Capacity of Entanglement in Quantum Systems

• Capacity of entanglement is defined both as the variance of a state’s modular Hamiltonian and as the one-shot entangling capability of a quantum process, quantifying entanglement fluctuations.
• In many-body physics and quantum field theory, it measures the spread of the entanglement spectrum, offering insights beyond conventional entanglement entropy.
• Operationally, it assesses the maximum entanglement that a CPTP map or unitary can generate, clearly differentiating state-based attributes from process-based capabilities.

Searching arXiv for recent and foundational papers on "capacity of entanglement" and "entangling capacity". Capacity of entanglement denotes two distinct but historically adjacent concepts in quantum theory. In quantum information, it can mean the one-shot entangling capacity of a dynamical process: the maximum entanglement that a bipartite CPTP map or unitary can generate in a single use, with local ancillas allowed (Campbell, 2010). In quantum many-body theory, quantum field theory, gravity, and random-state theory, it denotes the variance of the modular Hamiltonian or, equivalently, the second cumulant of the entanglement spectrum of a reduced density matrix (Boer et al., 2018). For a subsystem $A$ with reduced state $\rho_A$ and modular Hamiltonian $K_A=-\log\rho_A$, the latter notion is

$$C_E=\mathrm{Tr}(\rho_A K_A^2)-\big(\mathrm{Tr}(\rho_A K_A)\big)^2 =\left.\frac{\partial^2}{\partial n^2}\log \mathrm{Tr}\,\rho_A^n\right|_{n=1},$$

up to convention-dependent equivalent forms used in the literature (Boer et al., 2018, Wei, 2022, Khoshdooni et al., 13 May 2025). It measures the width of the entanglement spectrum rather than its mean, and is therefore complementary to entanglement entropy. The two usages share a common emphasis on quantifying entanglement beyond von Neumann entropy, but they address different objects: processes in the operational setting, states in the modular-fluctuation setting.

1. Definitions and competing usages

The state-based notion of capacity of entanglement is built from the Rényi generating function. For a bipartite pure state with reduced density matrix $\rho_A$, the entanglement entropy is

$$S_{EE}=-\mathrm{Tr}(\rho_A\log\rho_A)=\langle K_A\rangle,$$

whereas the capacity is the variance

$C_E=\langle K_A^2\rangle-\langle K_A\rangle^2$

of the modular Hamiltonian $K_A=-\log\rho_A$ (Boer et al., 2018, Khoshdooni et al., 13 May 2025). Equivalently,

$$C_E=\left.\frac{\partial^2}{\partial n^2}\log \mathrm{Tr}\,\rho_A^n\right|_{n=1},$$

and closely related refined-Rényi conventions are also common in gravity (Kawabata et al., 2021, Arias et al., 10 Mar 2026). In thermodynamic language, $\mathrm{Tr}\,\rho_A^n=\mathrm{Tr}\,e^{-nK_A}$ makes $\rho_A$0 analogous to inverse temperature, so $\rho_A$1 is the entanglement analogue of a heat capacity (Boer et al., 2018, Khoshdooni et al., 13 May 2025).

A notable qualitative feature is that $\rho_A$2 can vanish both for a separable state and for a maximally entangled state with flat entanglement spectrum. Entanglement entropy increases monotonically between those extremes, but capacity is zero at both ends and maximal at an intermediate partially entangled spectrum (Wei, 2022, Nandy, 2021). This already distinguishes it sharply from entropy-based entanglement diagnostics.

The operational usage, by contrast, concerns a channel or unitary rather than a state. For a bipartite CPTP map

$\rho_A$3

the unassisted entangling capacity with respect to an entanglement monotone $\rho_A$4 is

$\rho_A$5

where $\rho_A$6 denotes separable inputs across the relevant bipartition including local ancillas, while the assisted capacity is

$\rho_A$7

with optimization over all physical density operators (Campbell, 2010). This meaning is standard in quantum Shannon-style resource theory, but it is conceptually distinct from the modular-fluctuation notion.

