Imaginarity Testing in Quantum and AI Systems

• Imaginarity testing is a framework that defines and quantifies the role of imaginary components as a resource in quantum and computational systems.
• It employs resource-theoretic approaches including l₁-norm, Tsallis, and Rényi measures to distinguish free (real) states and operations.
• The methodologies support enhanced noise resilience in quantum protocols and provide benchmarks for evaluating context-faithfulness in language models.

Imaginarity testing encompasses a range of methodologies for quantifying, detecting, and operationalizing the resource corresponding to the presence of imaginary components in mathematical and physical systems—most notably in quantum mechanics and, more recently, in artificial intelligence and LLMs. The central objective is to define principled frameworks and concrete metrics that diagnose how and when systems depend on, exploit, or reveal the fundamental necessity of complex numbers (imaginary quantities) for their structure, performance, or expressivity.

1. Formal Resource-Theoretic Frameworks for Imaginarity

In quantum information theory, imaginarity is formalized by defining the set of free states as those whose density matrices are real in a fixed basis; states with nonreal matrix elements possess imaginarity as a resource (Hickey et al., 2018, Wu et al., 2021, Chen et al., 2022). A state ρ is "real" (free) if and only if ρ = ρ*, that is, all entries are real numbers. Operations (CPTP maps) are free if their Kraus operators are real-valued or, equivalently, if the channel's Choi state is real (Hickey et al., 2018, Chen et al., 10 May 2024).

The resource theory of imaginarity admits measures and monotones satisfying (R1) faithfulness (vanishing for real states or operations), (R2) monotonicity under real operations, and, where applicable, (R3) convexity. This structure mirrors analogous resource theories for entanglement and coherence, but with key distinctions in operational classes and conversion conditions.

For quantum operations, imaginarity testing is extended by defining free operations as those that neither detect nor create imaginarity; free superoperations are compositions with free operations (Wu et al., 11 Jun 2025). The deimaginarity map Δ, defined by Δ(ρ) = (ρ + ρᵀ)/2, plays a central role in classifying operations that are unable to manipulate the imaginary parts of a state.

2. Quantitative Measures of Imaginarity

Multiple imaginarity quantifiers have been developed, most of which require the minimization of a contractive "distance" between a given state and the set of free (real) states. Prominent examples include:

• l₁-Norm of Imaginarity:

$$M_1(\rho) = \sum_{i,j} |\operatorname{Im}\rho_{ij}|$$

This captures the sum of absolute imaginary values in the density matrix (Chen et al., 2022, Alok et al., 2 Dec 2024).

• Tsallis Relative $\alpha$-Entropy-Based Measure:

$$D_\alpha(\rho\|\sigma) = \frac{1 - \operatorname{tr}(\rho^\alpha \sigma^{1-\alpha})}{1 - \alpha}$$

The measure is:

$$\mathcal{M}_{T,\alpha}(\rho) = \min_{\sigma \in \mathcal{F}} \left\{1 - \left[\operatorname{tr}(\rho^\alpha \sigma^{1-\alpha})\right]^{1/\alpha}\right\}$$

(Guo et al., 14 Jan 2025, Xu, 2023)

• Sandwiched Rényi Relative Entropy Measure:

$$\mathcal{M}_{S,\alpha}(\rho) = \min_{\sigma \in \mathcal{F}} \frac{1}{\alpha-1} \left[\left\{\operatorname{tr}\left[(\rho^{\frac{1-\alpha}{2\alpha}}\sigma\rho^{\frac{1-\alpha}{2\alpha}})^\alpha \right]\right\}^{\frac{1}{1-\alpha}} - 1\right]$$

• Robustness of Imaginarity:

$$R(\rho) = \min\{ s \geq 0: \exists\ \tau,\ (\rho + s\tau)/(1 + s) \in \mathcal{F} \}$$

and, for the trace norm,

$$M_{\text{tr}}(\rho) = \frac{1}{2} \|\rho - \rho^T\|_1$$

(Hickey et al., 2018, Wu et al., 2021)

• Convex Roof Measures:

For a concave, decreasing $f(x)$ with $f(1) = 0$:

$$I_f(\rho) = \min \left\{ \sum_k p_k f(|\langle \psi^*_k | \psi_k \rangle|):\ \rho = \sum_k p_k |\psi_k\rangle\langle\psi_k| \right\}$$

(Du et al., 19 Nov 2024)

All these measures are zero on real states and monotonic under real (free) operations. For quantum operations, norm-based measures such as $M_c(\Theta)$, $M_d(\Theta)$, $M_{dc}(\Theta) = \|\Delta\Theta - \Theta\Delta\|$, and weight-based measures capturing the minimal required nonfree admixture are used (Wu et al., 11 Jun 2025).

