Operator Capacity: Variational Approaches & Applications

• Operator capacity is a multifaceted measure quantifying what an operator can preserve, create, or transmit through distinct variational constructions.
• Key methodologies include determinant-infimum in operator scaling, Choquet-type set functions in real-stable polynomial theory, and variance-based metrics in quantum information.
• Applications range from algorithm design and network optimization to quantum channel analysis, bridging theoretical insights with practical implementations.

Searching arXiv for recent and foundational papers related to “operator capacity” and closely related usages across operator scaling and quantum information. Operator capacity is a polysemous technical term. In the arXiv literature, it denotes several distinct variational constructions attached to operators, currents, channels, or operator-induced dynamics. These include determinant-based capacity for completely positive operators in operator scaling, capacity inequalities for linear operators acting on real-stable polynomials, set capacities generated by complex Hessian or fractional dissipative operators, entanglement-sensitive capacities associated with local or unitary operators, and channel- or network-capacity notions in which the relevant object is an operator system, a linear operator channel, or a service operator (Garg et al., 2015, Dhouib et al., 2015, Nandy, 2021, Patra, 2023). This suggests a shared structural theme—variational measurement of what an operator can preserve, create, regularize, or transmit—while also making clear that the term does not refer to a single universal invariant.

1. Taxonomic scope

A common source of confusion is terminological. In one body of work, operator capacity is a determinant-infimum on positive matrices; in another, it is a Choquet-type set function; in quantum-information settings it may be the variance of a modular Hamiltonian or the distance of a unitary from local unitaries; and in communications it may refer to capacities controlled by network operators or to channels defined by random linear operators. The underlying optimization variables, admissible objects, and operational meanings are therefore context-dependent.

Domain	Basic object	Representative capacity
Operator scaling	Completely positive operator $\mathcal T$	$\operatorname{cap}(\mathcal T)$ via determinants
Stable-polynomial theory	Linear operator $T$ on polynomials	$$\operatorname{cpc}_\alpha(P)=\inf_{x>0}P(x)/x^\alpha$$
Nonlinear potential theory	Hessian or dissipative operator	$\operatorname{cap}_{m,T}$, $\operatorname{cap}_{m,T}^{\phi}$, $C_{p,q}^{(\alpha)}$, $\operatorname{Cap}_{V,\infty}$
Quantum operators	Local operator or unitary $U$	$C_E(\rho_A)$, $\operatorname{cap}(\mathcal T)$0, $\operatorname{cap}(\mathcal T)$1
Quantum channels	Operator system or TRO-channel	$\operatorname{cap}(\mathcal T)$2 bounds, $\operatorname{cap}(\mathcal T)$3, $\operatorname{cap}(\mathcal T)$4, $\operatorname{cap}(\mathcal T)$5
Communications networks	Linear operator channel or service operator	$\operatorname{cap}(\mathcal T)$6, $\operatorname{cap}(\mathcal T)$7, leasing/sharing capacities

Representative formulations of these usages appear in operator scaling and real-stability theory (Garg et al., 2015, Gurvits et al., 2018), in pluripotential and parabolic analysis (Dhouib et al., 2015, Jiang et al., 2012), in quantum-information treatments of local operators and unitaries (Nandy, 2021, Patra, 2023), and in communications and network-operations models (Yang et al., 2011, Duan et al., 2012).

2. Determinant-based capacity in operator scaling

For a completely positive operator $\operatorname{cap}(\mathcal T)$8 with Kraus form $\operatorname{cap}(\mathcal T)$9, capacity is defined by

$T$0

In the square case $T$1, the normalization used in operator scaling is

$T$2

This quantity is the central potential function in Gurvits’s operator-scaling algorithm, where alternating left and right scalings drive row- and column-marginals toward the identity (Bez et al., 4 Aug 2025, Garg et al., 2015).

Several structural properties are fundamental. If

$T$3

with $T$4, then

$T$5

If $T$6 is trace-preserving or $T$7 is trace-preserving, then $T$8. The quantity

$T$9

measures distance to double stochasticity; when $$\operatorname{cpc}_\alpha(P)=\inf_{x>0}P(x)/x^\alpha$$0 is small, one proves that $$\operatorname{cpc}_\alpha(P)=\inf_{x>0}P(x)/x^\alpha$$1 is rank-non-decreasing and that capacity is close to $$\operatorname{cpc}_\alpha(P)=\inf_{x>0}P(x)/x^\alpha$$2. In the algorithmic analysis of alternating normalization, the progress estimate

$$\operatorname{cpc}_\alpha(P)=\inf_{x>0}P(x)/x^\alpha$$3

is the key monotonicity statement (Garg et al., 2015).

