Holographic Subregion Complexity

• Subregion complexity is a holographic measure that quantitatively captures the mixed-state complexity of a boundary subregion via volume (CV) or action (CA) prescriptions.
• It employs extremal surface techniques and variational methods to study dynamic, anisotropic, and phase-transition behaviors in gravitational systems.
• Key investigations address divergence structures, renormalization schemes, and ambiguous information-theoretic interpretations within mixed-state holography.

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Subregion complexity is a holographic notion of mixed-state complexity associated with a boundary subregion $A$, rather than with the full boundary state. In the most widely used “subregion CV” prescription, one first finds the Ryu–Takayanagi or Hubeny–Rangamani–Takayanagi surface $\gamma_A$ homologous to $A$, then defines the complexity by the volume of the codimension-one bulk region $\Gamma_A$ bounded by $A$ and $\gamma_A$, $\mathcal{C} = \mathrm{Vol}(\Gamma_A)/(8\pi R G)$ (Alishahiha, 2015). A complementary “subregion CA” prescription instead evaluates the on-shell action on the intersection of the Wheeler–DeWitt patch with the entanglement wedge, $\mathcal{W}[A] = {\rm WDW}\cap E[A]$, with $\mathcal{C}_A(A)=I(\mathcal{W}[A])/\pi$ (Auzzi et al., 2019). Across AdS$_3$/BTZ, boosted black branes, holographic superconductors, confining QCD-like models, non-conformal RG flows, anisotropic plasmas, evaporation/island models, and Vaidya quenches, subregion complexity has emerged as a geometric observable sensitive to entanglement wedge topology, horizon access, multipartite structure, and phase transitions (Ghosh et al., 2019, Chakraborty, 2019, Zhang, 2018, Chen et al., 2018, Cáceres et al., 2018, Bhattacharya et al., 2021, Goki et al., 2024).

1. Definitions and principal prescriptions

Subregion complexity is formulated as a mixed-state analogue of holographic complexity. The essential input is a boundary reduced density matrix $\gamma_A$0, encoded holographically by the entanglement wedge of $\gamma_A$1. In the static CV prescription, the relevant bulk region is the codimension-one region $\gamma_A$2 or $\gamma_A$3 whose boundary is $\gamma_A$4, and the complexity is proportional to its volume (Ghosh et al., 2019, Chakraborty, 2019). In the CA prescription, the relevant bulk region is instead $\gamma_A$5, and the complexity is the on-shell gravitational action of that intersection (Auzzi et al., 2019). A spacetime-volume variant, often called subregion CV 2.0, uses the spacetime volume of the same region $\gamma_A$6 rather than the action (Auzzi et al., 2019).

A standard distinction runs between full-state and subregion proposals. Full-state CV and CA attach complexity to an entire boundary time slice or full Wheeler–DeWitt patch. Subregion prescriptions instead depend on the RT/HRT surface and are therefore intrinsically sensitive to the geometry and topology of the entanglement wedge. This is already visible in AdS$\gamma_A$7, where a single interval, two disjoint intervals, and overlapping intervals lead to different structures for the corresponding mixed-state complexities (Auzzi et al., 2019).

The proposals also admit information-theoretic refinements. “Mutual complexity” is defined in direct analogy with mutual information,

$\gamma_A$8

For CV and CV 2.0, this quantity is always superadditive, hence negative, by geometric arguments; for CA it is not constrained a priori and can be either negative or positive depending on the ambiguity parameter $\gamma_A$9 (Auzzi et al., 2019). A further proposal identifies subregion complexity with purification complexity. In that setting, CV and CV 2.0 subregion complexity are argued to equal holographic purification complexity, whereas CA subregion complexity appears only to provide an upper bound, with explicit non-saturating examples coming from higher-genus black-hole interiors (Cáceres et al., 2018).