3. Imaginarity Decay under Noise and Channel Actions

Imaginarity is sensitive to decoherence and noise. Under paradigmatic noisy quantum channels—bit flip, phase damping, amplitude damping—imaginarity measures decay, quantifiable by

$$\Delta \mathcal{M} = \mathcal{M}(\rho_{\text{in}}) - \mathcal{M}(\rho_{\text{out}})$$

Each measure exhibits distinctive behavior. For example, the Tsallis relative $\alpha$-entropy measure ($\mathcal{M}_{T,\alpha}$) demonstrates slower decay under incrementing noise parameters than the Sandwiched Rényi measure, indicating greater robustness (less sensitivity to decoherence) (Guo et al., 14 Jan 2025). This makes the Tsallis measure particularly pertinent for resource management in noisy channels. It is also shown that the ordering of single-qubit states by imaginarity is preserved under bit-flip channels for ranges of $\alpha$ (Guo et al., 14 Jan 2025, Chen et al., 2022), a property crucial for reliable resource quantification post-noise.

4. Unitary-Invariant and Multi-State Imaginarity Witnesses

Moving beyond basis-dependent diagnostics, unitary-invariant imaginarity witnesses—such as the Bargmann invariants—characterize the resource independent of representation (Fernandes et al., 22 Mar 2024, Li et al., 20 Jul 2025):

$$\Delta_{123} = \langle \psi_1|\psi_2\rangle \langle \psi_2|\psi_3\rangle \langle \psi_3|\psi_1\rangle$$

The imaginary part of such an invariant witnesses genuine nonclassicality in the set. In single-qubit systems, all multi-state imaginarity can be fully captured by the collection of two-state overlaps (Gram matrix) and the phases of third-order Bargmann invariants (Li et al., 20 Jul 2025). For a set of states to be imaginariy-free (realizable in a real basis), their Bloch vectors must be coplanar; rank-based tests on the Gram matrix provide operationally accessible criteria.

These witnesses form the mathematical basis for experimental protocols, including interferometric tests, that are robust against basis misalignment and can be generalized—partially—to higher dimensional systems.

5. Applications: Quantum Information, Experimental Protocols, and Beyond

Quantitative imaginarity measures enable several operational and foundational applications:

• Resource Conversion and State Transformations: Conditions for single-shot and probabilistic conversion under physically consistent free operations are given explicitly by orderings of the imaginarity measure (typically monotonicity suffices for pure qubit states) (Hickey et al., 2018, Wu et al., 2021, Du et al., 19 Nov 2024).
• Channel Discrimination: Imaginarity provides a resource advantage in discrimination of quantum channels, especially in ancilla-free or distributed (LOCC) scenarios (Wu et al., 2023).
• Experimental Realizations: Imaginarity testing is manifest in experimental quantum photonics, including optimized linear optics for real-valued operations (fewer independent wave plates required), and in protocols on multipartite CV states using easily computable measures for Gaussian systems (Wu et al., 2021, Zhang et al., 10 Apr 2025).
• Order and Robustness in Open Systems: The invariance of imaginarity ordering among qubit states under bit-flip and other noise is established rigorously, supporting its use in error-resilient quantum information tasks (Guo et al., 14 Jan 2025, Chen et al., 2022).
• Foundational Studies: Imaginarity measures enable the exploration of quantum-classical boundaries (e.g., via real-vs-complex quantum mechanics), the necessity of complex numbers in quantum phenomena (e.g., neutrino oscillations) (Alok et al., 2 Dec 2024), and the intrinsic nonclassicality of higher-order invariants.

6. Comparative Properties and Theoretical Relationships

Different families of imaginarity measures admit mutual inequalities, for example,

$$\mathcal{M}^{\mathrm{R}}_{(\alpha)}(\rho) \leq \mathcal{M}^{\mathrm{R}}_{(\alpha, z)}(\rho) \leq \mathcal{M}^{\mathrm{T}}_{(\alpha)}(\rho)$$

for the Rényi, (α, z)-Rényi, and Tsallis-based measures, respectively (Chen et al., 31 Mar 2024). These relationships map the landscape of sensitivity and faithfulness across quantifiers.

