Distillable Imaginarity of Assistance

• DIA is a resource-theoretic measure quantifying the optimal distillation of quantum imaginarity from the imaginary components of density matrices under assisted protocols.
• It employs one-way classical communication with real operations to achieve fidelity benchmarks using the maximally imaginary pure state (imbit).
• Experimental protocols and analytical models reveal DIA's practical advantages in channel discrimination and distributed quantum networks.

Distillable imaginarity of assistance (DIA) is a resource-theoretic quantity in quantum information theory, representing the optimal amount of imaginarity—quantum coherence residing in the imaginary parts of density matrix elements—that can be distilled onto a subsystem (typically Bob’s qubit) with assistance from another party (Alice) using one-way classical communication and real operations. DIA has a rigorously defined operational meaning, mathematically precise protocols for optimal distillation, and concrete performance bounds and experimental demonstrations. Its theoretical analysis builds on the resource theory of imaginarity, where free states and operations are defined by the absence of complex coefficients, and the maximally imaginary pure state (“imbit”) $\ket{+_i} = (\ket{0} + i\ket{1})/\sqrt{2}$ underpins fidelity and conversion protocols. DIA is operationally and mathematically distinct from other quantum resources, such as entanglement or coherence, due to its restriction to real operations and its direct connection to the nonlocal advantage of quantum imaginarity (NAQI) (Wu et al., 2023, Wang et al., 16 Jan 2026).

1. Resource-Theoretic Framework

The resource theory of imaginarity begins by specifying a fixed orthonormal basis $\{\ket{i}\}$. States are classified as real if their matrix elements $\langle i|\rho|j\rangle$ are real ($\rho = \rho^T$). Free operations are completely positive maps whose Kraus operators have real-valued matrix elements. This framework is analogous to resource theories for coherence or entanglement but restricts allowable operations more strictly: only those which cannot generate or amplify imaginary entries are free (Wu et al., 2023).

Key measures of imaginarity include:

• Trace-norm of imaginarity: $$\mathcal{I}_{tr}(\rho) = \tfrac{1}{2}\|\rho-\rho^T\|_1$$
• Relative entropy of imaginarity: $\mathcal{I}_{re}(\rho)=S((\rho+\rho^T)/2)-S(\rho)$

Measures must vanish precisely for real states, be nonincreasing under real operations, and satisfy convexity. The “imbit” $\ket{+_i}$ is the unique maximally imaginary pure qubit state (up to basis relabeling) (Wu et al., 2023).

2. Formal Definition and Protocols for DIA

DIA is operationalized via protocols in bipartite settings, typically involving Alice (A) and Bob (B), who share $\rho_{AB}$:

• Local quantum-real operations and one-way classical communication (LQRCC) are the protocol class.
• Alice performs an arbitrary POVM $\{E_i\}$ on A, communicates result $i$ to Bob.
• Bob performs real operation $\Lambda_i$ on B, resulting in post-measurement state $\Lambda_i[\rho_{B|E_i}]$ with probability $p_i$.

The central figure of merit,

$$F_a(\rho_{AB}) = \max_{\{E_i,\Lambda_i\}} \sum_i p_i\,F(\Lambda_i[\rho_{B|E_i}],\,|\tilde{+}\rangle\langle\tilde{+}|),$$

quantifies the maximal achievable fidelity with respect to $\ket+_i$ on Bob's system. One-shot DIA is defined as $D_{ass}(\rho_{AB}) = 2F_a(\rho_{AB})-1$, which is strictly positive only if genuine assistance yields an advantage over single-party protocols. Asymptotic (many-copy) rates are captured by $I_d^A(\rho^{AB})$ (Wang et al., 16 Jan 2026, Wu et al., 2023).

3. Analytical Solutions for Qubit and Higher-Dimensional Systems

For general two-qubit states, one invokes the Bloch-matrix representation: $\rho^{AB} = \tfrac14\Big(\openone\otimes\openone+\sum_k a_k\,\sigma_k\otimes\openone+\sum_\ell b_\ell\,\openone\otimes\sigma_\ell+\sum_{k,\ell}E_{k\ell}\,\sigma_k\otimes\sigma_\ell\Big).$ Let $b_2 = \textrm{tr}[\rho(\openone\otimes\sigma_y)]$ and $\mathbf{s} = (E_{1\,2}, E_{2\,2}, E_{3\,2})$. Then (Wu et al., 2023): $$F_a(\rho^{AB}) = \frac12\Big(1 + \max\left\{|b_2|, \|\mathbf{s}\|\right\}\Big).$$

For pure states of arbitrary dimension with Bob’s target state $\sigma^B$, formulas involving angles derived from imaginarity measures (Chernoff–type divergences) yield sharp conversion probabilities and fidelities (see “Analytic solutions” section in (Wu et al., 2023)). These protocols require only projective measurement by Alice, classical bit communication, and real filterings by Bob to saturate the optimal bound.

