Quantum Texture-Based Resource Monotones

• Texture-based resource monotones are basis-dependent measures quantifying the inhomogeneity of a quantum state's matrix elements relative to a chosen basis.
• They are defined via logarithmic rugosity, leveraging state overlaps with free states, and generalized through fidelity-based and convex-roof formulations.
• Applications include operational gate identification in quantum circuits and establishing links with fixed-point theories, coherence, and purity.

Texture-based resource monotones are basis-dependent quantifiers of quantum-state “texture,” understood as the inhomogeneity of a state’s matrix-element distribution relative to a chosen computational or laboratory basis. In the original formulation, the unique textureless free state is the equal-superposition state, and the central monotone is the logarithmic “rugosity” obtained from the overlap with that free state. Subsequent work generalized this construction in two directions: from the fixed equal-superposition reference to an arbitrary pure reference state or a convex free set, and from the original grand-sum shortcut to fidelity-based and convex-roof formulations. These generalizations place texture within a broader class of fixed-point resource theories and connect it operationally to gate identification, especially the discrimination of CNOT-containing layers from single-qubit-only layers (Parisio, 2024, Greenwood et al., 26 Feb 2026).

1. Foundational definition and basis dependence

Fix a computational basis $B=\{|i\rangle\}_{i=1}^D$. The original texture theory singles out the equal-superposition state

$$|f_1\rangle = \frac{1}{\sqrt D}\sum_{i=1}^D |i\rangle,$$

with free state $f_1=|f_1\rangle\langle f_1|$. In the notation of the original texture work, the “grand sum” of matrix elements is

$$\Sigma(\rho;B)=\sum_{i,j=1}^D \rho_{ij}=D\langle f_1|\rho|f_1\rangle,$$

and the corresponding logarithmic monotone, called rugosity, is

$$R(\rho;B)=-\ln\!\left(\frac{\Sigma(\rho;B)}{D}\right)=-\ln\langle f_1|\rho|f_1\rangle.$$

For pure states $|\phi\rangle$, this reduces to

$$R(|\phi\rangle;B)=-\ln |\langle f_1|\phi\rangle|^2.$$

The resource is basis dependent: changing the basis changes $|f_1\rangle$, the grand sum, and hence the monotone itself (Parisio, 2024).

The generalized formulation replaces $|f_1\rangle$ by an arbitrary pure reference state $|\psi\rangle$. One then defines

$$|f_1\rangle = \frac{1}{\sqrt D}\sum_{i=1}^D |i\rangle,$$0

For pure inputs $$|f_1\rangle = \frac{1}{\sqrt D}\sum_{i=1}^D |i\rangle,$$1,

$$|f_1\rangle = \frac{1}{\sqrt D}\sum_{i=1}^D |i\rangle,$$2

This formulation preserves the overlap-based structure of the original rugosity while making explicit that the grand-sum shortcut is not essential (Greenwood et al., 26 Feb 2026).

The same body of work also emphasizes that the unique textureless state in the original theory is the only state whose density matrix has all entries equal in the chosen basis, and that $$|f_1\rangle = \frac{1}{\sqrt D}\sum_{i=1}^D |i\rangle,$$3 if and only if $$|f_1\rangle = \frac{1}{\sqrt D}\sum_{i=1}^D |i\rangle,$$4. Conversely, $$|f_1\rangle = \frac{1}{\sqrt D}\sum_{i=1}^D |i\rangle,$$5 whenever $$|f_1\rangle = \frac{1}{\sqrt D}\sum_{i=1}^D |i\rangle,$$6, as occurs for Fourier-basis states orthogonal to $$|f_1\rangle = \frac{1}{\sqrt D}\sum_{i=1}^D |i\rangle,$$7 (Parisio, 2024).

