Quantum Imaginarity: Quantitative Measures

• Quantitative Measures of Imaginarity are a resource theory that rigorously quantifies the imaginary components in quantum states and channels using real-coefficient constraints and free operations.
• Key metrics such as the ℓ1-norm, relative entropy, and robustness offer operational tests that satisfy conditions like faithfulness, monotonicity, and convexity.
• These measures enable practical applications in state conversion, noise characterization, and performance benchmarking across both finite-dimensional and continuous-variable (Gaussian) systems.

Quantitative Measures of Imaginarity

The resource theory of imaginarity formalizes the role of complex numbers—specifically, the imaginary components of quantum states and channels—as a quantifiable resource under operational constraints. In this framework, "free states" are those whose representation in a fixed orthonormal basis contains only real coefficients, and "free operations" are completely positive trace-preserving (CPTP) maps that cannot generate imaginary matrix elements from real input states. Quantitative imaginarity measures are functionals on quantum states or operations that satisfy faithfulness (vanishing if and only if the argument is “real”), monotonicity under free operations, and additional structural properties such as convexity or additivity. A mature landscape of such measures exists for both finite-dimensional systems and continuous-variable (Gaussian) states, as well as for dynamical quantum channels.

1. Resource-Theoretic Framework and Validity Conditions

Let $\mathcal{H}$ be a $d$-dimensional Hilbert space with a fixed orthonormal reference basis $\{|m\rangle\}_{m=1}^d$. The resource-free set comprises density operators with real coefficients: $$\mathcal{F} = \{\rho \in \mathcal{D}(\mathcal{H}) : \langle m|\rho|n\rangle \in \mathbb{R}\ \forall m,n\},$$ and free operations are those CPTP maps admitting all-real Kraus decompositions. For quantum operations, the analogous setting defines free channels as those whose Choi matrices are real in the computational basis (Wu et al., 11 Jun 2025, Chen et al., 2024).

A function $\mathcal{I}$ is a valid imaginarity measure (here, for states) if it satisfies:

• Faithfulness: $\mathcal{I}(\rho)=0$ if and only if $\rho \in \mathcal{F}$.
• Monotonicity: $\mathcal{I}(\Phi(\rho)) \le \mathcal{I}(\rho)$ for any free operation $\Phi$.
• Convexity/Strong Monotonicity: Convex combinations and averages under real Kraus maps do not increase $\mathcal{I}$. For channels, analogous properties apply under real (free) superchannels (Chen et al., 2024).

These axioms are operationally motivated by the structure of physically realizable quantum operations and constitute the backbone for constructing rigorous imaginarity quantifiers (Hickey et al., 2018, Wu et al., 2021, Han et al., 14 Mar 2026).

2. Principal Quantitative Measures for States

2.1 $\ell_1$-Norm Measure

The $\ell_1$-norm of imaginarity quantifies the total “weight” of the imaginary components of off-diagonal elements: $$\mathcal{I}_{\ell_1}(\rho) = \sum_{i \ne j} |\Im[\rho_{ij}]|.$$ This measure is basis-dependent and vanishes if and only if $\rho$ is real. For qubits, in Bloch form $$\rho = \frac12(I + x \sigma_x + y \sigma_y + z \sigma_z)$$, $\mathcal{I}_{\ell_1}(\rho) = 2|y|$ (Alok et al., 2024, Chen et al., 2022, Han et al., 14 Mar 2026).

2.2 Relative Entropy of Imaginarity

Defined as the entropic distinguishability between $\rho$ and its real part: $$\mathcal{I}_r(\rho) = S(\tfrac12(\rho+\rho^T)) - S(\rho),$$ where $S(\rho) = -\operatorname{Tr}[\rho \log \rho]$. For pure states $\rho = |\psi\rangle\langle\psi|$, a closed form is available based on $A = |\langle \psi^*|\psi\rangle|$ (Alok et al., 2024, Wu et al., 2020, Han et al., 14 Mar 2026).

2.3 Robustness of Imaginarity

Operationally interpretable as the minimum noise required to "de-imaginify" $\rho$: $$\mathcal{I}_{R}(\rho) = \min\{ s \ge 0: (\rho + s\tau)/(1+s) \in \mathcal{F},\ \tau \in \mathcal{D}(\mathcal{H})\} = \frac12\|\rho - \rho^T\|_1.$$ For qubits, this reduces again to $|y|$ in the Bloch sphere (Wu et al., 2021, Hickey et al., 2018, Han et al., 14 Mar 2026).

