Fixed-Point Resource Theories

• Fixed-point resource theories are defined by requiring every free state to remain invariant under all free operations, establishing a strict fixed-point condition.
• The framework employs a convex free set, fixed-point maps, and fidelity-based resource monotones to enable operational tasks like gate identification in quantum circuits.
• It unifies established resource theories such as coherence, purity, and athermality while highlighting the contrast between weak monotonicity and the failure of strong monotonicity in convex-roof measures.

Searching arXiv for the cited paper and the related imaginarity reference to ensure current, citable metadata. tool unavailable Fixed-point resource theories are a family of quantum resource theories in which the free operations are defined by a fixed-point condition on a designated free set of states. In the formulation developed in "Revisiting the Role of State Texture in Gate Identification and Fixed-Point Resource Theories" (Greenwood et al., 26 Feb 2026), the framework arises from a reconsideration of a gate-identification protocol for distinguishing controlled-NOT gates from single-qubit-only gates in universal quantum circuits using randomized input states, a protocol previously connected to the resource of state texture. The paper shows that a more general fidelity-based formulation succeeds for nearly all laboratory bases, extends the construction from single resourceless states to convex free sets via a convex-roof construction, and introduces a family of fixed-point resource theories encompassing fixed-point instances of state texture, genuine coherence, purity, and athermality (Greenwood et al., 26 Feb 2026).

1. Origin in state texture and gate identification

The immediate motivation for the fixed-point framework is a protocol for identifying controlled-NOT (CNOT) gates versus single-qubit-only gates in universal quantum circuits using randomized input states. That protocol had been shown to be intimately connected to the quantum resource of state texture. The 2026 analysis revisits that setting and demonstrates that a more general fidelity-based formulation succeeds for nearly all laboratory bases (Greenwood et al., 26 Feb 2026).

Within the same development, a broader family of quantum resource theories is considered, where a distinct resource theory can be defined for each choice of reference pure state. The paper states that this establishes core resource-theoretic requirements without the computational shortcut offered by the "grand sum" employed in the original formulation of state texture. It further remarks that proving monotonicity of the fidelity-based lower bound for arbitrary single-state or convex free sets does not rely on any "grand-sum" shortcut, but only on fidelity monotonicity and fixed-point conditions (Greenwood et al., 26 Feb 2026).

This suggests that the role of state texture in the original gate-identification problem is not treated as an isolated phenomenon. Rather, it is embedded into a more general fixed-point paradigm in which operational discrimination tasks and resource quantification are linked through fidelity to a convex set of resourceless states.

2. Formal specification of the framework

A fixed-point resource theory is specified by three ingredients (Greenwood et al., 26 Feb 2026).

First, there is a convex free set

$\mathcal{F}_0 \subset \mathcal{D}(\mathcal{H})$

of density operators that contain exactly the resourceless states. In particular, all pure states in $\mathcal{F}_0$ are taken to be mutually orthogonal.

Second, there is a family of completely positive trace-preserving maps, interpreted as free operations, such that every free state is a fixed point: $$\forall \sigma \in \mathcal{F}_0:\qquad \Lambda(\sigma)=\sigma.$$

Third, there is a resource monotone

$M:\mathcal{D}(\mathcal{H})\to \mathbb{R}_{\ge 0}$

satisfying the usual axioms:

• nonnegativity, $M(\rho)\ge 0$ for all $\rho$, and $M(\rho)=0$ iff $\rho\in\mathcal{F}_0$;
• closure / convexity, with $\mathcal{F}_0$ convex and closed;
• weak monotonicity, $M(\Lambda(\rho))\le M(\rho)$ for every free $\mathcal{F}_0$0.

The framework therefore differs from resource theories defined only by preservation of a free set. Here the stronger requirement is literal pointwise invariance of every free state under every free operation. That fixed-point condition is the structural feature from which the subsequent fidelity-based monotonicity result follows.

3. Structure of free operations

In a basis that diagonalizes $\mathcal{F}_0$1, each free operation admits a Kraus decomposition

$\mathcal{F}_0$2

with Kraus operators of block form

$\mathcal{F}_0$3

where

$\mathcal{F}_0$4

The matrices $\mathcal{F}_0$5 and $\mathcal{F}_0$6 permit only transitions out of the resourceful subspace (Greenwood et al., 26 Feb 2026).

Trace preservation and the fixed-point condition impose

$\mathcal{F}_0$7

and

$\mathcal{F}_0$8

The paper identifies this block-diagonal construction of fixed-point free operations as the novel technique of the work, referring to it as Eq. (11) in the paper. Its significance is twofold. Formally, it characterizes the admissible free dynamics compatible with exact invariance of the free set. Conceptually, it provides a single operator-level template under which multiple basis-dependent resource theories can be organized (Greenwood et al., 26 Feb 2026).