2. Resource-theoretic formulation

For a single reference free pure state $$|f_1\rangle = \frac{1}{\sqrt D}\sum_{i=1}^D |i\rangle,$$8, the free completely positive trace-preserving maps $$|f_1\rangle = \frac{1}{\sqrt D}\sum_{i=1}^D |i\rangle,$$9 are those that fix $f_1=|f_1\rangle\langle f_1|$0: $f_1=|f_1\rangle\langle f_1|$1 Equivalently, every Kraus operator satisfies

$f_1=|f_1\rangle\langle f_1|$2

In the decomposition $f_1=|f_1\rangle\langle f_1|$3, these Kraus operators have block form

$f_1=|f_1\rangle\langle f_1|$4

where $f_1=|f_1\rangle\langle f_1|$5 only couples states into $f_1=|f_1\rangle\langle f_1|$6, and $f_1=|f_1\rangle\langle f_1|$7 acts within $f_1=|f_1\rangle\langle f_1|$8 (Greenwood et al., 26 Feb 2026).

This single-reference theory extends naturally to fixed-point resource theories. A fixed-point resource theory is specified by an orthogonal set of free pure states $f_1=|f_1\rangle\langle f_1|$9 and by free CPTP maps $$\Sigma(\rho;B)=\sum_{i,j=1}^D \rho_{ij}=D\langle f_1|\rho|f_1\rangle,$$0 satisfying

$$\Sigma(\rho;B)=\sum_{i,j=1}^D \rho_{ij}=D\langle f_1|\rho|f_1\rangle,$$1

In the $$\Sigma(\rho;B)=\sum_{i,j=1}^D \rho_{ij}=D\langle f_1|\rho|f_1\rangle,$$2-diagonal basis, Kraus operators take the block-triangular form

$$\Sigma(\rho;B)=\sum_{i,j=1}^D \rho_{ij}=D\langle f_1|\rho|f_1\rangle,$$3

with $$\Sigma(\rho;B)=\sum_{i,j=1}^D \rho_{ij}=D\langle f_1|\rho|f_1\rangle,$$4 for each $$\Sigma(\rho;B)=\sum_{i,j=1}^D \rho_{ij}=D\langle f_1|\rho|f_1\rangle,$$5. These maps can destroy the resource by moving amplitude from $$\Sigma(\rho;B)=\sum_{i,j=1}^D \rho_{ij}=D\langle f_1|\rho|f_1\rangle,$$6 into $$\Sigma(\rho;B)=\sum_{i,j=1}^D \rho_{ij}=D\langle f_1|\rho|f_1\rangle,$$7 (Greenwood et al., 26 Feb 2026).

Several standard theories appear as fixed-point instances. In genuine coherence, the free states are diagonal in a chosen basis,

$$\Sigma(\rho;B)=\sum_{i,j=1}^D \rho_{ij}=D\langle f_1|\rho|f_1\rangle,$$8

and free maps preserve all incoherent states, $$\Sigma(\rho;B)=\sum_{i,j=1}^D \rho_{ij}=D\langle f_1|\rho|f_1\rangle,$$9; the Kraus operators are fully diagonal, and dephasing is free. In purity, the free set is the maximally mixed state $$R(\rho;B)=-\ln\!\left(\frac{\Sigma(\rho;B)}{D}\right)=-\ln\langle f_1|\rho|f_1\rangle.$$0, and depolarizing maps $$R(\rho;B)=-\ln\!\left(\frac{\Sigma(\rho;B)}{D}\right)=-\ln\langle f_1|\rho|f_1\rangle.$$1 are free and resource destroying. In athermality, the free set is the Gibbs state $$R(\rho;B)=-\ln\!\left(\frac{\Sigma(\rho;B)}{D}\right)=-\ln\langle f_1|\rho|f_1\rangle.$$2, and fixed-point maps preserve $$R(\rho;B)=-\ln\!\left(\frac{\Sigma(\rho;B)}{D}\right)=-\ln\langle f_1|\rho|f_1\rangle.$$3 (Greenwood et al., 26 Feb 2026).

