Revisiting the Role of State Texture in Gate Identification and Fixed-Point Resource Theories

Abstract

A protocol for identifying controlled-NOT (CNOT) gates versus single-qubit-only gates in universal quantum circuits using randomized input states was recently shown to be intimately connected to the quantum resource of state texture. Here we revisit this gate identification protocol and demonstrate that a more general fidelity-based formulation succeeds for nearly all laboratory bases. We then examine a broader family of quantum resource theories, where a distinct resource theory can be defined for each choice of reference pure state, establishing core resource-theoretic requirements without the computational shortcut offered by the "grand sum" employed in the original formulation of state texture. By extending from single "resourceless" states to convex sets via a convex-roof construction, we recover single-qubit measures of known resource theories such as imaginarity and coherence. Finally, we introduce a family of "fixed-point resource theories" that includes fixed-point instances of the theories of state texture, genuine coherence, purity, and athermality. For these fixed-point resource theories we show that, under free operations, the fidelity-based lower bound is weakly monotonic, while specific violations of strong monotonicity are found for the convex-roof logarithmic measure.

Figures (4)

• Figure 1: Illustrations of texture generalizations. (a) Single zero-resource state, with concentric curves marking equal fidelity (equal rugosity), covered in Sec. \ref{['sec:generalizing_quantum_state_texture']}. (b) Extension to a convex free set, with rugosity lower-bounded by the maximum fidelity to this set, covered in Sec. \ref{['sec:relationship_with_measures']}. (c) Imposing fixed-point constraints on free operations (free states are invariant when acted on by free operations), with the free set restricted to orthogonal extreme points (pure states in the free set are orthogonal to one another), discussed in Sec. \ref{['sec:fixed']}.
• Figure 2: Black axes define the CNOT basis with one basis ket $\ket{c}$ listed ($\ket{c'}$ omitted for clarity) and $\psi_\pm$ on the CNOT equator. $R_{z}(\nu_{2})$ rotates the pair $\psi_{\pm}$ along the CNOT equator (they remain antipodal) followed by a fixed tilt from $R_y(\pi/4)$ (resulting in the green dashed circle). The blue arrow indicates the tilt from the CNOT to the lab basis $U\psi_\pm$, also tilted by the fixed $R_y(\pi/4)$.
• Figure 3: Illustration of minimum (red) and maximum (blue) resource states for single-qubit implementations of (a) quantum state texture, (b) imaginarity, and (c) coherence. Definitions: $\ket{D}\equiv\frac{1}{\sqrt{2}}(\ket{0}+\ket{1})$ and $\ket{R}\equiv\frac{1}{\sqrt{2}}(\ket{0}+\mathrm{i}\ket{1})$.
• Figure 4: Convex-roof extension $\mathcal{R}_{\mathcal{F}_o}(\rho^{(i)})$ evaluated for initial state $\rho^{(0)}=\ket{4}\bra{4}$ after $i$ rounds of successive, randomly generated free operations $\Lambda^{(i)}$. $\mathcal{R}_{\mathcal{F}_o}(\rho^{(i)})$ decays monotonically with increasing number of operations for all dimensions $\dim\{\mathcal{F}_o\} \in\{ 1,2,3\}$ corresponding to blue, green, and purple colors respectively. Datapoints connected by dashed lines correspond to the statistical mode of the prediction of our differential evolution algorithm after 30 repeated trials. Error bars correspond to quantiles within the range [0.05, 0.95]. The numerical observation $\mathcal{R}_{\mathcal{F}_o}(\rho^{(i)}) \leq \mathcal{R}_{\mathcal{F}_o}(\rho^{(i-1)})$ is consistent with weak monotonicity.