Because both usages occur in the literature under the same phrase, disambiguation depends on context. In state-based QFT, gravity, and random-matrix work, “capacity of entanglement” almost always refers to modular variance (Boer et al., 2018, Wei, 2022, Okuyama, 2021). In process theory, network theory, and older negativity-based work, it typically refers to entangling power or throughput (Campbell, 2010, Vardoyan et al., 2023).

2. Modular-fluctuation capacity of states

For a reduced density matrix $\rho_A$8 with eigenvalues $\rho_A$9, the capacity can be written as

$K_A=-\log\rho_A$0

which makes explicit that it is the variance of entanglement “energies” $K_A=-\log\rho_A$1 sampled with weights $K_A=-\log\rho_A$2 (Wei, 2022, Shrimali et al., 2022). This identifies $K_A=-\log\rho_A$3 as a second cumulant of the entanglement spectrum. The first cumulant is the entanglement entropy.

The Rényi representation is often more useful in field theory and gravity. With

$K_A=-\log\rho_A$4

one has

$K_A=-\log\rho_A$5

in standard conventions (Khoshdooni et al., 13 May 2025, Mozaffar, 2024). This form underlies replica-trick calculations, random-state averages, and gravitational saddle analyses.

The same quantity is tightly connected to information geometry. Along the specific modular flow family $K_A=-\log\rho_A$6, the capacity equals the quantum Fisher information and the fidelity susceptibility in the corresponding direction in state space (Boer et al., 2018). This relation is specific to modular rescaling and does not hold for arbitrary deformations.

In Gaussian systems, $K_A=-\log\rho_A$7 can be expressed mode-by-mode in terms of one-body entanglement spectra. For bosonic Gaussian states with symplectic eigenvalues $K_A=-\log\rho_A$8,

$K_A=-\log\rho_A$9

while for fermionic Gaussian states with restricted correlation eigenvalues $$C_E=\mathrm{Tr}(\rho_A K_A^2)-\big(\mathrm{Tr}(\rho_A K_A)\big)^2 =\left.\frac{\partial^2}{\partial n^2}\log \mathrm{Tr}\,\rho_A^n\right|_{n=1},$$0,

$$C_E=\mathrm{Tr}(\rho_A K_A^2)-\big(\mathrm{Tr}(\rho_A K_A)\big)^2 =\left.\frac{\partial^2}{\partial n^2}\log \mathrm{Tr}\,\rho_A^n\right|_{n=1},$$1

(Khoshdooni et al., 13 May 2025, Arias et al., 2023). For fermionic Gaussian ensembles this structure leads to exact random-matrix averages and asymptotic volume laws (Huang et al., 2023).

A recurring interpretation is that $$C_E=\mathrm{Tr}(\rho_A K_A^2)-\big(\mathrm{Tr}(\rho_A K_A)\big)^2 =\left.\frac{\partial^2}{\partial n^2}\log \mathrm{Tr}\,\rho_A^n\right|_{n=1},$$2 signals a reduced density matrix close to maximally mixed on its support, with weak $$C_E=\mathrm{Tr}(\rho_A K_A^2)-\big(\mathrm{Tr}(\rho_A K_A)\big)^2 =\left.\frac{\partial^2}{\partial n^2}\log \mathrm{Tr}\,\rho_A^n\right|_{n=1},$$3-dependence of Rényi entropies and a comparatively flat entanglement spectrum (Mozaffar, 2024, Mozaffar, 2024). This theme appears in nonlocal theories, squeezed states, field-space entanglement, and certain black-hole island phases.

