Resource Nongenerating Operations

• Resource Nongenerating Operations (RNOs) are completely positive trace-preserving maps that preserve free states, forming the maximal set of free operations in quantum resource theories.
• They underpin various frameworks by ensuring that free inputs remain resource-free in domains such as coherence, thermodynamics, entanglement, and Gaussian/non-Gaussian theories.
• The approach unifies distinct operational classes—like self-dual maps, dephasing-covariant operations, and strictly incoherent operations—guiding the design of free superchannels and dynamical resources.

Searching arXiv for recent and foundational papers on resource nongenerating operations, coherence, and related quantum resource-theory frameworks. Resource nongenerating operations (RNOs) are completely positive trace-preserving maps that send every free state of a quantum resource theory to a free state. In the notation of affine resource theories, they form the maximal set $O_{\max}$ of free maps, while in generic convex theories they are written as $${\cal M}_R=\{\Lambda\in{\cal C}_{\cal H}\mid \forall\,\rho\in{\cal F}_R:\Lambda(\rho)\in{\cal F}_R\}$$ (Gour, 2016, Shi, 16 Sep 2025). The notion appears across coherence, thermodynamics, asymmetry, entanglement, Gaussian and non-Gaussian continuous-variable theories, and dynamical resource theories of channels. In that literature, RNOs serve both as a maximal benchmark—capturing every map that never creates the resource from free inputs—and as the basis for more structured subclasses such as self-dual maps, dephasing-covariant operations, strictly incoherent operations, Gaussian operations, and algebraically defined automorphism channels (Theurer et al., 2018, Diaz et al., 14 Jul 2025).

1. Definitional core and maximality

A standard definition fixes a convex set $F\subset{\cal D}({\cal H})$ of free states and declares a CPTP map $\Lambda$ to be resource non-generating when $\Lambda(F)\subseteq F'$, equivalently $\forall \sigma\in F,\ \Lambda(\sigma)\in F'$ (Ng et al., 2018). Gour’s single-shot framework denotes the maximal set of all such maps by $O_{\max}$, and states explicitly that any physically motivated free-operation set $O$ satisfies $O\subseteq O_{\max}$ (Gour, 2016). In affine resource theories, this maximality is structural rather than merely terminological: no larger CPTP set can satisfy the RNG condition.

Affine resource theories are those for which the free set is affine, equivalently ${\cal F}=V\cap H_{d,+,1}$, where $${\cal M}_R=\{\Lambda\in{\cal C}_{\cal H}\mid \forall\,\rho\in{\cal F}_R:\Lambda(\rho)\in{\cal F}_R\}$$0 is the real linear span of free states. The examples listed in this framework are athermality, coherence, and asymmetry under a compact group, while entanglement is explicitly excluded from the affine class (Gour, 2016). This matters because the single-shot conversion theory, the dual constructions, and the conditional-min-entropy monotones are developed precisely for this affine setting.

The same paper distinguishes a maximal set of RNOs from a self-dual subset. For affine input and output free sets, the self-dual set $${\cal M}_R=\{\Lambda\in{\cal C}_{\cal H}\mid \forall\,\rho\in{\cal F}_R:\Lambda(\rho)\in{\cal F}_R\}$$1 consists of maps $${\cal M}_R=\{\Lambda\in{\cal C}_{\cal H}\mid \forall\,\rho\in{\cal F}_R:\Lambda(\rho)\in{\cal F}_R\}$$2 satisfying simultaneously $${\cal M}_R=\{\Lambda\in{\cal C}_{\cal H}\mid \forall\,\rho\in{\cal F}_R:\Lambda(\rho)\in{\cal F}_R\}$$3 and $${\cal M}_R=\{\Lambda\in{\cal C}_{\cal H}\mid \forall\,\rho\in{\cal F}_R:\Lambda(\rho)\in{\cal F}_R\}$$4. When a resource-destroying map $${\cal M}_R=\{\Lambda\in{\cal C}_{\cal H}\mid \forall\,\rho\in{\cal F}_R:\Lambda(\rho)\in{\cal F}_R\}$$5 exists, $${\cal M}_R=\{\Lambda\in{\cal C}_{\cal H}\mid \forall\,\rho\in{\cal F}_R:\Lambda(\rho)\in{\cal F}_R\}$$6 coincides with the set of $${\cal M}_R=\{\Lambda\in{\cal C}_{\cal H}\mid \forall\,\rho\in{\cal F}_R:\Lambda(\rho)\in{\cal F}_R\}$$7-covariant operations (Gour, 2016). The literature therefore does not identify “free operations” with a unique operational class; rather, it repeatedly distinguishes the maximal non-generating set from smaller families with additional symmetry, duality, or implementation constraints.