2. Geometric formulation and perturbative machinery

In static asymptotically AdS geometries, $A$0 and the volume region $A$1 lie on a constant-time slice, and the problem reduces to extremal submanifolds in a Riemannian geometry. In time-dependent or rotating backgrounds, $A$2 and $A$3 need not lie on a single boundary time slice, and the extremal-surface problem becomes substantially harder (Ghosh et al., 2019). This difference underlies much of the technical literature on subregion complexity.

A general perturbative framework around pure AdS was developed using variational methods and an inhomogeneous Jacobi equation for the normal deformation of extremal surfaces. For the volume functional $A$4, the first variation of the volume of the complexity region bounded by $A$5 takes the form

$A$6

where $A$7 is the RT surface, $A$8 is its deformation vector, and $A$9 is the conormal of $\Gamma_A$0 inside $\Gamma_A$1 (Ghosh et al., 2019). The normal deformation satisfies an inhomogeneous Jacobi equation

$\Gamma_A$2

so the response of subregion complexity is reduced to solving a linear elliptic problem with Dirichlet data fixed by the boundary entangling surface (Ghosh et al., 2019).

This variational method yields explicit results for boosted black brane perturbations over pure AdS$\Gamma_A$3. For a disk subregion, the RT surface is a half-sphere, the complexity region is foliated by half-spheres, and the bulk and boundary terms cancel exactly at first order. The result is that the linear correction to holographic subregion complexity for a spherical region in AdS$\Gamma_A$4 vanishes, even though both the metric perturbation and the RT-surface deformation are nonzero individually (Ghosh et al., 2019). For strip subregions, by contrast, the correction is nonzero and depends on the orientation of the strip relative to the boost direction, exhibiting explicit anisotropy (Ghosh et al., 2019).

The same general logic reappears in dynamical settings. In Vaidya geometries the HRT surface is described by coupled functions $\Gamma_A$5 and $\Gamma_A$6, and the codimension-one volume bounded by the HRT surface can often be reduced to an effective one-dimensional variational problem. In the Einstein–Born–Infeld quench studied in AdS$\Gamma_A$7, the HRT surface and the volume surface are solved numerically, and the HRT profile itself solves the relevant extremal-volume equation for the subregion CV functional (Ling et al., 2018). In moving-plasma backgrounds dual to boosted black branes, the extremal HRT surface and the enclosed volume are instead obtained numerically by a finite difference method on the stationary but non-static bulk geometry (Goki et al., 2024).

3. Divergences, renormalization, and dependence on region geometry

The divergence structure of subregion complexity depends strongly on the proposal and on the entangling region. In subregion CA for a single interval in AdS$\Gamma_A$8 or BTZ, the complexity has a leading divergence linear in the interval length and a logarithmic contribution proportional to the entanglement entropy. For a single BTZ interval,

$\Gamma_A$9

while the corresponding subregion CV 2.0 result is

$A$0

In both cases this realizes a “linear + entropy + constant” structure, but that structure fails for two intervals in AdS$A$1, where the finite part acquires nontrivial dependence on the cross-ratio through logarithms and dilogarithms (Auzzi et al., 2019). This establishes that subregion complexity can encode multipartite geometric information not reducible to entanglement entropy alone (Auzzi et al., 2019).

In strip-based CV studies, the divergence pattern is often simpler. For the $A$2-dimensional holographic superconductor with fully backreacted AdS$A$3 dual, the strip complexity is

$A$4

so there is a single UV divergence proportional to $A$5, and the universal term remains finite across the phase transition (Chakraborty, 2019). In the $A$6-dimensional p-wave superconductor, the divergence is likewise state-independent at fixed interval size and scales as $A$7, allowing a finite “universal” piece $A$8 to be extracted numerically (Fujita, 2018).