3. Random states and typical behavior

Random pure states provide one of the cleanest settings in which the state-based capacity can be computed exactly. In the standard bipartite random pure-state model, with subsystem dimensions $$C_E=\mathrm{Tr}(\rho_A K_A^2)-\big(\mathrm{Tr}(\rho_A K_A)\big)^2 =\left.\frac{\partial^2}{\partial n^2}\log \mathrm{Tr}\,\rho_A^n\right|_{n=1},$$4 and $$C_E=\mathrm{Tr}(\rho_A K_A^2)-\big(\mathrm{Tr}(\rho_A K_A)\big)^2 =\left.\frac{\partial^2}{\partial n^2}\log \mathrm{Tr}\,\rho_A^n\right|_{n=1},$$5, the reduced density matrix follows a Wishart-Laguerre distribution. The exact finite-dimensional capacity was computed from $$C_E=\mathrm{Tr}(\rho_A K_A^2)-\big(\mathrm{Tr}(\rho_A K_A)\big)^2 =\left.\frac{\partial^2}{\partial n^2}\log \mathrm{Tr}\,\rho_A^n\right|_{n=1},$$6 using the replica method (Okuyama, 2021). In the large balanced limit, the capacity approaches

$$C_E=\mathrm{Tr}(\rho_A K_A^2)-\big(\mathrm{Tr}(\rho_A K_A)\big)^2 =\left.\frac{\partial^2}{\partial n^2}\log \mathrm{Tr}\,\rho_A^n\right|_{n=1},$$7

while the entropy grows as $$C_E=\mathrm{Tr}(\rho_A K_A^2)-\big(\mathrm{Tr}(\rho_A K_A)\big)^2 =\left.\frac{\partial^2}{\partial n^2}\log \mathrm{Tr}\,\rho_A^n\right|_{n=1},$$8 in the planar regime with $$C_E=\mathrm{Tr}(\rho_A K_A^2)-\big(\mathrm{Tr}(\rho_A K_A)\big)^2 =\left.\frac{\partial^2}{\partial n^2}\log \mathrm{Tr}\,\rho_A^n\right|_{n=1},$$9 fixed (Okuyama, 2021). Thus the ratio $\rho_A$0 vanishes for large Hilbert-space dimension even though the entropy remains large.

The exact random-pure-state averaging problem was later extended to the major random-state ensembles used in quantum information: Hilbert–Schmidt and Bures–Hall (Wei, 2022). There the average capacity for subsystem dimensions $\rho_A$1 is expressed in closed form in terms of digamma, trigamma, and the finite sum

$\rho_A$2

In the regime $\rho_A$3 with fixed $\rho_A$4, the ensemble averages approach universal constants: $\rho_A$5 (Wei, 2022). Since the average von Neumann entropy grows as $\rho_A$6, typical capacity remains $\rho_A$7 while entropy diverges logarithmically.

Fermionic Gaussian random states exhibit a different asymptotic structure. For both particle-number-constrained and unconstrained fermionic Gaussian ensembles, the average capacity scales extensively,

$\rho_A$8

for $\rho_A$9 with fixed $$S_{EE}=-\mathrm{Tr}(\rho_A\log\rho_A)=\langle K_A\rangle,$$0 (Huang et al., 2023). Here the volume-law coefficient is universal across the two ensembles. This contrast with the $$S_{EE}=-\mathrm{Tr}(\rho_A\log\rho_A)=\langle K_A\rangle,$$1 random pure-state result reflects the very different spectral structure of Gaussian-state ensembles.

These random-state results clarify a frequent misconception. Large entanglement entropy does not imply large capacity. In Haar-random pure states, entropy becomes large while capacity remains bounded (Okuyama, 2021, Wei, 2022). In random fermionic Gaussian states, both can scale extensively, but with distinct coefficients and spectral meaning (Huang et al., 2023). Capacity tracks fluctuation structure, not entanglement amount in any monotonic sense.

