Formal Resource Theory for Quantum Imaginarity

• The paper defines quantum imaginarity by identifying free states with real-valued density matrices and free operations that preserve these real structures.
• It introduces various imaginarity monotones, such as the ℓ₁-norm and relative entropy measures, to robustly quantify the resource under real operations.
• The framework sets clear conditions for state and channel transformations, with practical implications in quantum metrology, optics, and channel discrimination.

Formal resource-theoretic frameworks for imaginarity rigorously formalize the quantification and manipulation of “imaginarity”—the non-reality of quantum states and channels—by isolating the operational role of complex numbers in quantum theory. These frameworks are defined by the identification of free objects (states/operations that do not possess or generate imaginary components in a chosen basis), operational monotones, and structure theorems for state and channel transformations under the restricted set of real (imaginarity-free) operations.

1. Structure of the Resource Theory: Free States and Free Operations

The resource theory is defined relative to a fixed computational basis $\{|i\rangle\}$ on a Hilbert space $\mathcal{H}$. The fundamental elements are as follows:

• Free States ("Real States"): A density matrix $\rho$ is free if all its entries are real, i.e. $\rho_{ij} \in \mathbb{R}$ for all $i,j$. Equivalently, $\rho = \rho^T$ or under complex conjugation, $\rho = \rho^*$ (Kondra et al., 2022, Wu et al., 2021, Chen et al., 2022, Han et al., 14 Mar 2026).
• Free Operations (Real Quantum Operations, RIO): A completely positive trace-preserving (CPTP) map $\Lambda$ is free if it admits a Kraus decomposition $\Lambda(\cdot)=\sum_m K_m(\cdot)K_m^\dagger$ with all $\langle i|K_m|j\rangle \in \mathbb{R}$. Equivalently, these maps are "covariant under transpose", i.e., $\Lambda(\rho^T) = \Lambda(\rho)^T$ for all $\rho$ (Kondra et al., 2022, Wu et al., 2021, Wu et al., 2020, Hickey et al., 2018).
• Basis Dependence and Lack of Resource-Destroying Maps: Imaginarity is fundamentally basis dependent, and the resource theory lacks a completely positive resource-destroying map since the transpose map is not completely positive (Hickey et al., 2018).

In infinite-dimensional (Gaussian) systems, free Gaussian states are those whose means and covariance matrices have only real entries and certain vanishing conditions, while real Gaussian channels precisely preserve these properties (Xu, 2023).

2. Quantification: Imaginarity Monotones and Operational Measures

All imaginarity monotones must satisfy core axioms: faithfulness, monotonicity under free operations, strong monotonicity, and convexity (sometimes equivalently reformulated as direct-sum additivity for resource measures on both states and channels) (Kondra et al., 2022, Hickey et al., 2018, Chen et al., 2024).

Core Families of Monotones

Measure Family	Definition/Closed Form	Key Properties/Context
$\ell_1$-norm imaginarity	$I_{l_1}(\rho) = \sum_{i\ne j} |\Im\rho_{ij}|$	Faithful, convex, simple analytic form
Robustness	$$R(\rho) = \min \{s \geq 0: (\rho + s\tau)/(1+s) \in \mathcal{F}\}$$	$R(\rho) = \tfrac{1}{2}\|\rho - \rho^T\|_1$
Relative entropy	$S_I(\rho) = S((\rho+\rho^T)/2) - S(\rho)$	For pure $|\psi\rangle$: closed form
Geometric imaginarity	$$I_g(|\psi\rangle) = \frac{1 - |\langle\psi|\psi^*\rangle|}{2}$$	Convex-roof for mixed states
Tsallis relative $\alpha$-entropy	$$D_\alpha(\rho\|\sigma) = \frac{1 - \mathrm{Tr}(\rho^\alpha \sigma^{1-\alpha})}{1-\alpha}$$	Minimizer $\sigma = \rho^*$; closed form
Sandwiched Rényi,	$D_{S,\alpha}$, $D_{O,\alpha}$: various divergence measures	Used for finer quantification
Fidelity-based	$I_F(\rho) = 1 - F(\rho, \mathrm{Re}(\rho))$	Contractive, closed qubit formula
Unified $(\alpha,\beta)$-relative entropy	$$D_\alpha^\beta(\rho\|\sigma) = \frac{\mathrm{Tr}[\rho^\alpha \sigma^{1-\alpha}]^\beta-1}{(\alpha-1)\beta}$$	Generalizes previous measures

References: (Kondra et al., 2022, Wu et al., 2021, Xu, 2023, Wu et al., 11 Jun 2025, Guo et al., 14 Jan 2025, Du et al., 2024, Liu et al., 29 Oct 2025, Guo et al., 2024, Chen et al., 2024, Chen et al., 2022, Han et al., 14 Mar 2026).

These measures typically satisfy:

• Vanishing exactly on real states
• Monotonicity and strong monotonicity under real operations
• Convexity under mixtures, additivity under direct sum

The minimizer in divergence-based measures (relative entropy, Tsallis, etc.) is always $\sigma = \rho^*$, substantially simplifying their computation (Chen et al., 2024).