2. Resource-destroying and algebraic formulations

A second major formulation begins from a resource-destroying map $${\cal M}_R=\{\Lambda\in{\cal C}_{\cal H}\mid \forall\,\rho\in{\cal F}_R:\Lambda(\rho)\in{\cal F}_R\}$$8, defined as a CPTP projector onto the free set: $${\cal M}_R=\{\Lambda\in{\cal C}_{\cal H}\mid \forall\,\rho\in{\cal F}_R:\Lambda(\rho)\in{\cal F}_R\}$$9 and $F\subset{\cal D}({\cal H})$0 (Theurer et al., 2018). In this language, a CPTP map $F\subset{\cal D}({\cal H})$1 is resource-non-generating when

$F\subset{\cal D}({\cal H})$2

The same framework records two equivalent conditions: $F\subset{\cal D}({\cal H})$3 These correspond to generation-non-generating and detection-non-generating constraints, respectively. The formalism extends naturally to super-operations generated by free pre-processing, post-processing, and tensoring, and every free super-operation can be written in the normal form

$F\subset{\cal D}({\cal H})$4

with $F\subset{\cal D}({\cal H})$5 free (Theurer et al., 2018).

A more recent unification replaces the resource-destroying map by a preferred algebraic structure $F\subset{\cal D}({\cal H})$6, consisting of distinguished operators $F\subset{\cal D}({\cal H})$7 together with algebraic relations $F\subset{\cal D}({\cal H})$8 such as $F\subset{\cal D}({\cal H})$9, multiplication, or commutators (Diaz et al., 14 Jul 2025). In that approach, a CPTP map

$\Lambda$0

is an RNO when each Kraus operator $\Lambda$1, so that conjugation by $\Lambda$2 preserves both the set $\Lambda$3 and the defining algebraic relations. Two consistency results are then stated. Theorem 1 gives the free-to-free property, $\Lambda$4, for free states defined as minimal orbits under the compact subgroup $\Lambda$5. Theorem 2 states resource monotonicity: any valid resource measure $\Lambda$6 that is nonincreasing under unitaries in $\Lambda$7 also obeys $\Lambda$8 for every RNO (Diaz et al., 14 Jul 2025).

For Lie-algebra-based theories, the automorphism perspective leads to “Complexified Free Operations” (CFOs), namely channels whose Kraus operators lie in

$\Lambda$9

In entanglement theory with $\Lambda(F)\subseteq F'$0, this produces $\Lambda(F)\subseteq F'$1 and $\Lambda(F)\subseteq F'$2, so SLOCC channels become the RNOs of the theory when one chooses $\Lambda(F)\subseteq F'$3 (Diaz et al., 14 Jul 2025). The same work presents this as a general answer to the Barnum et al. open problem of defining SLOCC analogues for arbitrary Lie-algebra-based resource theories, and notes that recent works by Jackson–Caves show that these complexified transformations can be physically implemented by sequences of continuous isotropic weak measurements.

3. Coherence as the principal laboratory

The resource theory of coherence supplies the most detailed finite-dimensional examples. Fixing an incoherent basis $\Lambda(F)\subseteq F'$4, the free states are exactly the diagonal density operators,

$\Lambda(F)\subseteq F'$5

At the level of operations, the literature distinguishes several non-equivalent classes. Maximally incoherent operations (MIO) require only $\Lambda(F)\subseteq F'$6. Dephasing-covariant operations (DIO) satisfy

$\Lambda(F)\subseteq F'$7

and therefore never produce off-diagonal terms from diagonal inputs. Strictly incoherent operations (SIO) strengthen incoherence to a Kraus-wise and adjoint-Kraus-wise condition: $\Lambda(F)\subseteq F'$8 Equivalently, each SIO Kraus operator is either purely diagonal or purely off-diagonal in the incoherent basis (Du et al., 2020, Chitambar, 2017).