4. Conformal field theory, nonconformal deformations, and Gaussian many-body systems

In $$S_{EE}=-\mathrm{Tr}(\rho_A\log\rho_A)=\langle K_A\rangle,$$2-dimensional CFT, the leading universal behavior of capacity often coincides with that of entanglement entropy. For a single interval with

$$S_{EE}=-\mathrm{Tr}(\rho_A\log\rho_A)=\langle K_A\rangle,$$3

one obtains

$$S_{EE}=-\mathrm{Tr}(\rho_A\log\rho_A)=\langle K_A\rangle,$$4

and in the ground state on the line

$$S_{EE}=-\mathrm{Tr}(\rho_A\log\rho_A)=\langle K_A\rangle,$$5

(Arias et al., 2023). This leading equality also appears in the broader review of QFT examples (Boer et al., 2018).

The equality is not generic once subleading structure is resolved. For two disjoint intervals, entropy and capacity share the same leading logarithmic divergence but differ in finite cross-ratio-dependent terms (Arias et al., 2023). In the free Dirac case the difference is a constant, while in the compact boson it depends nontrivially on the cross-ratio through $$S_{EE}=-\mathrm{Tr}(\rho_A\log\rho_A)=\langle K_A\rangle,$$6 (Arias et al., 2023).

Massive deformations provide another sharp distinction. For the massive scalar in $$S_{EE}=-\mathrm{Tr}(\rho_A\log\rho_A)=\langle K_A\rangle,$$7 dimensions, the entropy contains a double-logarithmic zero-mode correction,

$$S_{EE}=-\mathrm{Tr}(\rho_A\log\rho_A)=\langle K_A\rangle,$$8

whereas

$$S_{EE}=-\mathrm{Tr}(\rho_A\log\rho_A)=\langle K_A\rangle,$$9

and therefore lacks the same $C_E=\langle K_A^2\rangle-\langle K_A\rangle^2$0 term (Arias et al., 2023). For the massive Dirac field, by contrast, the first mass correction agrees for entropy and capacity,

$C_E=\langle K_A^2\rangle-\langle K_A\rangle^2$1

(Arias et al., 2023). These examples show that the familiar CFT tracking of $C_E=\langle K_A^2\rangle-\langle K_A\rangle^2$2 and $C_E=\langle K_A^2\rangle-\langle K_A\rangle^2$3 is fragile under nonconformal perturbations.

Gaussian lattice systems make these distinctions concrete. In free fermionic chains, the capacity displays oscillatory finite-size corrections absent in $C_E=\langle K_A^2\rangle-\langle K_A\rangle^2$4,

$C_E=\langle K_A^2\rangle-\langle K_A\rangle^2$5

while the entropy has no corresponding oscillatory term (Arias et al., 2023). This places $C_E=\langle K_A^2\rangle-\langle K_A\rangle^2$6 closer to higher Rényi entropies than to the von Neumann entropy in its subleading sensitivity.

The review article also emphasized the relation of capacity to thermal heat capacity for spherical regions in CFT. Via the conformal map between a vacuum ball and a thermal state on $C_E=\langle K_A^2\rangle-\langle K_A\rangle^2$7, one finds

$C_E=\langle K_A^2\rangle-\langle K_A\rangle^2$8

for the associated hyperbolic thermal system (Boer et al., 2018). In higher-dimensional CFTs this leads to universal relations between the coefficients controlling $C_E=\langle K_A^2\rangle-\langle K_A\rangle^2$9 and $K_A=-\log\rho_A$0, although beyond the most symmetric cases the ratio is generally not universal and can depend on regularization or shape data (Boer et al., 2018).

5. Volume laws, squeezed states, and Lifshitz theories

Capacity of entanglement becomes particularly informative in systems with volume-law entanglement. In the vacuum of nonlocal scalar theories with Hamiltonian

$K_A=-\log\rho_A$1

the entropy scales as $K_A=-\log\rho_A$2 for $K_A=-\log\rho_A$3, whereas the capacity behaves as

$K_A=-\log\rho_A$4

and saturates for $K_A=-\log\rho_A$5 (Mozaffar, 2024). Thus both can be extensive in subsystem size, but the ratio $K_A=-\log\rho_A$6 becomes small at large nonlocality scale, suggesting a reduced density matrix close to maximally mixed (Mozaffar, 2024).

