Resource Theory of Information Preservability

• Resource Theory of Information Preservability is a framework that defines how quantum systems preserve informational content and nonclassical resources under free operations.
• It employs state and channel formulations with monotones and robustness measures to quantify preservation, linking operational tasks to thermodynamic and communication costs.
• The framework unifies various quantum measures such as coherence, entanglement, and informational completeness through resource-destroying maps and experimentally accessible protocols.

Searching arXiv for recent and foundational papers on information/resource preservability. Resource theories of information preservability study the ability of physical transformations to retain, rather than create, informational or nonclassical resourcefulness. In the quantum setting, this theme appears in two closely related forms. One takes information itself as the primitive resource, quantified by $I(\rho)=\ln d-S(\rho)$, with preservation analyzed through resource-destroying operations and their induced information loss. The other induces a channel resource theory from a state resource theory, so that the central object is a channel’s ability to avoid annihilating a resource such as purity, incompatibility, coherence, entanglement, or informational completeness (Costa et al., 2020, Hsieh, 2019).

1. General resource-theoretic formulation

The axiomatic starting point is a state resource theory specified by a resource $R$, a set of free states $\mathcal{F}_R$, and a set of free operations $\mathcal{O}_R$. Within this setting, resource preservability quantifies a free operation’s ability to preserve resourcefulness rather than its ability to generate resource. The key free channels in the induced channel theory are the resource-annihilating channels,

$$\mathcal{O}_R^N := \{\mathcal{E} \in \mathcal{O}_R \mid \mathcal{E}(\rho) \in \mathcal{F}_R,\ \forall\, \rho\},$$

that is, channels whose outputs are always free states (Hsieh, 2019).

This induces a channel resource theory whose objects are channels in $\mathcal{O}_R$, whose free channels are $\mathcal{O}_R^N$, and whose free super-channels are formed by pre- and post-processing by free operations together with tensoring by absolutely resource-annihilating auxiliary channels. A resource preservability monotone $P_R$ is then required to be monotonic under these free super-channels and to satisfy the tensorisation property

$$P_R(\mathcal{E}\otimes \mathcal{E}') \geq P_R(\mathcal{E}),$$

with equality when $\mathcal{E}'$ is absolutely resource-annihilating (Hsieh, 2019).

A central conceptual point is that preservability is distinct from resource generation. Free operations may degrade resourcefulness, but they need not annihilate it for every input. This distinction is the basis for treating preservation ability as a dynamical resource in its own right. The framework of (Hsieh, 2019) explicitly presents this as a channel resource theory induced by a state resource theory and gives what it calls the first systematic and general formulation of the resource preservation character of free operations.

2. Information as the primitive resource

An information-based unification identifies quantum information itself as the most primitive resource. In this approach, the informational content of a state is

$R$0

where $R$1 is the Hilbert-space dimension and $R$2 is the von Neumann entropy. The unique free state is the maximally mixed state $R$3, while pure states are maximally resourceful (Costa et al., 2020).

The basic dynamical mechanism is the resource-destroying operation $R$4, a CPTP map that fixes free states and maps resourceful states to zero resource. Interpolating between identity and full destruction yields the monitoring map

$R$5

If the resource quantifier $R$6 is convex, then

$R$7

so monitoring cannot increase the resource (Costa et al., 2020).

For information itself, the relevant maximally resource-destroying operations are projective measurement maps in maximally incompatible bases. In the formulation summarized in (Costa et al., 2020), applying such maps erases all local and correlated information and sends the state to $R$8. This turns information preservability into control over the leakage induced by monitoring: $R$9 leaves the state unchanged, while $\mathcal{F}_R$0 implements full destruction.

The same paper uses this informational perspective to unify several standard and nonstandard quantum resources. Coherence in basis $\mathcal{F}_R$1 is written as

$\mathcal{F}_R$2

and irreality of observable $\mathcal{F}_R$3 is

$\mathcal{F}_R$4

For pure states, entanglement is expressed as informational loss under dephasing in the Schmidt basis; discord is the mutual information destroyed by local measurement; and realism-based nonlocality is the change in irreality of one observable under remote measurement of another (Costa et al., 2020).

The unifying claim is therefore not merely terminological. Each resource quantifier is cast as a difference between the information in the original state and the information remaining after a tailored resource-destroying map. In this sense, preserving a resource becomes equivalent to preserving the relevant informational content against a specified destructive monitoring procedure.

3. Monotones, robustness, and operational meaning

Two general classes of preservability monotones have been developed. The first class is built from state resource monotones and quantifies the optimal relative or absolute amount of resource that can be retained by a channel, possibly with ancillas. The second class is distance-based: given a contractive distance $\mathcal{F}_R$5, one measures how far a channel is from the set of resource-annihilating channels. A particularly important case is the robustness-like monotone

$\mathcal{F}_R$6

which quantifies the amount of perturbation required to erase all preservability (Hsieh, 2019).