4. Physical Realization: Lossey Cavity Model and Off-Resonant Dynamics

Examining two qubits in a lossy cavity, the Hamiltonian in the rotating-wave approximation reads: $$H = \sum_{n=A,B} \omega_n \sigma_+^n \sigma_-^n + \sum_k \omega_k b_k^\dag b_k + (\alpha_A \sigma_+^A + \alpha_B \sigma_+^B) \sum_k g_k b_k + \text{h.c.}$$ Under large detuning $|\delta_n| \gg R\lambda$ (with $R = \alpha_T W/\lambda$, $\alpha_T^2 = \alpha_A^2 + \alpha_B^2$), the effective Hamiltonian simplifies after adiabatic elimination of the cavity: $$H_\text{eff} = \sum_{n=A,B} \frac{R^2 r_n^2}{\delta_n} \sigma_+^n \sigma_-^n + \frac{R^2 r_A r_B}{2 \delta_n}(\sigma_+^A \sigma_-^B + \sigma_-^A \sigma_+^B)$$ with $r_n = \alpha_n/\alpha_T$ (Wang et al., 16 Jan 2026).

This regime ensures both NAQI and DIA are conserved for long periods: assisted imaginarity fidelity behaves as $F_a(t) \approx \exp[-(R^2 / \delta)^2 \lambda t]$, with increasing $\delta$ yielding extended preservation. Initial product states can evolve to generate a maximal DIA via off-resonant virtual-photon exchange, demonstrated explicitly in the time-dependent amplitudes and optimal coupling ratios ($r_A^2 \simeq 0.17$ in “bad cavity” and $r_A^2 \simeq 0.16$ in “good cavity”). Off-resonant protocols outperform resonant counterparts by suppressing decoherence and leakage of imaginary coherence.

5. Operational Significance, Channel Discrimination, and Experimental Demonstration

DIA directly quantifies the operational quantum advantage of steering Bob’s system into the maximally imaginary state using real-only operations on his side but quantum measurements on Alice’s. A value $F_a > \frac12$ or $D_{ass} > 0$ certifies nonclassicality and resource utility (Wu et al., 2023).

Key applications:

• Channel discrimination without ancilla: Imaginary probes and measurements can perfectly discriminate certain real channels where all-real protocols fail ($p_\textrm{succ}=1$ for imaginarity vs.\ $p_\textrm{succ}=1/2$ for real-only).
• Channel discrimination with ancilla: Entangled inputs allow real LOCC measurements to achieve ideal performance, showing the role of imaginary resources can be simulated via entanglement.

Experimental protocols employ photonic qubits prepared via SPDC, with modular setups implementing Alice’s measurements and Bob’s real filtering. Data precisely match theoretical bounds on $F_a$, confirming the resource-theoretic predictions and the operational superiority of imaginarity in discrimination tasks (Wu et al., 2023).

6. Comparison with Related Resource Theories and Practical Advantages

Unlike coherence or entanglement, imaginarity is strictly linked to complex-number structure and is nontrivial only when complex phases are operationally accessible. Real-only resource theories fail to exploit certain quantum advantages unless boosted by either entanglement or assisted imaginarity distillation. The presence of assistance strictly extends the class of achievable quantum processes, with explicit performance separation demonstrated for channel and state discrimination (Wu et al., 2023).

A plausible implication is that in distributed quantum networks where subsystems are limited to real operations, inter-party assistance can unlock full quantum advantages attributed to complex structure.

7. Enhancement Strategies, Longevity, and Limits

DIA is optimized in physical implementations by tuning system parameters for large, symmetric detunings, which suppress environmental leakage and irreversibility. Resonant interaction regimes are suboptimal—DIA decays quickly due to enhanced decoherence. Passive protection and active generation of DIA are best realized via combined tuning of qubit frequencies and couplings, including the use of symmetric opposite detunings to minimize cavity-mediated decoherence. This parameter regime allows for nearly ideal, long-lived distribution and distillation of imaginary-number coherence in quantum networks (Wang et al., 16 Jan 2026).

The longevity of DIA, scalable protocol performance, and the discriminability enhancement directly tie the resource-theoretic analysis to practical, experimentally verifiable operational advantages—thus positioning DIA as a central quantity in distributed quantum protocols sensitive to complex-number quantum resources.