Nonlocal Advantage of Quantum Imaginarity (NAQI)

• NAQI is a quantum nonlocality phenomenon where bipartite correlations enable local measurements to produce conditional states with imaginarity exceeding single-qubit limits.
• Its operational utility is demonstrated in tasks like quantum steering, assisted distillation, and ancilla-free channel discrimination, while showing robustness against noise.
• NAQI occupies a unique niche between steering and Bell nonlocality, exhibiting monogamy constraints and requiring minimal measurement settings for effective implementation.

The Nonlocal Advantage of Quantum Imaginarity (NAQI) is a recently established form of quantum nonlocality, formally characterized by the ability of bipartite quantum correlations to generate—via local measurements on one party (Alice)—conditional quantum states on the other party (Bob) whose average “imaginarity” exceeds the maximal value achievable by any uncorrelated single-qubit quantum state. Imaginarity refers to the resource content of imaginary-valued off-diagonal matrix elements in a given basis, and its operational utility has been demonstrated for quantum steering, assisted distillation, and ancilla-free channel discrimination. NAQI is strictly stronger than steering but weaker than Bell nonlocality, and displays distinctive features such as robust performance under noise and monogamy-type tradeoffs in multipartite settings.

1. Quantification of Quantum Imaginarity

Imaginarity is a basis-dependent quantifier of "quantumness," corresponding to the irreducible imaginary parts in the density matrix of a quantum state. Several measures are employed for imaginarity in a fixed orthonormal basis $\mathcal{B}=\{|a\rangle\}$ (Wei et al., 2024, Datta et al., 23 Aug 2025):

• $\ell_1$-norm of imaginarity: $${\mathscr I}^{\,l_1}_\mathcal{B}(\rho) = \sum_{a,b} |\Im\,\rho_{ab}|$$.
• Relative entropy of imaginarity: $${\mathscr I}^{\,r}_\mathcal{B}(\rho) = S\bigl(\Delta(\rho)\bigr) - S(\rho)$$, where $\Delta(\rho)=\frac{1}{2}(\rho+\rho^T)$.
• Robustness of imaginarity: $$\mathscr{I}_R(\rho)=\min_\tau \{\,s\ge0: (\rho+s\,\tau)/(1+s)\in\mathcal{R}\}:= \tfrac{1}{2}\|\rho-\rho^T\|_1$$, where $\mathcal{R}$ is the set of real-valued density matrices.
• Other resource monotones include fidelity-based measures (maximal fidelity with the "maximally imaginary" state $|+i\rangle$), geometric, and concurrence-like monotones (Wu et al., 2023).

For a single qubit state $\rho=\frac{1}{2}(\openone+\vec{n}\cdot\vec{\sigma})$, $\mathscr{I}_R^x(\rho)=|n_y|$, $\mathscr{I}_R^y(\rho)=|n_x|$, and $\mathscr{I}_R^z(\rho)=|n_y|$, highlighting the purely imaginary off-diagonal contributions in each Pauli basis (Datta et al., 23 Aug 2025).

A key property is the complementarity relation in dimension 2: for three mutually unbiased bases (the Pauli bases), $$\sum_{i=1}^3 \mathscr{I}_i^\gamma(\rho)\le\mathscr{I}_\gamma$$, with $\mathscr{I}_{l_1}=\sqrt{5}$ and $\mathscr{I}_r\approx2.027$ (Wei et al., 2024).

2. Formal Definition and Operationalization of NAQI

NAQI is formalized in bipartite systems where measurements on Alice allow Bob to obtain conditional states with a total average imaginarity (over mutually unbiased bases) exceeding the single-qubit complementarity bound (Wei et al., 2024, Wang et al., 16 Jan 2026). Explicitly, for a two-qubit state $\rho_{AB}$ and measurement scenario:

$$\mathcal{N}_\alpha(\rho_{AB}) = \max_{\{\Pi_i,\mathcal{M}_i\}} \sum_{i=1}^3\sum_{a=\pm} p_{B|\Pi_i^a} \mathcal{I}_\alpha^{\mathcal{M}_i}\left(\rho_{B|\Pi_i^a}\right)$$

NAQI is demonstrated if $$\mathcal{N}_\alpha(\rho_{AB}) > \mathcal{I}_{\alpha,\mathcal{B}}$$, where $\mathcal{I}_{\alpha,\mathcal{B}}$ is the single-qubit bound (e.g., $\sqrt5$ for the trace-norm measure) (Wang et al., 16 Jan 2026). In the two-setting scenario (Datta et al., 23 Aug 2025), the steering functional reduces to:

$$I_2(\rho_{AB}) = \sum_{a}p(a|y)\mathscr{I}_R^x(\rho_{B|a,y}) + \sum_{a}p(a|x)\mathscr{I}_R^y(\rho_{B|a,x}) \le \sqrt{2}.$$

A violation, i.e., $I_2(\rho_{AB}) > \sqrt{2}$, certifies NAQI under minimal measurement resources.