More elaborate renormalization schemes appear in confining and non-conformal theories. In the Einstein–Maxwell–dilaton holographic QCD model, the “renormalized holographic complexity density” is defined by

$A$9

with $\gamma_A$0 typically chosen as the disconnected-surface complexity in the same background (Zhang, 2018). In the five-dimensional Einstein–scalar RG-flow model, one instead compares directly to the AdS result via

$\gamma_A$1

and also studies a finite connected–disconnected difference

$\gamma_A$2

which isolates the disentangling transition (Asadi, 2020). These constructions are not equivalent, but they share the aim of removing the universal near-boundary divergence so that geometric and phase-structural information can be compared across states.

4. Dynamics, anisotropy, and phase-structure diagnostics

Under a thermal quench in Vaidya-AdS, subregion complexity shows a characteristic non-monotonic evolution: it increases at early time, reaches a maximum, decreases, and then saturates (Chen et al., 2018). When the strip is large enough and the quench is fast enough, the evolution becomes discontinuous in $\gamma_A$3 with $\gamma_A$4, with a sudden drop caused by a transition of the extremal entanglement surface (Chen et al., 2018). In Vaidya spacetime with de Sitter boundary and spherical subregions, the same basic pattern is sharpened into four stages: almost linear growth, slowing growth, rapid descent after a maximum, and final saturation. The saturation time depends almost only on the subregion radius,

$\gamma_A$5

which is linear for small $\gamma_A$6 and logarithmically divergent as $\gamma_A$7 approaches the cosmological horizon (Zhang, 2019).

Charged quenches change this pattern. In Einstein–Born–Infeld theory, large charge not only modifies the continuous-versus-discontinuous structure of the evolution but can also wash out the second stage of linear growth that appears for sufficiently large strip size in lower-dimensional examples (Ling et al., 2018). The same study argues that subregion CV complexity can be treated as a probe of the black-hole interior, because the enclosed codimension-one bulk region tracks how deeply the HRT surface and its associated volume enter the post-quench interior (Ling et al., 2018).

Static phase transitions produce equally distinctive signatures. In the fully backreacted holographic superconductor dual to AdS$\gamma_A$8, subregion complexity detects the same critical temperatures as the free energy and entanglement entropy. For the $\gamma_A$9 superconductor, the transition is second order and the complexity is continuous with a kink at $\mathcal{C} = \mathrm{Vol}(\Gamma_A)/(8\pi R G)$0; for the $\mathcal{C} = \mathrm{Vol}(\Gamma_A)/(8\pi R G)$1 superconductor, the transition is first order and the complexity has a finite jump at $\mathcal{C} = \mathrm{Vol}(\Gamma_A)/(8\pi R G)$2, together with an “S”-curve multivalued structure as a function of strip width (Chakraborty, 2019). In the $\mathcal{C} = \mathrm{Vol}(\Gamma_A)/(8\pi R G)$3-dimensional p-wave superconductor, the universal term remains finite across the phase transition, but its temperature dependence changes qualitatively with the ratio $\mathcal{C} = \mathrm{Vol}(\Gamma_A)/(8\pi R G)$4: it increases as temperature is lowered when $\mathcal{C} = \mathrm{Vol}(\Gamma_A)/(8\pi R G)$5, whereas for $\mathcal{C} = \mathrm{Vol}(\Gamma_A)/(8\pi R G)$6 it becomes non-monotonic and multi-valued, and it exhibits a discontinuous jump as a function of the interval size when the RT surface wraps almost the entire spatial circle (Fujita, 2018).