The determinant-infimum also has direct complexity-theoretic content. For the square completely positive operator $$\operatorname{cpc}_\alpha(P)=\inf_{x>0}P(x)/x^\alpha$$4, one has $$\operatorname{cpc}_\alpha(P)=\inf_{x>0}P(x)/x^\alpha$$5 if and only if $$\operatorname{cpc}_\alpha(P)=\inf_{x>0}P(x)/x^\alpha$$6 does not decrease rank on any psd $$\operatorname{cpc}_\alpha(P)=\inf_{x>0}P(x)/x^\alpha$$7, and Gurvits’s theorem identifies this with non-commutative invertibility of $$\operatorname{cpc}_\alpha(P)=\inf_{x>0}P(x)/x^\alpha$$8. This is the bridge from capacity to deterministic polynomial-time algorithms for the non-commutative singularity problem and non-commutative polynomial identity testing (Garg et al., 2015).

A later regularity theorem substantially strengthens continuity information. There exists an exponent $$\operatorname{cpc}_\alpha(P)=\inf_{x>0}P(x)/x^\alpha$$9, depending only on $\operatorname{cap}_{m,T}$0, such that on every compact $\operatorname{cap}_{m,T}$1 there is $\operatorname{cap}_{m,T}$2 with

$\operatorname{cap}_{m,T}$3

for all $\operatorname{cap}_{m,T}$4. The proof reduces capacity to an infimum of weighted sums of exponentials and then uses the Bennett–Bez–Buschenhenke–Cowling–Flock theorem together with Lipschitz dependence of the determinant coefficients $\operatorname{cap}_{m,T}$5 on $\operatorname{cap}_{m,T}$6 (Bez et al., 4 Aug 2025). A plausible implication is that perturbative analyses of operator scaling can be formulated with genuinely Hölder, rather than inverse-logarithmic, stability.

3. Capacity-preserving operators in real-stable polynomial theory

A different notion of capacity appears for polynomials with nonnegative coefficients. If

$\operatorname{cap}_{m,T}$7

and $\operatorname{cap}_{m,T}$8, the $\operatorname{cap}_{m,T}$9-capacity is

$\operatorname{cap}_{m,T}^{\phi}$0

This capacity is finite exactly when $\operatorname{cap}_{m,T}^{\phi}$1 lies in the Newton polytope of $\operatorname{cap}_{m,T}^{\phi}$2, and it behaves naturally under products, scaling, diagonalization, and external fields (Gurvits et al., 2018).

The central question is how much a linear operator $\operatorname{cap}_{m,T}^{\phi}$3 that preserves real stability can shrink capacity. In the bounded-degree setting, if $\operatorname{cap}_{m,T}^{\phi}$4 has symbol

$\operatorname{cap}_{m,T}^{\phi}$5

with nonnegative coefficients and real stability in $\operatorname{cap}_{m,T}^{\phi}$6, then for every real-stable $\operatorname{cap}_{m,T}^{\phi}$7 and nonnegative vectors $\operatorname{cap}_{m,T}^{\phi}$8,

$\operatorname{cap}_{m,T}^{\phi}$9

The unbounded-degree analogue replaces the prefactor by $C_{p,q}^{(\alpha)}$0 and uses the transcendental symbol $C_{p,q}^{(\alpha)}$1. In both cases the bounds are tight (Gurvits et al., 2018).

This framework unifies several lower-bound arguments in combinatorics. By choosing $C_{p,q}^{(\alpha)}$2 to be a permanent or matching polynomial and $C_{p,q}^{(\alpha)}$3 to be a differentiation operator extracting the relevant coefficient, one recovers the Van der Waerden bound, Schrijver’s inequality, and Csikvári’s lower bound settling Friedland’s lower matching conjecture. The significance of the capacity-preservation theorem is therefore methodological: it converts coefficient extraction into a symbol-capacity computation, reducing many counting lower bounds to a common analytic template (Gurvits et al., 2018).