For one qubit, Du and Bai give a canonical SIO representation with at most four Kraus operators, two diagonal and two anti-diagonal: $\Lambda(F)\subseteq F'$9 with $\forall \sigma\in F,\ \Lambda(\sigma)\in F'$0 and $\forall \sigma\in F,\ \Lambda(\sigma)\in F'$1 (Du et al., 2020). For bistochastic SIOs, the same paper derives a structural theorem expressing every such channel as a convex combination of Pauli- and phase-operator channels, together with cross-terms intertwining $\forall \sigma\in F,\ \Lambda(\sigma)\in F'$2 and $\forall \sigma\in F,\ \Lambda(\sigma)\in F'$3 through the phase operator $\forall \sigma\in F,\ \Lambda(\sigma)\in F'$4. In Bloch form, $\forall \sigma\in F,\ \Lambda(\sigma)\in F'$5, the action is affine but unital,

$\forall \sigma\in F,\ \Lambda(\sigma)\in F'$6

with $\forall \sigma\in F,\ \Lambda(\sigma)\in F'$7, $\forall \sigma\in F,\ \Lambda(\sigma)\in F'$8, and $\forall \sigma\in F,\ \Lambda(\sigma)\in F'$9. The $O_{\max}$0-plane is therefore mixed by a $O_{\max}$1 block, while the $O_{\max}$2-component is scaled independently.

Another operation-level viewpoint, developed in work on nonclassicality of operations, defines “classical” maps by the commutation relation

$O_{\max}$3

and quantifies failure of this relation through

$O_{\max}$4

The decomposition of this relative entropy into “distinguishing-power” and “generating-power” terms makes explicit that operation-level resourcefulness in coherence is not exhausted by state-generation alone (Meznaric, 2013).

4. Convertibility, monotones, and asymptotic behavior

In single-shot affine resource theories, Gour derives necessary and sufficient conditions for conversion under the maximal set $O_{\max}$5. The central statement is that $O_{\max}$6 under an RNO exists if and only if a complete family of conditional-min-entropy monotones satisfies

$O_{\max}$7

for all allowed choices of $O_{\max}$8 (Gour, 2016). In the same framework, the monotones are faithful, additive on tensor products, and monotonic under any RNO. For coherence under MIO, the paper gives the explicit form of the associated bipartite state $O_{\max}$9 built from $O$0, a free basis, and auxiliary states with uniform diagonal.

A separate asymptotic picture emerges for coherence under DIO. Chitambar shows that any two states $O$1 and $O$2 are asymptotically interconvertible under DIO at the unique rate

$O$3

In particular, distillation to the maximally coherent bit occurs at rate $O$4, formation from that bit has inverse rate $O$5, and the theory is fully reversible (Chitambar, 2017). The same work lifts the construction to maximally correlated entanglement via MCDC, where the corresponding rate becomes the ratio of relative entropies of entanglement.

Thermodynamic resource theory gives another canonical RNO example. Taking the free set to be the Gibbs state $O$6, a thermal operation has the form

$O$7

subject to the energy-conservation condition $O$8, and by construction preserves $O$9 (Ng et al., 2018). For block-diagonal states, state conversion is characterized by thermo-majorization. For catalytic thermal operations, the family of generalized free energies

$O\subseteq O_{\max}$0

obeys $O\subseteq O_{\max}$1 for all $O\subseteq O_{\max}$2, yielding the “second laws of quantum thermodynamics.”

One-qubit SIOs also exhibit explicit operational consequences. Du and Bai show that repeated application of a qubit SIO is relaxing iff all Bloch-matrix eigenvalues satisfy $O\subseteq O_{\max}$3 and $O\subseteq O_{\max}$4. They note in particular that the bit-flip $O\subseteq O_{\max}$5 and depolarizing $O\subseteq O_{\max}$6 channels are relaxing whenever $O\subseteq O_{\max}$7, and they derive the image of a Bloch vector under all stochastic SIOs as a cylinder determined by

$O\subseteq O_{\max}$8

For a “Pauli-only” subclass with $O\subseteq O_{\max}$9, the reachable set becomes a cuboid (Du et al., 2020).

Recent work extends convertibility statements in the maximal-RNO setting. In generic convex resource theories, pure-to-pure conversions in theories such as coherence under maximally incoherent operations and entanglement under non-entangling operations are governed by majorization, and Theorem II.1 gives the sufficient criterion

${\cal F}=V\cap H_{d,+,1}$0

in terms of generalized robustness and geometric measure (Shi, 16 Sep 2025).

5. Dynamical theories and free superchannels

The resource-non-increasing framework of Yang and Yu promotes the non-generation principle from states to channels. Given a static resource measure ${\cal F}=V\cap H_{d,+,1}$1, a channel ${\cal F}=V\cap H_{d,+,1}$2 is non-selective RNI when ${\cal F}=V\cap H_{d,+,1}$3 for all ${\cal F}=V\cap H_{d,+,1}$4, and selective RNI when ${\cal F}=V\cap H_{d,+,1}$5 for every Kraus decomposition ${\cal F}=V\cap H_{d,+,1}$6 (Yang et al., 2022). Their key theorem states that the set of RNI-free operations coincides with the static free-operation set. For dynamical coherence, this yields: without post-selection, the free maps are MIO; with post-selection, the free maps are IO.