4. Fidelity-based quantification and weak monotonicity

The fidelity between states $\mathcal{F}_0$9 and $$\forall \sigma \in \mathcal{F}_0:\qquad \Lambda(\sigma)=\sigma.$$0 is defined as

$$\forall \sigma \in \mathcal{F}_0:\qquad \Lambda(\sigma)=\sigma.$$1

The associated fidelity-based resource quantifier is

$$\forall \sigma \in \mathcal{F}_0:\qquad \Lambda(\sigma)=\sigma.$$2

For fixed-point resource theories, this quantity satisfies weak monotonicity under free operations: $$\forall \sigma \in \mathcal{F}_0:\qquad \Lambda(\sigma)=\sigma.$$3 The proof proceeds exactly through the fixed-point property and Uhlmann-fidelity monotonicity (Greenwood et al., 26 Feb 2026):

1. For every $$\forall \sigma \in \mathcal{F}_0:\qquad \Lambda(\sigma)=\sigma.$$4, $$\forall \sigma \in \mathcal{F}_0:\qquad \Lambda(\sigma)=\sigma.$$5.
2. Uhlmann’s fidelity is monotonic under CPTP maps:

$$\forall \sigma \in \mathcal{F}_0:\qquad \Lambda(\sigma)=\sigma.$$6

1. Hence

$$\forall \sigma \in \mathcal{F}_0:\qquad \Lambda(\sigma)=\sigma.$$7

2. Maximizing over $$\forall \sigma \in \mathcal{F}_0:\qquad \Lambda(\sigma)=\sigma.$$8 yields

$$\forall \sigma \in \mathcal{F}_0:\qquad \Lambda(\sigma)=\sigma.$$9

and therefore

$M:\mathcal{D}(\mathcal{H})\to \mathbb{R}_{\ge 0}$0

This is presented as Theorem 1 in the paper. A common misconception in resource-theoretic constructions is that a fidelity-based expression automatically inherits all stronger monotonicity properties once weak monotonicity is established. The fixed-point analysis does not support that inference. It proves weak monotonicity for $M:\mathcal{D}(\mathcal{H})\to \mathbb{R}_{\ge 0}$1, but a different behavior emerges for the convex-roof logarithmic measure.

5. Convex-roof logarithmic measure and failure of strong monotonicity

For each pure state $M:\mathcal{D}(\mathcal{H})\to \mathbb{R}_{\ge 0}$2, let

$M:\mathcal{D}(\mathcal{H})\to \mathbb{R}_{\ge 0}$3

The convex-roof logarithmic measure is then defined by

$M:\mathcal{D}(\mathcal{H})\to \mathbb{R}_{\ge 0}$4

The paper shows that $M:\mathcal{D}(\mathcal{H})\to \mathbb{R}_{\ge 0}$5 need not satisfy strong monotonicity (Greenwood et al., 26 Feb 2026). The explicit construction uses a $M:\mathcal{D}(\mathcal{H})\to \mathbb{R}_{\ge 0}$6-dimensional free set

$M:\mathcal{D}(\mathcal{H})\to \mathbb{R}_{\ge 0}$7

and a pure input state

$M:\mathcal{D}(\mathcal{H})\to \mathbb{R}_{\ge 0}$8

A two-Kraus free operation $M:\mathcal{D}(\mathcal{H})\to \mathbb{R}_{\ge 0}$9, diagonal in the computational basis, is chosen so that outcome 1 acts as a filter that boosts the overlap with the resourceful subspace and outcome 2 projects onto $M(\rho)\ge 0$0. Explicitly,

$M(\rho)\ge 0$1

If

$M(\rho)\ge 0$2

denotes the postselected pure state for outcome $M(\rho)\ge 0$3 with probability $M(\rho)\ge 0$4, then for suitable choices of $M(\rho)\ge 0$5 and $M(\rho)\ge 0$6,

$M(\rho)\ge 0$7

Thus $M(\rho)\ge 0$8 fails the strong-monotonicity condition

$M(\rho)\ge 0$9

This is Theorem 2 of the paper. The result is significant because it separates two frequently conflated notions: weak monotonicity of a state functional under the overall free channel, and strong monotonicity under selective free operations conditioned on Kraus outcomes. In the fixed-point setting, the former is guaranteed for the fidelity lower bound, whereas the latter can fail for the convex-roof logarithmic extension.

6. Recovery of established resource theories

By appropriate choice of the free set $\rho$0, the fixed-point formalism recovers several familiar resources (Greenwood et al., 26 Feb 2026).

For imaginarity, with

$\rho$1

a pure state $\rho$2 has

$\rho$3

where $\rho$4 picks out the largest eigenvalue of the real part. Mixed-state imaginarity follows by convex roof. The paper explicitly associates this case with Ref. Wu et al. 2021.

For coherence in a fixed basis, or speakable coherence, with

$\rho$5

a pure state

$\rho$6

satisfies

$\rho$7

Its single-qubit convex-roof extension can be solved analytically by noting that the function

$\rho$8

is convex and applying Jensen’s lemma.

For purity, where the resource is deviation from the maximally mixed state, the free set is

$\rho$9

and one recovers the usual geometric measure of purity: $M(\rho)=0$0

For athermality, the free set is the Gibbs state

$M(\rho)=0$1

with

$M(\rho)=0$2

and its convex-roof extension quantifies thermodynamic work potential.

All of these satisfy the fixed-point condition

$M(\rho)=0$3

under physically motivated free operations, such as energy-preserving maps for athermality (Greenwood et al., 26 Feb 2026). A plausible implication is that the fixed-point framework is less a single resource theory than a unifying template for resource theories whose free states are intended to remain exactly invariant under the admissible dynamics.