4. Capacities induced by analytic operators

In pluripotential theory, Dhouib and Elkhadhra introduced the relative $C_{p,q}^{(\alpha)}$4-capacity associated to a closed $C_{p,q}^{(\alpha)}$5-positive current $C_{p,q}^{(\alpha)}$6 of bidimension $C_{p,q}^{(\alpha)}$7, $C_{p,q}^{(\alpha)}$8. For a Borel set $C_{p,q}^{(\alpha)}$9,

$\operatorname{Cap}_{V,\infty}$0

and equivalently

$\operatorname{Cap}_{V,\infty}$1

The capacity is monotone in $\operatorname{Cap}_{V,\infty}$2 and in $\operatorname{Cap}_{V,\infty}$3, continuous under increasing unions, and characterizes $\operatorname{Cap}_{V,\infty}$4-pluripolar sets by the condition $\operatorname{Cap}_{V,\infty}$5. When $\operatorname{Cap}_{V,\infty}$6 and $\operatorname{Cap}_{V,\infty}$7, it reduces to the classical $\operatorname{Cap}_{V,\infty}$8-relative Monge–Ampère capacity (Dhouib et al., 2015).

The same paper develops Cegrell-type classes $\operatorname{Cap}_{V,\infty}$9, $U$0, and $U$1 for negative $U$2-subharmonic functions and proves continuity of the complex Hessian operator on decreasing sequences in these classes. It also establishes a Xing-type comparison principle for $U$3 and introduces an $U$4-potential current $U$5 defined locally by convolution with a Riesz-type kernel,

$U$6

which specializes to the classical Lelong–Skoda local potential when $U$7 (Dhouib et al., 2015).

A weighted refinement replaces the constant barrier $U$8 by a negative $U$9-subharmonic weight $C_E(\rho_A)$0. For compact $C_E(\rho_A)$1,

$C_E(\rho_A)$2

This weighted $C_E(\rho_A)$3-capacity is monotone in the set and in the weight, continuous under exhaustion, and subadditive. It is linked to a weighted $C_E(\rho_A)$4-extremal function $C_E(\rho_A)$5, and it characterizes Cegrell classes through finiteness criteria: $C_E(\rho_A)$6 if and only if $C_E(\rho_A)$7, while $C_E(\rho_A)$8 if and only if $C_E(\rho_A)$9 for every compact $\operatorname{cap}(\mathcal T)$00 (Elaini et al., 2019).

For the fractional dissipative operator

$\operatorname{cap}(\mathcal T)$01

the associated $\operatorname{cap}(\mathcal T)$02-capacity of a compact set $\operatorname{cap}(\mathcal T)$03 is

$\operatorname{cap}(\mathcal T)$04

where $\operatorname{cap}(\mathcal T)$05 is the Duhamel potential. This capacity is monotone, $\operatorname{cap}(\mathcal T)$06-subadditive, translation-invariant in $\operatorname{cap}(\mathcal T)$07, and compatible with parabolic scaling. The capacity of a parabolic ball has sharp asymptotics, and these estimates yield the Hausdorff-dimension bound

$\operatorname{cap}(\mathcal T)$08

for the blow-up set $\operatorname{cap}(\mathcal T)$09 in the subcritical regime (Jiang et al., 2012).

Rakotoson’s potential capacity addresses Schrödinger-type equations with singular potentials. For a compact $\operatorname{cap}(\mathcal T)$10,

$\operatorname{cap}(\mathcal T)$11

If $\operatorname{cap}(\mathcal T)$12, where $\operatorname{cap}(\mathcal T)$13 is the set of irregular points of $\operatorname{cap}(\mathcal T)$14, then every bounded Radon measure $\operatorname{cap}(\mathcal T)$15 satisfying $\operatorname{cap}(\mathcal T)$16 yields a unique very-weak solution $\operatorname{cap}(\mathcal T)$17; if $\operatorname{cap}(\mathcal T)$18, no such solution exists. A density theorem shows that $\operatorname{cap}(\mathcal T)$19 is dense in $\operatorname{cap}(\mathcal T)$20 whenever $\operatorname{cap}(\mathcal T)$21 (Rakotoson, 2018).