3. State and Channel Transformations under Real Operations

State Conversion (Single-Copy)

• Deterministic pure-state conversion: $|\psi\rangle \to |\phi\rangle$ by real CPTP iff $I_g(|\psi\rangle) \geq I_g(|\phi\rangle)$, or equivalently $$|\langle\psi|\psi^*\rangle| \leq |\langle\phi|\phi^*\rangle|$$ (Kondra et al., 2022, Wu et al., 2021).
• Probabilistic pure-to-mixed conversion: Maximal success probability $$P(|\psi\rangle \to \rho) = \min\left[ \frac{I_g(|\psi\rangle)}{I_g(\rho)}, 1\right]$$; analogous formulas exist for other geometric-like monotones (Kondra et al., 2022, Guo et al., 2024).
• Approximate conversions with fidelity constraint: $P_f$ and $F_p$ are given by explicit expressions in terms of geometric or geometric-like imaginarity (Kondra et al., 2022, Guo et al., 2024).
• For general states: Necessary and sufficient conditions for state transformation under RIO correspond to conditions on conditional min-entropy for auxiliary states and can be checked via semidefinite programming (Kondra et al., 2022).

Channel Imaginarity and Superchannels

• Imaginarity of Channels: Free (real) channels are those whose Choi matrix is real. Quantification uses robustness, trace-norm, and sandwiched Rényi–based measures on Choi matrices (Chen et al., 2024, Wu et al., 11 Jun 2025).
• Axiomatic Structure: Channel monotones must be faithful, monotone under real superchannels, convex, and additive under direct sums (Chen et al., 2024).
• Operational meaning: Imaginarity in channels allows for discrimination or other tasks not possible with real (free) channels, mirroring the situation for states (Wu et al., 2020, Chen et al., 2024).

4. Relation to Other Quantum Resource Theories

• Coherence Relation: Imaginarity is formally a special case of coherence, with the resource being imaginary (instead of general complex) coherence in off-diagonals. However, unlike standard coherence, imaginarity resource theory is closed under both completely resource non-generating and stochastically resource non-generating operations and lacks a CPTP resource-destroying map (Hickey et al., 2018).
• Entanglement: Real entanglement monotones exist which detect entanglement not visible to standard LOCC protocols. For example, a state separable in the conventional sense may be “real-entangled” under restriction to real operations (Kondra et al., 2022).

5. Operational and Physical Applications

• Physical Demonstrations: In neutrino oscillation, imaginarity quantifies the nonclassical features that remain even in the absence of complex phases due to CP violation (Alok et al., 2024).
• Metrology, State Discrimination and Channel Discrimination: Imaginarity can enable perfect local state discrimination where complex measurements outperform real ones, and enhances the discrimination power in quantum channels (Wu et al., 2020, Wu et al., 2023).
• Optical Implementation Complexity: Real optical operations require fewer non-fixed waveplate settings compared to general complex operations—a factor-of-two savings asymptotically (Wu et al., 2021).

6. Mathematical and Structural Properties

• Monotone Hierarchies and Ordering: Imaginarity measures have explicit ordering relationships, e.g., $I_F \leq I_G \leq 1 - (1 - I_F)^2$, and the order of single-qubit states is preserved under bit-flip and certain noisy channels (Guo et al., 14 Jan 2025, Liu et al., 29 Oct 2025).
• Decay under Noise: Imaginarity monotones (e.g., geometric-like, Tsallis, sandwiched Rényi) differ in robustness under quantum channels, with Tsallis-based measures exhibiting greater robustness (slower decay) (Han et al., 14 Mar 2026, Guo et al., 2024, Guo et al., 14 Jan 2025).
• Convex-roof and Least-Imaginarity Constructions: Mixed-state quantifiers are often extended from pure-state monotones by convex roof minimization or by minimal cost of generating the target state via real operations from pure resources (Du et al., 2024).

7. Extensions and Generalizations

• Unifying Divergence Families: The unified $(\alpha,\beta)$-relative entropy gives rise to families of monotones that subsume previous quantities. Key properties include superadditivity under direct sums and subadditivity under tensor product (Wu et al., 11 Jun 2025, Chen et al., 2024).
• Gaussian Continuous-Variable Systems: Imaginarity measures have analytic formulas for Gaussian states in terms of covariance matrices and means, and free Gaussian operations are explicitly characterized by constraints on phase-space parameters (Xu, 2023, Xu, 2023).

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In summary, formal resource-theoretic frameworks for imaginarity provide a rigorous and comprehensive structure for analyzing the operational role of the imaginary part of quantum states, both at the level of states and channels. They unify many quantifiers (robustness, relative entropy, Tsallis, geometric-type, and fidelity-based), offer explicit monotonicity and convexity properties, and yield necessary and sufficient conversion conditions—particularly in the single-qubit and Gaussian regimes—while being applicable across diverse physical settings such as quantum optics, metrology, and particle physics (Kondra et al., 2022, Wu et al., 2020, Wu et al., 2021, Xu, 2023, Guo et al., 2024, Wu et al., 11 Jun 2025, Han et al., 14 Mar 2026).