For Gaussian states in CV systems, the measure $\mathcal{I}^{G_n}$ relies only on displacement vectors and covariance matrices and is computationally preferable to fidelity-based or Tsallis-based measures for multi-mode settings (Zhang et al., 10 Apr 2025). Importantly, the new measure supports multipartite scenarios and satisfies all required axioms for genuine multi-mode resource monotones.

7. Imaginarity in AI and LLMs: Synthetic Scenarios and Benchmarks

In modern AI evaluation, imaginarity testing is repurposed to diagnose the disentangling of linguistic and parametric (world knowledge) abilities in LLMs. Synthetic or “imaginary” datasets—using fictitious entities and facts—neutralize the confounding influence of pretrained world knowledge, exposing model capabilities in text comprehension, hypothetical reasoning, and abstraction (Basmov et al., 9 Apr 2024). Notably, models excel at simple factual queries but exhibit severe performance degradation on modal and conditional (hypothetical) scenarios within imaginary contexts, indicating fundamental challenges for context-faithfulness and semantic displacement.

Similarly, “imaginary question answering” (IQA) assesses the “shared imagination space” of LLMs, demonstrating that models, even in the absence of any factual grounding, exhibit consistent, above-chance mutual answerability of hallucinated questions—underscoring latent homogeneity in hallucination patterns and suggesting implications for model creativity and ensemble methods (Zhou et al., 23 Jul 2024).

Finally, synthetic benchmarks such as Hyperphantasia quantify the ability of multimodal models to internally simulate (“mentally visualize”) visual phenomena, revealing large gaps between model and human performance in tasks requiring visual pattern completion and simulation—another variant of imaginarity testing (Sepehri et al., 16 Jul 2025).

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Table: Representative Imaginarity Measures

Measure Family	Functional Form	Reference(s)
l₁-norm	$$M_1(\rho) = \sum_{i,j} |\operatorname{Im}\rho_{ij}|$$	(Chen et al., 2022)
Tsallis relative $\alpha$-entropy	$$D_\alpha(\rho\|\sigma) = \frac{1 - \operatorname{tr}(\rho^\alpha \sigma^{1-\alpha})}{1-\alpha}$$, $$\mathcal{M}_{T,\alpha}(\rho) = \min_{\sigma \in \mathcal{F}} (1 - [\operatorname{tr}(\rho^\alpha \sigma^{1-\alpha})]^{1/\alpha})$$	(Guo et al., 14 Jan 2025)
Sandwiched Rényi entropy	$$F_\alpha(\sigma\|\rho) = \frac{ \ln \operatorname{tr} (\rho^{\frac{1-\alpha}{2\alpha}} \sigma \rho^{\frac{1-\alpha}{2\alpha}})^\alpha }{\alpha-1}$$; see full text	(Guo et al., 14 Jan 2025)
Robustness	$$R(\rho) := \min \{ s: (\rho+s\tau)/(1+s) \in \mathcal{F} \}$$	(Hickey et al., 2018)
Convex roof (pure states)	$I_f(|\psi\rangle) = f(|\langle\psi^*|\psi\rangle|)$	(Du et al., 19 Nov 2024)
Gaussian measure $\mathcal{I}^{G_n}$	$$1 - \frac{ \det \nu }{ \det (A_{11}) \det (A_{22}) } + h(|d_2|+|d_4|+...+|d_{2n}|)$$	(Zhang et al., 10 Apr 2025)

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Summary

Imaginarity testing provides a rigorous, multidimensional methodology for quantifying the operational essence of complex numbers as fundamental resources in physics, information science, and emerging AI systems. Through the definition of principled resource theories, robust families of distance and entropy-based measures, and experimentally implementable invariants and witnesses, imaginarity is identified not only as a foundational mathematical property but also as a quantifiable and utilizable quantum (and computational) resource. The field encompasses significant implications for resource manipulation and noise resilience in quantum systems and supplies critical benchmarks for testing and advancing the context-faithfulness, abstraction, and creative potential of large models in artificial intelligence.