---
title: Formal Resource Theory for Quantum Imaginarity
url: https://www.emergentmind.com/topics/formal-resource-theoretic-frameworks-for-imaginarity
type: topic
---

# Formal Resource Theory for Quantum Imaginarity

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, $\rho = \rho^*$ [2210.15820, 2103.01805, 2210.14443, 2603.13980].

- **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 $\rho$ [2210.15820; 2103.01805; 2007.14847; 1801.05123].

- **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 [1801.05123].

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 [2307.14116].

## 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) [2210.15820, 1801.05123, 2405.06222].

### 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: [2210.15820, 2103.01805, 2311.12547, 2506.09799, 2501.07775, 2411.12215, 2510.25313, 2410.20879, 2404.00637, 2210.14443, 2603.13980].

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 [2404.00637].

## 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 $|\langle\psi|\psi^*\rangle| \leq |\langle\phi|\phi^*\rangle|$ [2210.15820, 2103.01805].
- **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 [2210.15820, 2410.20879].
- **Approximate conversions with fidelity constraint**: $P_f$ and $F_p$ are given by explicit expressions in terms of geometric or geometric-like imaginarity [2210.15820, 2410.20879].
- **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 [2210.15820].

### 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 [2405.06222, 2506.09747].
- **Axiomatic Structure**: Channel monotones must be faithful, monotone under real superchannels, convex, and additive under direct sums [2405.06222].
- **Operational meaning**: Imaginarity in channels allows for discrimination or other tasks not possible with real (free) channels, mirroring the situation for states [2007.14847, 2405.06222].

## 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 [1801.05123].
- **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 [2210.15820].

## 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 [2412.01871].
- **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 [2007.14847, 2301.04782].
- **Optical Implementation Complexity**: Real optical operations require fewer non-fixed waveplate settings compared to general complex operations—a factor-of-two savings asymptotically [2103.01805].

## 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 [2501.07775, 2510.25313].
- **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) [2603.13980, 2410.20879, 2501.07775].
- **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 [2411.12215].

## 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 [2506.09799, 2404.00637].
- **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 [2311.12547, 2307.14116].

---

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 [2210.15820, 2007.14847, 2103.01805, 2311.12547, 2410.20879, 2506.09799, 2603.13980].

Source: https://www.emergentmind.com/topics/formal-resource-theoretic-frameworks-for-imaginarity