5. Entanglement-sensitive capacities of operators

In the study of local operator excitations, the capacity of entanglement is defined for a reduced density matrix $\operatorname{cap}(\mathcal T)$22 by the variance of the modular Hamiltonian $\operatorname{cap}(\mathcal T)$23: $\operatorname{cap}(\mathcal T)$24 Equivalently,

$\operatorname{cap}(\mathcal T)$25

For locally excited states, one studies the excess quantity

$\operatorname{cap}(\mathcal T)$26

In free, massless fermionic theory and free $\operatorname{cap}(\mathcal T)$27 Yang–Mills theory in four spacetime dimensions, $\operatorname{cap}(\mathcal T)$28 vanishes for $\operatorname{cap}(\mathcal T)$29, rises to a single maximum at $\operatorname{cap}(\mathcal T)$30, and then decays or approaches a nonzero constant depending on the operator sector (Nandy, 2021).

The early-time peak is quantitatively rigid. Numerically,

$\operatorname{cap}(\mathcal T)$31

in all cases analyzed, independent of $\operatorname{cap}(\mathcal T)$32, the spin parameter $\operatorname{cap}(\mathcal T)$33, or the number $\operatorname{cap}(\mathcal T)$34 of scalar insertions. The Page time is defined by the stationarity condition $\operatorname{cap}(\mathcal T)$35, and the normalized Page time

$\operatorname{cap}(\mathcal T)$36

depends only on operator data. Explicit examples are

$\operatorname{cap}(\mathcal T)$37

At late times, uncharged fermions and scalar insertions give $\operatorname{cap}(\mathcal T)$38, while fermions with $\operatorname{cap}(\mathcal T)$39 approach $\operatorname{cap}(\mathcal T)$40. The time dependence parallels the microcanonical and canonical replica-wormhole patterns discussed in the black-hole information paradox (Nandy, 2021).

A distinct but related notion is the entangling capacity of a unitary $\operatorname{cap}(\mathcal T)$41. Using a unitarily invariant metric $\operatorname{cap}(\mathcal T)$42 on states, one defines

$\operatorname{cap}(\mathcal T)$43

and then

$\operatorname{cap}(\mathcal T)$44

This is a minimax distance from $\operatorname{cap}(\mathcal T)$45 to the manifold $\operatorname{cap}(\mathcal T)$46 of local unitaries. The dual maximin quantity

$\operatorname{cap}(\mathcal T)$47

satisfies $\operatorname{cap}(\mathcal T)$48 (Patra, 2023).

The dual quantity admits a Schmidt-coefficient interpretation. If $\operatorname{cap}(\mathcal T)$49, then

$\operatorname{cap}(\mathcal T)$50

where $\operatorname{cap}(\mathcal T)$51 is the minimal largest Schmidt coefficient. Hence $\operatorname{cap}(\mathcal T)$52 if and only if the operator Schmidt rank of $\operatorname{cap}(\mathcal T)$53 exceeds $\operatorname{cap}(\mathcal T)$54, and $\operatorname{cap}(\mathcal T)$55 for Schmidt rank $\operatorname{cap}(\mathcal T)$56. For generalized control operators

$\operatorname{cap}(\mathcal T)$57

maximal entanglement generation occurs if and only if there is some $\operatorname{cap}(\mathcal T)$58 such that the vectors $\operatorname{cap}(\mathcal T)$59 are mutually orthogonal. The capacity $\operatorname{cap}(\mathcal T)$60 is invariant under left and right multiplication by local unitaries, satisfies $\operatorname{cap}(\mathcal T)$61, is subadditive under composition, and is continuous in the unitary metric (Patra, 2023).

6. Operator systems, channel capacities, and superactivation

Quantum channels provide another setting in which operator-structured capacity questions arise. Given a channel $\operatorname{cap}(\mathcal T)$62 with Kraus operators $\operatorname{cap}(\mathcal T)$63, its non-commutative confusability graph is the operator system

$\operatorname{cap}(\mathcal T)$64

The quantum complexity of an operator system $\operatorname{cap}(\mathcal T)$65 is

$\operatorname{cap}(\mathcal T)$66

and the sub-complexity is

$\operatorname{cap}(\mathcal T)$67

These quantities bound zero-error communication: $\operatorname{cap}(\mathcal T)$68 For suitable families, $\operatorname{cap}(\mathcal T)$69 and $\operatorname{cap}(\mathcal T)$70 outperform the previously known quantum Lovász-$\operatorname{cap}(\mathcal T)$71 upper bound (Levene et al., 2017).