These monotones admit explicit operational interpretations. In the resource theory of athermality, max-relative-entropy-based preservability is linked to the minimal bath size required to thermalize all outputs of a channel. The same robustness-type quantity also bounds the one-shot classical capacity in thermodynamically constrained communication scenarios. In that setting, the excess of the one-shot classical capacity over a resourceless baseline quantifies the resource-based communication advantage conferred by preservability (Hsieh, 2019, Hsieh, 2020).

The thermodynamic interpretation is especially notable. For Gibbs-preserving, coherence non-generating channels, one-shot classical capacity is upper bounded by coherence preservability plus a bath-size term associated with an incoherent version of the channel. For coherence-annihilating channels, the bath size required to thermalize outputs gives an upper bound on the transmitted classical information, yielding what (Hsieh, 2020) describes as a dynamical analogue of Landauer’s principle. In this formulation, bath size functions as the thermodynamic cost of erasing transmitted information.

The same general framework also gives an erasure-cost interpretation. The robustness-like monotone $\mathcal{F}_R$7 is identified with the erasure cost for resource preservability, namely the minimal number of randomizing operations required to destroy preservability completely (Hsieh, 2019). This places information preservation, communication capacity, and thermodynamic cost within a common dynamical resource-theoretic vocabulary.

4. Dynamical theories of informational non-equilibrium and incompatibility

A direct specialization to information occurs in the dynamical resource theory of informational non-equilibrium preservability. Here the only free state is the maximally mixed state $\mathcal{F}_R$8, and the resource is any deviation from it, i.e. informational non-equilibrium or purity. The static theory uses noisy operations and majorization, while the dynamical theory treats channels as the objects and takes the state-preparation channel of the maximally mixed state as the free channel (Stratton et al., 2023).

Allowed operations are resource non-generating super-channels of the form

$\mathcal{F}_R$9

with $\mathcal{O}_R$0 and $\mathcal{O}_R$1 noisy operations. For qubit unital channels, convertibility is characterized by local permutations of the Choi-state eigenvalues; for $\mathcal{O}_R$2-dimensional Weyl-covariant channels, the analogous characterization uses local cyclic permutations. The induced preorder also has an operational characterization through a state discrimination game with Bell measurements, and the Holevo capacity is a monotone of the theory. Channels that better preserve informational non-equilibrium have higher classical communication capacity (Stratton et al., 2023).

A complementary development treats measurement incompatibility as the preserved resource. In the dynamical resource theory of incompatibility preservability, the free channels are incompatibility-annihilating channels $\mathcal{O}_R$3: channels that always map any measurement assemblage to statistics reproducible by a jointly measurable assemblage. Allowed operations consist of preprocessing via quantum filters, passage through the noisy channel, and conditional postprocessing, combined into deterministic allowed operations

$\mathcal{O}_R$4

which cannot create incompatibility preservability from a free channel (Hsieh et al., 2024).

The basic quantifier is the incompatibility preservability robustness

$\mathcal{O}_R$5

which vanishes exactly on incompatibility-annihilating channels and is monotonic under allowed operations. Its operational interpretation is given by the entanglement-assisted filter game, for which

$\mathcal{O}_R$6

equals the maximal relative advantage over incompatibility-annihilating channels. The same game also yields a necessary and sufficient criterion for channel conversion under the allowed operations (Hsieh et al., 2024).

A useful synthesis is that informational non-equilibrium preservability is formulated in the Schrödinger picture in terms of state purity, whereas incompatibility preservability explicitly generalizes dynamical resource theory to a Heisenberg-picture resource.

5. Measurement informativeness and informational completeness

The preservation of information can also be studied at the level of measurements themselves. A resource theory of quantum measurements takes equivalence classes of POVMs as the resource objects and free transformations as informationally degrading classical post-processing. These free operations include “making up outcomes” by stochastic splitting and “confusing outcomes” by deterministic merging, both represented by column-stochastic maps

$\mathcal{O}_R$7

The terminal free class consists of POVMs whose elements are all proportional to the identity, i.e. measurements that return random noise and no system information (Guff et al., 2019).

Within this order structure, standard information-gain measures are resource monotones, including Shannon mutual information, Buscemi’s mutual information, Banaszek’s information gain, and the robustness of measurement. Quantum state discrimination provides an operational task whose success probability is monotonic under free transformations, and the family of state discrimination games is complete for the preorder. A distinctive structural result is that catalysis and purification are impossible in this theory: if $\mathcal{O}_R$8, then $\mathcal{O}_R$9 (Guff et al., 2019).

