{\rm Alpha} \, ,$$1

This pattern appears in both the CA and CV analyses (Balushi et al., 2020).

For rotating BTZ, CA and CV complexity of formation are linear in the temperature in the grand canonical ensemble and diverge with the same structure in the critical angular-velocity limit. In that limit the holographic divergences take the form

$$a(s)=C_{\rm shape}^s-C_{\rm shape}^{\rm Alpha} \, ,$$2

up to proposal-dependent prefactors (Bernamonti et al., 2021). The same work emphasizes that, in the CA computation, inclusion of the null-boundary counterterm is crucial for both the linear-in-$$a(s)=C_{\rm shape}^s-C_{\rm shape}^{\rm Alpha} \, ,$$3 behavior and the divergence structure at high angular velocity (Bernamonti et al., 2021).

Counterterms also matter at the level of renormalization. In Einstein gravity, new timelike and null counterterms can remove the UV divergences of the on-shell action on the Wheeler–DeWitt patch. However, for AdS–Schwarzschild black holes these null counterterms do not change the complexity of formation: they cancel the same divergences in the black-hole and vacuum contributions, so the finite vacuum-subtracted quantity is unchanged (Akhavan et al., 2019). This establishes a limited form of scheme independence for the relative observable, even though the un-subtracted complexity itself is regulator-sensitive.

5. Positivity, negativity, and model dependence

The assumption that the vacuum is the least complex state is not generally stable under extensions of the holographic setup. When compact directions are included, the vacuum-subtracted maximal volume can become negative, and explicit asymptotically AdS$$a(s)=C_{\rm shape}^s-C_{\rm shape}^{\rm Alpha} \, ,$$4 supergravity solutions exhibit arbitrarily negative complexity of formation (Engelhardt et al., 2021). The same work shows that both natural extensions of the CV prescription to compact spaces—the full-space volume and the reduced noncompact volume times the compact-space volume—admit such negative examples (Engelhardt et al., 2021).

These constructions rely on the compact directions and on relevant scalar primaries, realized as tachyonic scalar condensates above the Breitenlohner–Freedman bound. A plausible implication is that maximal-volume complexity, in its standard vacuum-subtracted form, is not a positive-semidefinite distance from the vacuum state once compact directions and relevant deformations are treated explicitly. The same paper further finds examples in which complexity decreases at late times, including both single-sided geometries and two-sided cosmological wormholes (Engelhardt et al., 2021).

Model dependence also appears in holographic superconductors. In the CV analysis of Einstein–Maxwell–complex scalar systems, the superconducting phase always has a smaller complexity than the unstable normal phase below the critical temperature. At low temperature, the complexity of formation scales as $$a(s)=C_{\rm shape}^s-C_{\rm shape}^{\rm Alpha} \, ,$$5, where $$a(s)=C_{\rm shape}^s-C_{\rm shape}^{\rm Alpha} \, ,$$6 depends on the scalar mass $$a(s)=C_{\rm shape}^s-C_{\rm shape}^{\rm Alpha} \, ,$$7, the charge $$a(s)=C_{\rm shape}^s-C_{\rm shape}^{\rm Alpha} \, ,$$8, and the dimension $$a(s)=C_{\rm shape}^s-C_{\rm shape}^{\rm Alpha} \, ,$$9; for $\mathcal{C}=\frac{I_{\text{WDW}}}{\pi \hbar} \, ,$0, one finds the universal result $\mathcal{C}=\frac{I_{\text{WDW}}}{\pi \hbar} \, ,$1, independent of $\mathcal{C}=\frac{I_{\text{WDW}}}{\pi \hbar} \, ,$2 (Yang et al., 2019). Here complexity of formation behaves in a manner reminiscent of free energy across the phase transition, but the precise scaling is controlled by the IR geometry of the condensed phase (Yang et al., 2019).

6. Other scientific usages of “formation complexity”

Outside holography, the phrase is attached to different but structurally related questions about the emergence of organized structure from simple antecedents. In relational $\mathcal{C}=\frac{I_{\text{WDW}}}{\pi \hbar} \, ,$3-body theory, shape complexity is a scale-invariant quantity built from the root-mean-square and mean-harmonic lengths,

$\mathcal{C}=\frac{I_{\text{WDW}}}{\pi \hbar} \, ,$4

and the complexity of formation is identified with the excess above the minimum attained by the maximally uniform configuration, namely the shape age $\mathcal{C}=\frac{I_{\text{WDW}}}{\pi \hbar} \, ,$5 (Barbour et al., 2024). In that framework the Janus point is the unique minimum of $\mathcal{C}=\frac{I_{\text{WDW}}}{\pi \hbar} \, ,$6, and increasing structural richness corresponds to increasing distance from maximal uniformity (Barbour et al., 2024).

In astrochemistry, the problem is formulated as the formation of molecules of increasing complexity under conditions far from standard laboratory chemistry. Two complementary approaches are emphasized: a bottom-up route from simple interstellar species and a top-down route tracing precursors of biochemically relevant molecules. The decisive mechanisms include ion-molecule reactions, neutral-radical chemistry, dissociative recombination, photodissociation, cosmic-ray-induced chemistry, and grain-surface processes such as Langmuir–Hinshelwood and Eley–Rideal reactions (Becker, 2013). This suggests a broader comparative theme: in both holography and astrochemistry, “formation complexity” is controlled by a subtraction against a simpler reference regime and by the pathways that bridge that reference to a structured state.

A related cosmological usage appears in studies of the first stars, where the transition from a nearly homogeneous early universe to a highly structured one is described as an evolution toward increasing complexity. The formation of Population III stars marks a primary transition event in that progression, with disk fragmentation, radiative feedback, magnetic fields, dark matter annihilation scenarios, and baryon–dark-matter streaming velocities contributing to the complexity of the outcome (Klessen, 2018). Here again the operative contrast is between an initially simple configuration and the cost, dynamics, or pathway by which structure emerges.

7. Current status

Within the literature represented here, the most developed meaning of complexity of formation is the holographic one: a vacuum-subtracted relative complexity for thermofield double states and their bulk duals. Its principal established properties are the static entropy law in high-temperature $\mathcal{C}=\frac{I_{\text{WDW}}}{\pi \hbar} \, ,$7 settings, the temperature-independent $\mathcal{C}=\frac{I_{\text{WDW}}}{\pi \hbar} \, ,$8 behavior, the transition to thermodynamic-volume control for large rotating black holes, the robustness of the relative quantity against certain UV counterterms, and the failure of positivity once compact directions and relevant primaries are incorporated (Chapman et al., 2016).

At the same time, the broader record shows that the phrase is portable across disciplines only at a structural level. In shape dynamics it measures distance from maximal uniformity; in astrochemistry it indexes convergent routes to molecular complexity; in early-universe astrophysics it marks the emergence of nonlinear structure from homogeneous initial conditions (Barbour et al., 2024). The common theme is therefore not a single invariant definition but a family of reference-state-based measures and narratives for how ordered or entangled structure is formed from simpler antecedents.