2.4 Geometric and Convex-Roof Measures

The pure-state geometric measure: $$\mathcal{I}_g(|\psi\rangle) = 1 - \max_{|\phi\rangle \in \mathcal{F}} |\langle \phi | \psi \rangle|^2 = \frac12 (1 - A)$$ with convex-roof extension to mixed states. The geometric-like measure improves noise resilience by

$$\mathcal{M}_{gl}(\rho) = 1 - \sqrt{\frac{1 + \sqrt{F(\rho,\rho^T)}}{2}}$$

where $F$ denotes Uhlmann fidelity (Guo et al., 2024, Du et al., 2024). State-conversion probabilities are given by the ratios of geometric(-like) measures.

2.5 Distance-Based and Divergence Measures

Tsallis and sandwiched Rényi relatives offer one-parameter families: $$I_{T, \alpha}(\rho) = 1 - \operatorname{Tr}(\rho^{\alpha} (\rho^*)^{1-\alpha}),$$ with $I_{R, \alpha, z}(\rho)$ using two-parameter $\alpha$-$z$ Rényi divergences and minimizing over real states, leading to explicit closed forms (Xu, 2023, Guo et al., 14 Jan 2025, Chen et al., 2024).

Operator-based divergences rely on Tsallis relative operator entropy and operator means, e.g.,

$$\mathcal{M}^O_{\lambda}(\delta) = 1 - \operatorname{Tr}(\delta \sharp_{\lambda} \delta^*)$$

for invertible $\delta$ and corresponding operator mean $\sharp_{\lambda}$ (Chen et al., 2024).

2.6 Fidelity-Based Measures

Defined via the Uhlmann fidelity between $\rho$ and its real-part state: $$M_{\Re}(\rho) = 1 - F(\rho, \Re(\rho)),\quad \Re(\rho) = \tfrac12(\rho+\rho^*),$$ location-wise directly comparable to geometric and trace-norm measures, particularly for qubits (Liu et al., 29 Oct 2025).

3. Imaginarity Measures for Quantum Channels

Quantifying the resourcefulness of channels requires a generalization to the space of quantum operations.

3.1 Robustness and Trace-Norm Measures

For a CPTP map $\Phi$ with Choi matrix $X_\Phi$: $$R_I(\Phi) = \min \{ s \ge 0: (X_\Phi + s X_\Gamma)/(1+s) \in \text{RC}_{A \to B}\},$$

$T_I(\Phi) = \| X_\Phi - X_\Phi^* \|_1.$

Both are faithful, monotonic under real superchannels, and convex (Chen et al., 2024, Wu et al., 11 Jun 2025). The norm-based measure quantifies non-commutativity with the de-imaginarization map.

3.2 Entropy-Based Channel Measures

Using the sandwiched Rényi or Umegaki relative entropy for Choi matrices: $$S_I^\alpha(\Phi) = D_\alpha(X_\Phi \| X_\Phi^*),\quad S_I(\Phi) = S(X_\Phi \| X_\Phi^*)$$ are additive, strongly monotonic, and admit discrimination-theoretic operational meanings (Chen et al., 2024).

3.3 Weight-Based Measures

Defined as the minimal fraction of resourceful operations needed in convex decomposition of the channel: $$I_\text{weight}(\Phi) = \min\{ s \ge 0: \Phi = (1-s)\Psi + s\Lambda,\, \Psi\ \text{free}\}.$$ This measure is convex and enjoys clear resource-theoretic justification (Wu et al., 11 Jun 2025).

4. Imaginarity Measures in Continuous-Variable (Gaussian) Systems

For $n$-mode Gaussian states with displacement vector $\bar d$ and covariance matrix $\nu$: $$\mathcal{I}^{G_n}(\rho) = 1 - \frac{\det(\nu)}{\det(A_{11})\det(A_{22})} + h\bigl(\| Q_n' P_n \bar d \|_1\bigr),$$ leveraging block structure and the phase-space representation for efficient and faithful quantification. This measure is readily computable and satisfies all expected monotonicity and faithfulness criteria. It meaningfully extends to multipartite Gaussian resources (Zhang et al., 10 Apr 2025).