In confining and anisotropic theories, subregion complexity has been proposed as a phase diagnostic. In the holographic QCD model with two different warp factors, the renormalized density $\mathcal{C} = \mathrm{Vol}(\Gamma_A)/(8\pi R G)$7 is discontinuous at a critical strip width $\mathcal{C} = \mathrm{Vol}(\Gamma_A)/(8\pi R G)$8 in confinement-like phases but continuous in deconfinement phases; as a function of temperature and chemical potential, it jumps across first-order transitions and stays continuous across crossover-like behavior (Zhang, 2018). In the anisotropic Einstein–dilaton–Maxwell model with a Van der Waals-like small/large black-hole transition, the thermodynamically stable branches are empirically the branches with smaller subregion complexity at fixed temperature, and anisotropy decreases the holographic subregion complexity relative to the isotropic case (Ali-Akbari et al., 2021). In the non-conformal Einstein–scalar model, relative subregion complexity is a measure of non-conformality both at zero and finite temperature, decreases monotonically with strip size along the RG flow, and exhibits a finite jump at the connected–disconnected disentangling transition (Asadi, 2020).

A more kinematic anisotropy appears in a moving strongly coupled plasma dual to boosted black branes. There, temperature, velocity, and subregion length all increase the CV subregion complexity, and in $\mathcal{C} = \mathrm{Vol}(\Gamma_A)/(8\pi R G)$9 the complexity diverges as the boost approaches the relativistic limit, with the divergence characterized by $\mathcal{W}[A] = {\rm WDW}\cap E[A]$0 (Goki et al., 2024). This identifies a robust orientation- and boost-sensitive sector of subregion complexity that complements the perturbative strip/disk results in boosted AdS$\mathcal{W}[A] = {\rm WDW}\cap E[A]$1 (Ghosh et al., 2019).

5. Information-theoretic interpretations

Several lines of work interpret subregion complexity as an information-theoretic mixed-state quantity rather than merely a geometric volume. One proposal identifies it with purification complexity. In holographic purifications of a boundary mixed state, the space of admissible purifications includes different geodesic completions of the entanglement wedge and different choices of cutoff surface. Within CV and CV 2.0, subregion complexity is argued to equal holographic purification complexity exactly; within CA, subregion complexity appears only as an upper bound on purification complexity, and higher-genus black-hole interiors provide explicit counterexamples to saturation (Cáceres et al., 2018). This makes the volume-based proposals structurally closer to an optimization over purifications than the action-based proposal (Cáceres et al., 2018).

Subregion CA also exposes a distinct class of ambiguity questions. In AdS$\mathcal{W}[A] = {\rm WDW}\cap E[A]$2 and BTZ, the null-boundary counterterm scale $\mathcal{W}[A] = {\rm WDW}\cap E[A]$3 controls both divergent and finite parts of the action. The sign of mutual action complexity for two intervals depends on $\mathcal{W}[A] = {\rm WDW}\cap E[A]$4, and the study finds a critical value $\mathcal{W}[A] = {\rm WDW}\cap E[A]$5 such that for $\mathcal{W}[A] = {\rm WDW}\cap E[A]$6 mutual action complexity is always superadditive, while for $\mathcal{W}[A] = {\rm WDW}\cap E[A]$7 it changes sign with the interval separation (Auzzi et al., 2019). This dependence is one of the clearest indications that the field-theoretic meaning of subregion CA remains scheme-dependent (Auzzi et al., 2019).

A different information-theoretic perspective arises in the geometric secret-sharing model of Hawking radiation. There, the complete radiation $\mathcal{W}[A] = {\rm WDW}\cap E[A]$8 exhibits the usual Page transition, but the subregion complexity of a single radiation exit $\mathcal{W}[A] = {\rm WDW}\cap E[A]$9 jumps only at a later “secret-sharing time” $\mathcal{C}_A(A)=I(\mathcal{W}[A])/\pi$0, when the minimal surface for $\mathcal{C}_A(A)=I(\mathcal{W}[A])/\pi$1 switches to one with access to a partial island (Bhattacharya et al., 2021). The minimal surfaces for $\mathcal{C}_A(A)=I(\mathcal{W}[A])/\pi$2 and $\mathcal{C}_A(A)=I(\mathcal{W}[A])/\pi$3 do not cover the entire island region available to $\mathcal{C}_A(A)=I(\mathcal{W}[A])/\pi$4; they access it only partially. This inaccessibility is interpreted in terms of “classical” Markov recovery, with the inaccessible bulk region serving as a geometric measure of the secret-sharing structure (Bhattacharya et al., 2021).