For finite-dimensional TRO-channels,

$\operatorname{cap}(\mathcal T)$72

one has single-letter formulas

$\operatorname{cap}(\mathcal T)$73

and therefore

$\operatorname{cap}(\mathcal T)$74

If $\operatorname{cap}(\mathcal T)$75 is a symbol perturbation, the perturbed channel $\operatorname{cap}(\mathcal T)$76 satisfies

$\operatorname{cap}(\mathcal T)$77

with analogous estimates for $\operatorname{cap}(\mathcal T)$78, $\operatorname{cap}(\mathcal T)$79, and the strong converse capacities. The proofs use operator-space and complex-interpolation methods on the Stinespring TRO (Gao et al., 2016).

Zero-error capacity also interacts sharply with non-commutative operator graphs. Shirokov’s graph $\operatorname{cap}(\mathcal T)$80, generated by matrices $\operatorname{cap}(\mathcal T)$81, yields channels $\operatorname{cap}(\mathcal T)$82 whose one-shot zero-error capacity vanishes when $\operatorname{cap}(\mathcal T)$83, and in fact

$\operatorname{cap}(\mathcal T)$84

However, pairing inverse parameters produces superactivation: $\operatorname{cap}(\mathcal T)$85 At $\operatorname{cap}(\mathcal T)$86, the graph becomes abelian and the associated algebra degenerates to the direct sum of four one-dimensional irreducible representations of the Klein group, changing the zero-error behavior qualitatively (Amosov et al., 2015). A plausible implication is that algebraic non-commutativity can suppress single-channel zero-error transmission even when tensor products restore a commutative coding structure.

7. Linear operator channels and network-operator capacity management

In finite-field network coding, a linear operator channel (LOC) is

$\operatorname{cap}(\mathcal T)$87

with $\operatorname{cap}(\mathcal T)$88, $\operatorname{cap}(\mathcal T)$89, and $\operatorname{cap}(\mathcal T)$90. Its Shannon capacity is

$\operatorname{cap}(\mathcal T)$91

while the subspace-coding capacity $\operatorname{cap}(\mathcal T)$92 is obtained by encoding and decoding through column spaces. The central structural question is when $\operatorname{cap}(\mathcal T)$93. Sufficient and necessary conditions are formulated through unique subspace degradation, row-space symmetry, and Markov constraints such as

$\operatorname{cap}(\mathcal T)$94

If the LOC is degraded, then $\operatorname{cap}(\mathcal T)$95; if the relevant Markov conditions fail, $\operatorname{cap}(\mathcal T)$96 can be strictly less than $\operatorname{cap}(\mathcal T)$97 (Yang et al., 2011).

The broadcast analogue is the constant-dimension multiplicative linear operator broadcast channel (CMLOBC), whose input alphabet is the Grassmannian $\operatorname{cap}(\mathcal T)$98 and whose output law is determined by rank-deficiency distributions $\operatorname{cap}(\mathcal T)$99 and $T$00. Degradation holds if and only if

$T$01

Even in the degraded case, time sharing need not exhaust the boundary of the capacity region. In the smallest nontrivial example $T$02, the superposition boundary is strictly concave if

$T$03

strictly convex if the inequality is reversed, and linear when equality holds (Andrianov et al., 2010).

In engineering usage, the phrase can also refer to capacity controlled by communication-service operators. In a femtocell market model, the macrocell operator chooses a wholesale price $T$04 and leasing amount $T$05, while the femtocell operator chooses a retail price $T$06 and actual lease $T$07. The equilibrium exhibits two regimes separated by a unique threshold $T$08: when $T$09, the macrocell leases positive spectrum $T$10; when $T$11, it withholds all spectrum, so $T$12 (Duan et al., 2012). This directly formalizes the tradeoff between efficiency gains from femtocells and competitive cannibalization.

A stochastic capacity-sharing model for public-transit operators treats first-stage committed pooled capacity $T$13 and second-stage recourse flows under disruption scenarios. Applied to the Randstad network, the grand coalition of four operators reduces the baseline expected passenger-minutes cost from $T$14 million p-min to $T$15 million p-min, a $T$16 improvement in overall network performance over not having any risk-pooling contract in place. Cooperative-game allocations such as the Shapley value, nucleolus, and $T$17-value then quantify bargaining power as a function of contributed capacity and network structure (Pantelidis et al., 2020).

Across these communications examples, operator capacity is operational rather than purely variational: it measures achievable rates, region boundaries, or allocable infrastructure resources. This contrasts with determinant-based and potential-theoretic usages, but preserves the same formal pattern of optimization under structural constraints.