Fidelity-based [Xu 2023] and Tsallis-entropy Gaussian measures, while conceptually similar, require more computational resources for multi-mode systems.

5. Operational Interpretation, Ordering, and Dynamical Behavior

Most imaginarity measures order single-qubit and higher-dimensional states identically on the set of pure states, and their ordering is often preserved under real quantum channels such as bit-flip and phase-flip (Chen et al., 2022, Guo et al., 14 Jan 2025). Explicit formulas for the decay of imaginarity under various noisy channels (dephasing, amplitude damping, etc.) reveal generic monotonic loss of resource, with Tsallis-$\alpha$ measures often exhibiting higher decay rates (Han et al., 14 Mar 2026, Guo et al., 14 Jan 2025).

Maximal imaginary states (e.g., $|+i\rangle$ for qubits) convert via free operations to all other states, serving as resource-theoretic analogues of maximally entangled or coherent states. Extensions to separable states and channels implicate the unique role of imaginarity in quantum protocols (Han et al., 14 Mar 2026).

For pure-state conversion and approximate distillation, conversion probabilities are fully characterized by suitable ratios of imaginality monotones (e.g., geometric or geometric-like measures) (Du et al., 2024, Guo et al., 2024).

6. Witnesses, Complementarity, and Mathematical Relations

Unitary-invariant Bargmann invariants provide basis-independent witnesses of imaginarity, detecting the resource presence via nonzero imaginary parts of cyclic trace invariants for tuples of pure or mixed states. For sets of three pure states, triplet invariants require full quantum positivity; for four states, pairwise overlaps suffice combined with semi-definite positivity checks. These witnesses are experimentally accessible but are not full monotones under free operations (Fernandes et al., 2024).

Complementarity relations constrain the distribution of imaginarity across mutually unbiased bases (MUBs); for example, no qubit can be maximally imaginary in all three Pauli bases, and explicit bounds tie imaginarity in each basis to state purity (Liu et al., 29 Oct 2025).

Quantitative relations between measures are dimension-dependent: e.g., the relative entropy is always less than the trace-norm, which is bounded above by the $\ell_1$-norm. Closed-form inequalities can be established for both qubits and higher dimensions (Han et al., 14 Mar 2026).

7. Applications and Extensions

Quantitative imaginarity is deeply relevant to quantum information tasks where complex phases are crucial: discrimination advantages, quantum speed limits, communication, and robust state preparation and transformation (Xuan et al., 8 Nov 2025, Wu et al., 2020). In neutrino systems, nonzero imaginarity appears even with real mixing matrices and maximizes when probabilities are non-deterministic, indicating that intrinsic quantum dynamics—not solely CP-violating phases—generate resourceful imaginarity (Alok et al., 2024). The multipartite extension to Gaussian systems reveals imaginarity as a form of continuous-variable correlation (Zhang et al., 10 Apr 2025).

These measures also enable the quantitative benchmarking of experimental protocols (e.g., waveplate count in optics), and inform the design of imaginarily robust quantum information processing schemes (Wu et al., 2021).

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References:

• (Alok et al., 2024) Quantifying Imaginarity in Neutrino Systems
• (Hickey et al., 2018) Quantifying the Imaginarity of Quantum Mechanics
• (Wu et al., 2021) Resource theory of imaginarity: Quantification and state conversion
• (Du et al., 2024) Quantifying imaginarity in terms of pure-state imaginarity
• (Fernandes et al., 2024) Unitary-invariant witnesses of quantum imaginarity
• (Guo et al., 14 Jan 2025) Quantifying the imaginarity via different distance measures
• (Zhang et al., 10 Apr 2025) An easily computable measure of Gaussian quantum imaginarity
• (Liu et al., 29 Oct 2025) Imaginarity measures induced by real part states and the complementarity relations
• (Xuan et al., 8 Nov 2025) Quantum-imaginarity-based quantum speed limit
• (Han et al., 14 Mar 2026) Revisited Quantification of the Resource Theory of Imaginarity
• (Chen et al., 2024) Imaginarity of quantum channels: Refinement and Alternative
• (Wu et al., 11 Jun 2025) Quantifying imaginarity of quantum operations
• (Xu, 2023) Quantifying the imaginarity of quantum states via Tsallis relative entropy
• (Xu, 2024) Coherence and imaginarity of quantum states
• (Reiffenstein, 2022) A quantitative formula for the imaginary part of a Weyl coefficient