The same work defines channel quantifiers by distance to the free dynamical set,

${\cal F}=V\cap H_{d,+,1}$7

and by maximal violation of the static monotonicity condition,

${\cal F}=V\cap H_{d,+,1}$8

Theorem 6 establishes faithfulness, convexity, and strong monotonicity for ${\cal F}=V\cap H_{d,+,1}$9 and $${\cal M}_R=\{\Lambda\in{\cal C}_{\cal H}\mid \forall\,\rho\in{\cal F}_R:\Lambda(\rho)\in{\cal F}_R\}$$00, while Theorem 7 gives the analogous properties for the violation-based measure. For the amplitude-damping channel, an analytic expression is obtained for the $${\cal M}_R=\{\Lambda\in{\cal C}_{\cal H}\mid \forall\,\rho\in{\cal F}_R:\Lambda(\rho)\in{\cal F}_R\}$$01-based dynamical total-coherence measure, and the diamond-norm distance from MIO acquires an operational meaning through a channel-discrimination advantage (Yang et al., 2022).

A complementary framework treats operations themselves as the resource carriers. Free super-operations are generated by pre-processing, post-processing, and tensoring with free maps, and the coherence-detection measures

$${\cal M}_R=\{\Lambda\in{\cal C}_{\cal H}\mid \forall\,\rho\in{\cal F}_R:\Lambda(\rho)\in{\cal F}_R\}$$02

and

$${\cal M}_R=\{\Lambda\in{\cal C}_{\cal H}\mid \forall\,\rho\in{\cal F}_R:\Lambda(\rho)\in{\cal F}_R\}$$03

are faithful, convex, and monotone under free super-operations (Theurer et al., 2018). The latter also has a single-shot discrimination interpretation,

$${\cal M}_R=\{\Lambda\in{\cal C}_{\cal H}\mid \forall\,\rho\in{\cal F}_R:\Lambda(\rho)\in{\cal F}_R\}$$04

The 2025 dynamical RNO framework makes the superchannel structure explicit. A free superchannel has the form

$${\cal M}_R=\{\Lambda\in{\cal C}_{\cal H}\mid \forall\,\rho\in{\cal F}_R:\Lambda(\rho)\in{\cal F}_R\}$$05

with $${\cal M}_R=\{\Lambda\in{\cal C}_{\cal H}\mid \forall\,\rho\in{\cal F}_R:\Lambda(\rho)\in{\cal F}_R\}$$06 and $${\cal M}_R=\{\Lambda\in{\cal C}_{\cal H}\mid \forall\,\rho\in{\cal F}_R:\Lambda(\rho)\in{\cal F}_R\}$$07, the absolutely RNO maps that remain RNO under tensoring with any other RNO (Shi, 16 Sep 2025). The same paper formulates axioms (P1–P4) for channel monotones, introduces a distance-based faithful monotone $${\cal M}_R=\{\Lambda\in{\cal C}_{\cal H}\mid \forall\,\rho\in{\cal F}_R:\Lambda(\rho)\in{\cal F}_R\}$$08, studies the $${\cal M}_R=\{\Lambda\in{\cal C}_{\cal H}\mid \forall\,\rho\in{\cal F}_R:\Lambda(\rho)\in{\cal F}_R\}$$09-destruction cost $${\cal M}_R=\{\Lambda\in{\cal C}_{\cal H}\mid \forall\,\rho\in{\cal F}_R:\Lambda(\rho)\in{\cal F}_R\}$$10 of erasing dynamical resources, bounds this cost by smooth channel robustness, and derives the one-shot communication bound

$${\cal M}_R=\{\Lambda\in{\cal C}_{\cal H}\mid \forall\,\rho\in{\cal F}_R:\Lambda(\rho)\in{\cal F}_R\}$$11

6. Representative domains and conceptual scope

RNOs appear in a wide range of resource theories, often with theory-specific implementations of the same non-generation principle.