Squeezed states of scalar fields furnish another volume-law regime. For Gaussian squeezed states of a free scalar, the capacity obeys a volume law in the large-squeezing limit (Mozaffar, 2024). In the all-modes-squeezed regime,

$K_A=-\log\rho_A$7

while

$K_A=-\log\rho_A$8

(Mozaffar, 2024). In the continuum,

$K_A=-\log\rho_A$9

so both are extensive but

$$C_E=\left.\frac{\partial^2}{\partial n^2}\log \mathrm{Tr}\,\rho_A^n\right|_{n=1},$$0

for large squeezing (Mozaffar, 2024). Mode-by-mode, entropy keeps growing with squeezing whereas capacity saturates, again indicating a flattening entanglement spectrum.

The broader study of volume-law settings confirmed the same pattern in several other systems, including field-space entanglement between interacting scalar theories (Mozaffar, 2024). There, strong coupling can drive entropy to diverge while the capacity saturates to a finite density, so $$C_E=\left.\frac{\partial^2}{\partial n^2}\log \mathrm{Tr}\,\rho_A^n\right|_{n=1},$$1 in the strong-coupling limit (Mozaffar, 2024).

Lifshitz theories replace Lorentz invariance by anisotropic scaling $$C_E=\left.\frac{\partial^2}{\partial n^2}\log \mathrm{Tr}\,\rho_A^n\right|_{n=1},$$2, $$C_E=\left.\frac{\partial^2}{\partial n^2}\log \mathrm{Tr}\,\rho_A^n\right|_{n=1},$$3, and reveal further departures from relativistic behavior (Khoshdooni et al., 13 May 2025). In bosonic Lifshitz theories, capacity grows logarithmically with subsystem size and increases with dynamical exponent $$C_E=\left.\frac{\partial^2}{\partial n^2}\log \mathrm{Tr}\,\rho_A^n\right|_{n=1},$$4, but more slowly than the entropy. In the massless Dirichlet case,

$$C_E=\left.\frac{\partial^2}{\partial n^2}\log \mathrm{Tr}\,\rho_A^n\right|_{n=1},$$5

so the relativistic $$C_E=\left.\frac{\partial^2}{\partial n^2}\log \mathrm{Tr}\,\rho_A^n\right|_{n=1},$$6 leading equality is recovered only in the Lorentz-invariant limit (Khoshdooni et al., 13 May 2025). In fermionic Lifshitz theories, by contrast, the small-mass leading coefficient of $$C_E=\left.\frac{\partial^2}{\partial n^2}\log \mathrm{Tr}\,\rho_A^n\right|_{n=1},$$7 is essentially $$C_E=\left.\frac{\partial^2}{\partial n^2}\log \mathrm{Tr}\,\rho_A^n\right|_{n=1},$$8-independent (Khoshdooni et al., 13 May 2025). The contrast underscores that the relation between entropy and capacity is highly model dependent once the entanglement spectrum departs from relativistic CFT structure.

6. Dynamics, RG diagnostics, and gravity

In out-of-equilibrium many-body settings, capacity typically obeys the same broad kinematic pattern as entanglement entropy but with quantitatively different mode weights. After global quenches in free bosonic and fermionic chains, both $$C_E=\left.\frac{\partial^2}{\partial n^2}\log \mathrm{Tr}\,\rho_A^n\right|_{n=1},$$9 and $\mathrm{Tr}\,\rho_A^n=\mathrm{Tr}\,e^{-nK_A}$0 show initial linear growth followed by saturation, and both admit quasiparticle formulas,

$\mathrm{Tr}\,\rho_A^n=\mathrm{Tr}\,e^{-nK_A}$1

(Arias et al., 2023). The slopes, however, differ because the capacity density $\mathrm{Tr}\,\rho_A^n=\mathrm{Tr}\,e^{-nK_A}$2 weights modes differently from the entropy density $\mathrm{Tr}\,\rho_A^n=\mathrm{Tr}\,e^{-nK_A}$3 (Arias et al., 2023). The same distinction persists in Lifshitz quenches, where slow modes play an especially strong role in bosonic theories (Khoshdooni et al., 13 May 2025).