A closely related development concerns informational completeness in the Heisenberg picture. For a measurement $$\mathcal{O}_R^N := \{\mathcal{E} \in \mathcal{O}_R \mid \mathcal{E}(\rho) \in \mathcal{F}_R,\ \forall\, \rho\},$$0, informational completeness is quantified by

$$\mathcal{O}_R^N := \{\mathcal{E} \in \mathcal{O}_R \mid \mathcal{E}(\rho) \in \mathcal{F}_R,\ \forall\, \rho\},$$1

This measure is faithful and unitarily invariant. For a qubit SIC-POVM,

$$\mathcal{O}_R^N := \{\mathcal{E} \in \mathcal{O}_R \mid \mathcal{E}(\rho) \in \mathcal{F}_R,\ \forall\, \rho\},$$2

and this value upper-bounds all qubit minimal informationally complete measurements, with equality if and only if the measurement is SIC (Ghai et al., 2 Jun 2026).

For channels, informational-completeness preservability is defined by

$$\mathcal{O}_R^N := \{\mathcal{E} \in \mathcal{O}_R \mid \mathcal{E}(\rho) \in \mathcal{F}_R,\ \forall\, \rho\},$$3

This quantity is faithful, invariant under unitary pre- and post-processing, and monotonic under post-processing order and related channel transformations. For qubit channels with affine Bloch representation parameters $$\mathcal{O}_R^N := \{\mathcal{E} \in \mathcal{O}_R \mid \mathcal{E}(\rho) \in \mathcal{F}_R,\ \forall\, \rho\},$$4 and $$\mathcal{O}_R^N := \{\mathcal{E} \in \mathcal{O}_R \mid \mathcal{E}(\rho) \in \mathcal{F}_R,\ \forall\, \rho\},$$5,

$$\mathcal{O}_R^N := \{\mathcal{E} \in \mathcal{O}_R \mid \mathcal{E}(\rho) \in \mathcal{F}_R,\ \forall\, \rho\},$$6

Moreover, it is bounded above by the absolute output coherence,

$$\mathcal{O}_R^N := \{\mathcal{E} \in \mathcal{O}_R \mid \mathcal{E}(\rho) \in \mathcal{F}_R,\ \forall\, \rho\},$$7

thereby linking preservation of informational completeness to guaranteed output coherence (Ghai et al., 2 Jun 2026).

6. Experimental, conceptual, and broader formulations

Preservability has also been developed as an experimentally accessible property of physical processes. For two-qubit experimental processes, entanglement preservability is quantified by composition $$\mathcal{O}_R^N := \{\mathcal{E} \in \mathcal{O}_R \mid \mathcal{E}(\rho) \in \mathcal{F}_R,\ \forall\, \rho\},$$8 and robustness $$\mathcal{O}_R^N := \{\mathcal{E} \in \mathcal{O}_R \mid \mathcal{E}(\rho) \in \mathcal{F}_R,\ \forall\, \rho\},$$9, together with a fidelity benchmark that compares an experimental process against the best incapable process, where incapable means entanglement-annihilating for all inputs. These quantities are experimentally feasible because they require only local measurements on single qubits and preparations of separable states, and they extend channel-resource-theoretic ideas to non-trace-preserving CP processes (Chen et al., 2020).

More broadly, resource theories need not be formulated solely in terms of density operators. A generalized framework based on specification spaces replaces exact state descriptions by sets of possible states,

$\mathcal{O}_R$0

with transformations acting elementwise,

$\mathcal{O}_R$1

This framework introduces approximation structures, robustness, stability, embeddings between coarse and fine descriptions, and a top-down derivation of subsystem structure from commutation relations among transformations (Rio et al., 2015). This suggests that a resource theory of information preservability can, in principle, be extended beyond exact quantum-state descriptions to epistemic or coarse-grained descriptions of what is known about a system.

Several misconceptions are clarified by the literature. First, preservability is not synonymous with generation: a free channel can preserve resourcefulness for some inputs without being able to create it. Second, preservability orders are generally partial rather than total; measurement resourcefulness, for example, is not fully captured by majorization alone (Guff et al., 2019). Third, structural phenomena depend strongly on the chosen theory: catalysis and purification are impossible in the measurement setting, whereas in general catalytic resource theories the asymptotic and one-shot costs of erasing resource are governed by regularized relative entropy of resource and smooth max-relative entropy, respectively (Anshu et al., 2017). This suggests a natural route for asymptotic formulations of information erasure and information preservability, although that extension is not itself the main result of (Anshu et al., 2017).

Taken together, these developments define a family of closely related theories in which “information preservability” names a dynamical capability: the ability of a process to prevent informational or nonclassical structure from being annihilated by noise, monitoring, coarse-graining, or classical post-processing. The family includes state-based and channel-based formulations, Schrödinger- and Heisenberg-picture resources, operational games, thermodynamic interpretations, and experimentally accessible quantifiers.