Resource theory	Free states or structure	RNOs / free operations
Thermodynamics	$${\cal M}_R=\{\Lambda\in{\cal C}_{\cal H}\mid \forall\,\rho\in{\cal F}_R:\Lambda(\rho)\in{\cal F}_R\}$$12	Thermal operations preserving the Gibbs state (Ng et al., 2018)
Type-independent common-cause nonclassicality	Resources simulable by a classical common-cause model	LOSR, the convex hull of product local CPTP maps (Schmid et al., 2019)
Non-Gaussianity of operations	Gaussian states	Gaussian operations $${\cal M}_R=\{\Lambda\in{\cal C}_{\cal H}\mid \forall\,\rho\in{\cal F}_R:\Lambda(\rho)\in{\cal F}_R\}$$13, i.e. maps that preserve Gaussianity even on subsystems of larger Gaussian states (Zhuang et al., 2018)
Lie-algebra-based QRTs	Preferred algebraic structure $${\cal M}_R=\{\Lambda\in{\cal C}_{\cal H}\mid \forall\,\rho\in{\cal F}_R:\Lambda(\rho)\in{\cal F}_R\}$$14 and minimal $${\cal M}_R=\{\Lambda\in{\cal C}_{\cal H}\mid \forall\,\rho\in{\cal F}_R:\Lambda(\rho)\in{\cal F}_R\}$$15-orbits	Automorphism channels; for complexification, CFOs with Kraus operators in $${\cal M}_R=\{\Lambda\in{\cal C}_{\cal H}\mid \forall\,\rho\in{\cal F}_R:\Lambda(\rho)\in{\cal F}_R\}$$16 (Diaz et al., 14 Jul 2025)

In the type-independent LOSR theory, an LOSR map is written as

$${\cal M}_R=\{\Lambda\in{\cal C}_{\cal H}\mid \forall\,\rho\in{\cal F}_R:\Lambda(\rho)\in{\cal F}_R\}$$17

equivalently $${\cal M}_R=\{\Lambda\in{\cal C}_{\cal H}\mid \forall\,\rho\in{\cal F}_R:\Lambda(\rho)\in{\cal F}_R\}$$18, and it is exactly the family of RNOs because it never creates nonclassicality from classical common-cause resources (Schmid et al., 2019). In the resource theory of non-Gaussian operations, free operations are Gaussian operations $${\cal M}_R=\{\Lambda\in{\cal C}_{\cal H}\mid \forall\,\rho\in{\cal F}_R:\Lambda(\rho)\in{\cal F}_R\}$$19, and the entanglement-assisted non-Gaussianity-generating power $${\cal M}_R=\{\Lambda\in{\cal C}_{\cal H}\mid \forall\,\rho\in{\cal F}_R:\Lambda(\rho)\in{\cal F}_R\}$$20 is faithful and invariant under Gaussian pre-processing, post-processing, and tensoring. This framework classifies non-Gaussian operations into a finite-nG class $${\cal M}_R=\{\Lambda\in{\cal C}_{\cal H}\mid \forall\,\rho\in{\cal F}_R:\Lambda(\rho)\in{\cal F}_R\}$$21 and a diverging-nG class $${\cal M}_R=\{\Lambda\in{\cal C}_{\cal H}\mid \forall\,\rho\in{\cal F}_R:\Lambda(\rho)\in{\cal F}_R\}$$22 (Zhuang et al., 2018).

An operationally distinct but closely related construction is “Local Operations on Physical wires” (LOP). There the free maps are generated by four elemental moves: permutations on the wire, phase-diagonals on the wire, observed operations on local quantum systems, and classical-to-quantum forwarding. In the single-wire setting, every free map has Kraus operators

$${\cal M}_R=\{\Lambda\in{\cal C}_{\cal H}\mid \forall\,\rho\in{\cal F}_R:\Lambda(\rho)\in{\cal F}_R\}$$23

and the theory recovers the relation $${\cal M}_R=\{\Lambda\in{\cal C}_{\cal H}\mid \forall\,\rho\in{\cal F}_R:\Lambda(\rho)\in{\cal F}_R\}$$24 (Egloff et al., 2018). This unifies coherence on wires, discord-like resources in the one-wire setting, and multipartite entanglement in a common operational picture.

A recurrent misconception is that “RNO” names a single universally accepted free-operation class. The literature instead presents several non-equivalent choices built on the same principle: maximal RNG maps, self-dual RNG maps, DIO, SIO, LOSR, Gaussian operations, and automorphism-based CFOs (Gour, 2016, Du et al., 2020, Diaz et al., 14 Jul 2025). This suggests that RNOs are best understood as a general non-generation criterion whose concrete operational realization depends on the free-state geometry, the presence or absence of a resource-destroying map, and the physical restrictions one wishes to impose.