Locally excited states exhibit a still more distinctive signature. In free massless fermion theory and free Yang–Mills theory in four dimensions, the excess capacity $\mathrm{Tr}\,\rho_A^n=\mathrm{Tr}\,e^{-nK_A}$4 develops a universal early-time peak with height

$\mathrm{Tr}\,\rho_A^n=\mathrm{Tr}\,e^{-nK_A}$5

whereas the entropy grows monotonically toward its late-time value (Nandy, 2021). The paper introduced a normalized “Page time” $\mathrm{Tr}\,\rho_A^n=\mathrm{Tr}\,e^{-nK_A}$6, defined by the time at which this peak occurs, and argued that $\mathrm{Tr}\,\rho_A^n=\mathrm{Tr}\,e^{-nK_A}$7 is characteristic of the inserted operator (Nandy, 2021). This sharply illustrates the capacity’s sensitivity to partial entanglement structure.

Candidate RG diagnostics based on capacity have also been investigated. Defining

$\mathrm{Tr}\,\rho_A^n=\mathrm{Tr}\,e^{-nK_A}$8

one finds numerically monotonic behavior in several relativistic free models under mass-driven RG flow (Arias et al., 2023). However, this monotonicity is not universal: in Lifshitz theories with $\mathrm{Tr}\,\rho_A^n=\mathrm{Tr}\,e^{-nK_A}$9, the corresponding

$\rho_A$00

fails to be monotonic in both bosonic and fermionic models (Khoshdooni et al., 13 May 2025). This suggests that capacity-based $\rho_A$01-functions are not genuine nonrelativistic RG monotones, even when entropy-based ones remain well behaved.

In gravity and black-hole information theory, capacity becomes a refined probe of replica-wormhole and island physics. In semiclassical dilaton gravity, the capacity can be written in terms of refined Rényi derivatives, and a general gravitational formula was derived: $\rho_A$02 (Kawabata et al., 2021). This differs structurally from the generalized entropy formula because it depends on the $\rho_A$03 deformation of the replica saddle, not only on the $\rho_A$04 QES data (Kawabata et al., 2021, Arias et al., 10 Mar 2026).

This sensitivity makes the capacity especially sharp at Page transitions. In toy models of Hawking radiation, it can show either a peak near the Page time or a discontinuity when the dominant saddle switches, even when the entropy remains continuous (Kawabata et al., 2021, Kawabata et al., 2021). In the microcanonical end-of-the-world brane model, the capacity peaks at the Page point with value

$\rho_A$05

while in moving-mirror or semiclassical saddle-switch settings it can jump across the transition (Kawabata et al., 2021).

The RST gravity analysis sharpened this further. For a single interval, the generalized capacity is time independent,

$\rho_A$06

parallel to the generalized entropy (Arias et al., 10 Mar 2026). For two intervals on the island saddle, however, the global replica solution produces an interaction term

$\rho_A$07

so after Lorentzian continuation the capacity remains time dependent even when the entropy is already on a Page plateau (Arias et al., 10 Mar 2026). This is a concrete example in which higher modular cumulants remain dynamical after entropy has saturated.

7. Entangling capacity of processes and related operational capacities

The process-based notion of capacity of entanglement studies how much entanglement a quantum operation can create. In the framework of “Optimal Entangling Capacity of Dynamical Processes” (Campbell, 2010), a bipartite CPTP map $\rho_A$08 acts on $\rho_A$09 with arbitrary local ancillas allowed. The key distinction is between unassisted entangling capacity, optimized over separable inputs, and assisted entangling capacity, optimized over all inputs but subtracting the input entanglement: $\rho_A$10 (Campbell, 2010).