3. Hierarchy and Relations to Other Nonlocal Phenomena

NAQI occupies a distinctive position in the quantum nonlocality hierarchy. The region of NAQI sits strictly between steering and full Bell nonlocality. Explicitly, for families such as Werner states and mixtures of Bell states, NAQI holds on parameter intervals strictly narrower than those sufficient for steering, but not as restrictive as Bell inequality violations (Wei et al., 2024). For instance, the NAQI threshold for the trace-norm measure on Werner states is $p>0.7454$, compared to $p>1/2$ for steering and $p>1/\sqrt{2}$ for CHSH violation (Wei et al., 2024, Datta et al., 23 Aug 2025).

NAQI implies steerability: any state exhibiting NAQI admits no local-hidden-state (LHS) model; however, not all steerable states are NAQI-capable (Wei et al., 2024). This establishes NAQI as a strictly stronger signature than quantum steering.

4. Monogamy and Multipartite Constraints

A fundamental feature of NAQI is the monogamy or exclusion property in multipartite states. For pure three-qubit states $|\Psi_{ABC}\rangle$, at most one of the reduced two-qubit pairs can exhibit NAQI simultaneously (Wei et al., 2024). In the two-setting steering framework, the sum of imaginarity violations across pairs is bounded:

$I_2(\rho_{AB}) + I_2(\rho_{AC}) \le 2\sqrt{2}$

(Datta et al., 23 Aug 2025). This extends the monogamy of nonlocal correlations to the domain of imaginarity-based nonlocality and constrains the distribution of imaginarity as a nonlocal resource.

5. Robustness, Resource Efficiency, and Measurement Overhead

NAQI-based criteria display high robustness against white noise, decoherence, and unsharp measurements. For Werner states, NAQI persists down to $v>1/\sqrt{2}$ (optimal steering threshold) (Datta et al., 23 Aug 2025). Under unsharp measurements with sharpness $\eta$, the same threshold applies, $I_2=2\eta>\sqrt{2}$ iff $\eta>1/\sqrt{2}$.

The measurement protocol for NAQI is resource-efficient relative to traditional steering and Bell tests. In the minimal two-setting scenario, only two dichotomic observables for Alice and two mutually unbiased bases for Bob are needed, requiring as few as four joint measurement statistics and depending on only four real state parameters (Datta et al., 23 Aug 2025). By contrast, standard steering–CHSH tests or NAQC (three-setting coherence steering) require more measurement settings and state parameters.

6. NAQI in Distributed and Dynamical Scenarios

NAQI has operational significance in distributed quantum protocols, dynamical open quantum systems, and channel discrimination. In two-qubit systems interacting with a lossy cavity, off-resonant regimes with large symmetric detunings greatly enhance and preserve NAQI and the related Distillable Imaginarity of Assistance (DIA), even over long time scales (Wang et al., 16 Jan 2026). The underlying mechanism is suppression of decoherence via virtual-photon exchange; effective dipole–dipole interactions mediate persistence or generation of imaginarity correlations. For appropriately tuned detunings and couplings, one can dynamically generate NAQI from initial product states or maintain near-maximal NAQI in time.

Beyond state preparation, NAQI operates as a resource in ancilla-free discrimination of quantum channels and in assisted distillation—where Alice helps Bob locally distill imaginarity via classical communication and local operations (LQRCC protocols) (Wu et al., 2023). Imaginarity enables success probabilities and fidelities inaccessible when restricted to real operations, and the performance has been experimentally verified in photonic platforms.

7. Experimental Realization and Applications

Experimental demonstrations include photonic entanglement sources, state preparation via interferometric schemes, and quantum state tomography for imaginarity extraction (Wu et al., 2023). Alice's optimal POVM measurement and Bob's application of real operations followed by measurement in an imaginary basis can realize assisted distillation protocols, as predicted theoretically. In both state-assisted and channel-discrimination tasks, the advantage over protocols restricted to real resources is observed with high agreement to theoretical predictions.

The operational advantage of imaginarity extends to lifting locality bounds in LOCC discrimination of Choi states, as well as enhancing ancilla-free channel discrimination (Wu et al., 2023). This establishes imaginarity as a genuinely nonlocal quantum resource—with unique status distinct from entanglement, coherence, and quantum discord—in both theoretical foundations and experimental implementations.