3. Fidelity-based and convex-roof texture monotones

The most direct generalization from a single free state to a convex free set $$R(\rho;B)=-\ln\!\left(\frac{\Sigma(\rho;B)}{D}\right)=-\ln\langle f_1|\rho|f_1\rangle.$$4 is the fidelity-based lower bound

$$R(\rho;B)=-\ln\!\left(\frac{\Sigma(\rho;B)}{D}\right)=-\ln\langle f_1|\rho|f_1\rangle.$$5

where the Uhlmann fidelity is

$$R(\rho;B)=-\ln\!\left(\frac{\Sigma(\rho;B)}{D}\right)=-\ln\langle f_1|\rho|f_1\rangle.$$6

For pure $$R(\rho;B)=-\ln\!\left(\frac{\Sigma(\rho;B)}{D}\right)=-\ln\langle f_1|\rho|f_1\rangle.$$7, this becomes

$$R(\rho;B)=-\ln\!\left(\frac{\Sigma(\rho;B)}{D}\right)=-\ln\langle f_1|\rho|f_1\rangle.$$8

so it coincides with the pure-state rugosity. Using the joint concavity of $$R(\rho;B)=-\ln\!\left(\frac{\Sigma(\rho;B)}{D}\right)=-\ln\langle f_1|\rho|f_1\rangle.$$9 and the convexity and monotonicity of $|\phi\rangle$0, this quantity is convex in $|\phi\rangle$1 and is a bona fide resource monotone (Greenwood et al., 26 Feb 2026).

A second extension is the convex-roof logarithmic texture measure. First define, for pure states,

$|\phi\rangle$2

Then extend to mixed states by

$|\phi\rangle$3

This construction preserves the logarithmic form and does not assume a grand-sum shortcut (Greenwood et al., 26 Feb 2026).

A distinct line of development introduces a two-parameter $|\phi\rangle$4-$|\phi\rangle$5 Rényi-based texture monotone. With $|\phi\rangle$6 as the unique free state, the quantity

$|\phi\rangle$7

is a valid texture measure for

$|\phi\rangle$8

In that parameter regime it satisfies non-negativity, monotonicity under free operations, and convexity. The same work establishes unitary invariance under $|\phi\rangle$9-preserving unitaries and the tensor-product bounds

$$R(|\phi\rangle;B)=-\ln |\langle f_1|\phi\rangle|^2.$$0

(Chen et al., 8 Apr 2026).

These three constructions—rugosity, fidelity lower bound, and $$R(|\phi\rangle;B)=-\ln |\langle f_1|\phi\rangle|^2.$$1-$$R(|\phi\rangle;B)=-\ln |\langle f_1|\phi\rangle|^2.$$2 Rényi texture—share a common structural feature: each quantifies texture through a privileged overlap with a free state or free set. This suggests that overlap-based geometry, rather than the grand sum specifically, is the central organizing principle of the theory.

4. Monotonicity results and fixed-point phenomena

For fixed-point free sets $$R(|\phi\rangle;B)=-\ln |\langle f_1|\phi\rangle|^2.$$3, define

$$R(|\phi\rangle;B)=-\ln |\langle f_1|\phi\rangle|^2.$$4

If $$R(|\phi\rangle;B)=-\ln |\langle f_1|\phi\rangle|^2.$$5 is free and fixes every $$R(|\phi\rangle;B)=-\ln |\langle f_1|\phi\rangle|^2.$$6, then fidelity data processing gives

$$R(|\phi\rangle;B)=-\ln |\langle f_1|\phi\rangle|^2.$$7

Maximizing over $$R(|\phi\rangle;B)=-\ln |\langle f_1|\phi\rangle|^2.$$8 and applying $$R(|\phi\rangle;B)=-\ln |\langle f_1|\phi\rangle|^2.$$9 yields

$|f_1\rangle$0

so the fidelity-based lower bound is weakly monotonic under free operations. Nonnegativity follows because $|f_1\rangle$1, with equality if and only if $|f_1\rangle$2 (Greenwood et al., 26 Feb 2026).

By contrast, strong monotonicity can fail for the convex-roof logarithmic measure. In the one-reference case $|f_1\rangle$3 in dimension $|f_1\rangle$4, consider the pure state

$|f_1\rangle$5

for which

$|f_1\rangle$6

With diagonal Kraus operators

$|f_1\rangle$7

the postselected outputs are $|f_1\rangle$8, having rugosity $|f_1\rangle$9, with probability

$|f_1\rangle$0

and $|f_1\rangle$1, having rugosity $|f_1\rangle$2, with probability $|f_1\rangle$3. Strong monotonicity would require

$|f_1\rangle$4

but this is violated whenever

$|f_1\rangle$5

An explicit family of violations is obtained for $|f_1\rangle$6, which yields violations for $|f_1\rangle$7 (Greenwood et al., 26 Feb 2026).