A central result is resource independence for broad classes of logarithmic decomposition-based monotones. For log-robustness, one has

$\rho_A$11

for every CPTP map (Campbell, 2010). For log-negativity, the paper proved an exact formula for all two-qubit unitaries. If $\rho_A$12 has operator Schmidt coefficients $\rho_A$13, then

$\rho_A$14

equal to the log-negativity of the Choi state (Campbell, 2010). This gives an operational interpretation to Choi-state entanglement and shows that prior entanglement does not improve the one-shot log-negativity capacity for all two-qubit unitaries.

The same theme reappears in negativity-based thesis work, where ancilla-assisted entangling capacity is bounded in terms of the non-PPT content of the partially transposed operation (Kao, 2020). There the capacity is explicitly a one-shot process capability, not a modular-spectrum fluctuation.

A related but distinct operational usage arises in network theory. In quantum networks with probabilistic link generation and probabilistic entanglement swapping, the “overall bipartite entanglement capacity” denotes the expected number of end-to-end EPR pairs deliverable per unit time between a source–destination pair (Vardoyan et al., 2023). For a realized network state $\rho_A$15, the state capacity is

$\rho_A$16

and the overall capacity is

$\rho_A$17

(Vardoyan et al., 2023). This is a throughput metric rather than an entanglement-spectrum quantity, but the shared terminology often causes confusion.

The contrast with the state-based modular capacity is therefore fundamental. Process/network capacities quantify entanglement generation capability of channels, Hamiltonians, or networks (Campbell, 2010, Shrimali et al., 2022, Vardoyan et al., 2023). State-based capacity quantifies entanglement-spectrum fluctuations already present in a reduced state (Boer et al., 2018, Wei, 2022).

8. Conceptual significance and recurrent themes

Across its different literatures, capacity of entanglement is valuable because it refines entanglement characterization beyond entropy alone. In the state-based setting, it distinguishes between states with similar $\rho_A$18 but different entanglement-spectrum widths. Maximally entangled EPR-like structure can carry entropy with zero capacity, whereas partially entangled or thermally broad spectra generate nonzero modular variance (Nandy, 2021, Boer et al., 2018).

A second recurrent theme is that $\rho_A$19 often shares the leading scaling of entropy—such as logarithmic scaling in $\rho_A$20-dimensional CFT or volume-law scaling in suitable excited, nonlocal, or field-space-entangled systems—while differing strongly in subleading terms or coefficients (Arias et al., 2023, Mozaffar, 2024). This makes it especially sensitive to structure that the entropy averages away.

A third theme is that $\rho_A$21 is naturally a susceptibility. In gravity it probes subleading replica saddles, partially connected wormholes, and first-order replica backreaction (Okuyama, 2021, Kawabata et al., 2021, Arias et al., 10 Mar 2026). In dynamics it highlights early-time partial entanglement and can peak at operator-specific times (Nandy, 2021). In RG studies it can behave monotonically in relativistic settings yet fail beyond Lorentz invariance (Khoshdooni et al., 13 May 2025).

A final theme concerns holography. In simple Einstein-gravity settings one often finds $\rho_A$22, especially for spherical regions (Boer et al., 2018). Many volume-law Gaussian and nonlocal states instead satisfy $\rho_A$23, which has been interpreted as evidence against a simple classical holographic dual, though this remains a diagnostic rather than a theorem (Mozaffar, 2024, Mozaffar, 2024).

Taken together, these results establish capacity of entanglement as a technically rich and conceptually distinct observable. Whether it refers to modular-Hamiltonian variance of states or one-shot entangling capability of processes, it exposes structure invisible to entanglement entropy alone.