The anisotropic phase-transition model adds yet another proposal: an informational and computational interpretation of stability. There the authors propose that, among coexisting black-hole branches, the branch that needs less information to be specified is the stable one, and equivalently that stability corresponds to non-decreasing computational resource as temperature increases (Ali-Akbari et al., 2021). This is not a general theorem, but it illustrates how subregion complexity is being used as more than a geometric diagnostic.

6. Structural results, open problems, and controversies

A recurring structural result is that subregion complexity can behave more rigidly than entanglement entropy in symmetric situations and more richly in multipartite ones. The exact cancellation of the first-order correction for spherical regions in boosted AdS$\mathcal{C}_A(A)=I(\mathcal{W}[A])/\pi$5 suggests a first-law-like structure for subregion complexity in highly symmetric setups (Ghosh et al., 2019). By contrast, two-interval subregion action complexity in AdS$\mathcal{C}_A(A)=I(\mathcal{W}[A])/\pi$6 depends on cross-ratios through finite logarithmic and dilogarithmic terms that are not reducible to entropy or mutual information, demonstrating that subregion complexity carries genuinely multipartite data (Auzzi et al., 2019).

Another persistent theme is the role of entangling-surface topology. In AdS$\mathcal{C}_A(A)=I(\mathcal{W}[A])/\pi$7-based models, including the p-wave superconductor and the secret-sharing construction, discontinuous jumps in complexity occur precisely when the dominant extremal surface changes topology or homology class, such as when a geodesic starts wrapping the horizon or when an infalling geodesic through an end-of-the-world brane becomes dominant (Fujita, 2018, Bhattacharya et al., 2021). In confining and non-conformal models the analogous connected–disconnected transition shows up as a finite jump in the renormalized complexity (Zhang, 2018, Asadi, 2020).

The most explicit controversy concerns CA. On the one hand, subregion CA yields analytic control in AdS$\mathcal{C}_A(A)=I(\mathcal{W}[A])/\pi$8 and BTZ and reveals a tight connection between complexity, entropy, and multipartite geometry (Auzzi et al., 2019). On the other hand, the relation to purification complexity fails generically in CA, the ambiguity scale $\mathcal{C}_A(A)=I(\mathcal{W}[A])/\pi$9 remains without a settled field-theoretic interpretation, and higher-genus wormhole examples force a trilemma: either those geometries are not holographic, or CA must be modified, or subregion CA is not dual to purification complexity (Cáceres et al., 2018). No analogous obstruction is reported for CV or CV 2.0 (Cáceres et al., 2018).

Methodologically, several open directions recur across the literature. The inhomogeneous Jacobi formalism already provides second-order equations and a natural route to shape deformations and higher-order corrections, but the explicit applications remain mostly first order (Ghosh et al., 2019). Time-dependent studies have established robust qualitative patterns—early growth, maxima, decreases, saturation, and jump phenomena—but their analytic control is still limited outside highly symmetric regimes (Chen et al., 2018, Zhang, 2019, Ling et al., 2018). Higher dimensions, more general subregion shapes, and alternative prescriptions such as subregion CA in dynamical or non-conformal backgrounds remain active problems (Chakraborty, 2019, Asadi, 2020).

Taken together, these results present subregion complexity as a family of mixed-state observables whose geometric definition is simple but whose physical interpretation remains proposal-dependent. In the volume-based prescriptions it has become a precise probe of entanglement wedge geometry, phase structure, anisotropy, RG flow, partial-island access, and purification. In the action-based prescription it is equally rich, but also more ambiguous, more sensitive to regulator choices, and more directly entangled with unresolved questions about what holographic complexity is meant to compute (Auzzi et al., 2019, Cáceres et al., 2018).