The same work notes that strong monotonicity is sensitive to the choice of free operations and basis, and that in the present fixed-point framework it fails generically for logarithmic convex-roof measures. Whether $|f_1\rangle$8 is always weakly monotonic for all fixed-point theories remains an open question; numerical evidence suggests yes in the tested settings (Greenwood et al., 26 Feb 2026).

5. Operational role in gate identification

Texture-based monotones were first linked to gate identification through a protocol for distinguishing universal circuit layers that contain at least one CNOT from layers composed only of single-qubit gates. The protocol uses identically prepared Haar-random single-qubit inputs and texture measurements on the output qubits, without tomography or ancillary systems (Parisio, 2024).

In the generalized formulation, one works in a laboratory basis that specifies both the reference state $|f_1\rangle$9 and its Hadamard transform $|\psi\rangle$0. For a single layer acting on identically prepared Haar-random qubits, the average grand sum obeys

$|\psi\rangle$1

for single-qubit-only gates. If a CNOT is present, and the two inputs are $|\psi\rangle$2, then in an unknown basis $|\psi\rangle$3 the reduced control and target states satisfy

$|\psi\rangle$4

and similarly

$|\psi\rangle$5

Any deviation from $|\psi\rangle$6 in either control or target indicates the presence of a CNOT (Greenwood et al., 26 Feb 2026).

The generalized analysis shows that the protocol succeeds in “nearly all laboratory bases.” Failure occurs only if $|\psi\rangle$7 lies on a measure-zero great circle on the Bloch sphere characterized by

$|\psi\rangle$8

and

$|\psi\rangle$9

which forces all four averages to equal $$|f_1\rangle = \frac{1}{\sqrt D}\sum_{i=1}^D |i\rangle,$$00. Thus, in the continuous limit with $$|f_1\rangle = \frac{1}{\sqrt D}\sum_{i=1}^D |i\rangle,$$01 drawn arbitrarily, the success probability is unity, with measure-zero exceptions (Greenwood et al., 26 Feb 2026).

The original gate-identification paper states the same operational content in a more concrete two-setting form. In the chosen reference basis and its Fourier basis, wires not participating in any CNOT satisfy $$|f_1\rangle = \frac{1}{\sqrt D}\sum_{i=1}^D |i\rangle,$$02 and $$|f_1\rangle = \frac{1}{\sqrt D}\sum_{i=1}^D |i\rangle,$$03, while CNOT-participating wires have at least one of these quantities deviating from $$|f_1\rangle = \frac{1}{\sqrt D}\sum_{i=1}^D |i\rangle,$$04 by at least $$|f_1\rangle = \frac{1}{\sqrt D}\sum_{i=1}^D |i\rangle,$$05, a basis-independent separation that supports complete layer characterization (Parisio, 2024). This operational role is significant because it identifies texture not merely as a static state property but as a directly measurable witness of entangling action in unknown circuit layers.

6. Relations to other resource theories, detection methods, and broader generalizations

With appropriate choices of free sets, the convex-roof texture formalism recovers known single-qubit measures. For imaginarity, taking the free set to be the real density matrices in a fixed basis gives, for pure states,

$$|f_1\rangle = \frac{1}{\sqrt D}\sum_{i=1}^D |i\rangle,$$06

where $$|f_1\rangle = \frac{1}{\sqrt D}\sum_{i=1}^D |i\rangle,$$07. For a mixed qubit with Bloch vector $$|f_1\rangle = \frac{1}{\sqrt D}\sum_{i=1}^D |i\rangle,$$08,

$$|f_1\rangle = \frac{1}{\sqrt D}\sum_{i=1}^D |i\rangle,$$09

For coherence in a fixed computational basis, one has

$$|f_1\rangle = \frac{1}{\sqrt D}\sum_{i=1}^D |i\rangle,$$10

and for a mixed qubit,

$$|f_1\rangle = \frac{1}{\sqrt D}\sum_{i=1}^D |i\rangle,$$11

In the qubit case these coincide with standard geometric coherence forms (Greenwood et al., 26 Feb 2026).

A different texture-derived monotone uses basis optimization. If $$|f_1\rangle = \frac{1}{\sqrt D}\sum_{i=1}^D |i\rangle,$$12 is the basis-dependent texture, then over all orthonormal bases

$$|f_1\rangle = \frac{1}{\sqrt D}\sum_{i=1}^D |i\rangle,$$13

Their difference

$$|f_1\rangle = \frac{1}{\sqrt D}\sum_{i=1}^D |i\rangle,$$14

and the scaled quantity

$$|f_1\rangle = \frac{1}{\sqrt D}\sum_{i=1}^D |i\rangle,$$15

define a valid purity monotone under unital maps in any dimension. The same work also introduces non-local texture, which for bipartite pure states satisfies

$$|f_1\rangle = \frac{1}{\sqrt D}\sum_{i=1}^D |i\rangle,$$16

equal to the geometric measure of entanglement, and extends this viewpoint to coherence, non-stabilizerness, and multipartite genuine entanglement through convex-roof constructions (Patra et al., 18 Jul 2025).

Texture can also be detected by witnesses. A Hermitian operator $$|f_1\rangle = \frac{1}{\sqrt D}\sum_{i=1}^D |i\rangle,$$17 is a texture witness if $$|f_1\rangle = \frac{1}{\sqrt D}\sum_{i=1}^D |i\rangle,$$18 and there exists at least one texture state $$|f_1\rangle = \frac{1}{\sqrt D}\sum_{i=1}^D |i\rangle,$$19 such that $$|f_1\rangle = \frac{1}{\sqrt D}\sum_{i=1}^D |i\rangle,$$20. A universal construction is

$$|f_1\rangle = \frac{1}{\sqrt D}\sum_{i=1}^D |i\rangle,$$21

Among the examples, the witness

$$|f_1\rangle = \frac{1}{\sqrt D}\sum_{i=1}^D |i\rangle,$$22

detects all texture states because

$$|f_1\rangle = \frac{1}{\sqrt D}\sum_{i=1}^D |i\rangle,$$23

with strict negativity for every $$|f_1\rangle = \frac{1}{\sqrt D}\sum_{i=1}^D |i\rangle,$$24. Additional witness families include

$$|f_1\rangle = \frac{1}{\sqrt D}\sum_{i=1}^D |i\rangle,$$25

and

$$|f_1\rangle = \frac{1}{\sqrt D}\sum_{i=1}^D |i\rangle,$$26

the latter detecting phase-sensitive off-diagonal structure and, for $$|f_1\rangle = \frac{1}{\sqrt D}\sum_{i=1}^D |i\rangle,$$27, the sign of the imaginary part of off-diagonal entries (Chen et al., 8 Apr 2026).

In a broader resource-theoretic setting, the abstract framework of universally-combinable resource theories shows how monotones arise by pullback through order-preserving maps. Within that framework, a texture-based monotone can be instantiated either by a set-based feature map with sup/inf root monotones, or by a tuple-based map combined with a contraction or information-content monotone. The same framework covers “distance to the free set” constructions, commuting-map constructions of the form $$|f_1\rangle = \frac{1}{\sqrt D}\sum_{i=1}^D |i\rangle,$$28, and weight or robustness measures derived from the convex-alignment 3-contraction (Gonda et al., 2019). This suggests that fidelity-based texture monotones belong to a wider monotone-construction paradigm rather than constituting an isolated resource-specific device.

Several limitations remain explicit. Texture measures are basis dependent and sensitive to the choice of reference state or free set; convex-roof computations in higher dimensions remain challenging; strong monotonicity fails in general for the logarithmic convex-roof measure; and asymptotic continuity is not addressed in the $$|f_1\rangle = \frac{1}{\sqrt D}\sum_{i=1}^D |i\rangle,$$29-$$|f_1\rangle = \frac{1}{\sqrt D}\sum_{i=1}^D |i\rangle,$$30 Rényi construction. These limitations coexist with a notable strength of the framework: the same overlap-based formalism accommodates single-reference texture, fixed-point theories, coherence, imaginarity, purity, athermality, and operational gate-detection tasks within a common mathematical language (Greenwood et al., 26 Feb 2026, Chen et al., 8 Apr 2026, Patra et al., 18 Jul 2025).