Superadditivity of One-Shot Quantum Capacity

• Superadditivity is the phenomenon where combining quantum channel uses with entangled inputs yields a one-shot capacity surpassing the sum of independent capacities.
• The approach employs smoothed coherent information and entropic bounds to quantify non-classical capacity enhancements in finite-resource quantum communication.
• Examples like dephrasure and Platypus channels demonstrate that joint coding strategies unlock higher transmission rates, emphasizing the need for multipartite entanglement in protocol design.

Superadditivity of one-shot quantum capacity is a non-classical phenomenon in quantum information theory wherein the maximal rate of reliable quantum communication achievable by coding over multiple uses of a quantum channel can strictly exceed the sum of the corresponding rates for each single use. This effect arises from exploiting entanglement and nonlocality in quantum codes, leading to scenarios where collective coding across channel uses or among network participants gives higher capacities than any strategy that treats channel uses independently. This behavior starkly contrasts with classical Shannon theory, where capacities are always additive under parallel channel use. The phenomenon is quantitatively tied to the nonadditivity of coherent information—a central entropic functional in quantum capacity formulas—and underpins both theoretical limitations and methodological advances in finite-resource quantum communication, quantum error correction, and quantum networks.

1. One-Shot Quantum Capacity: Definitions and Non-Additivity

The one-shot quantum capacity, denoted $Q_{\mathrm{ent}}(\Phi; \varepsilon)$ for a quantum channel $\Phi$ and error tolerance $\varepsilon$, is defined as the largest number of qubits (measured as $\log m$) that can be reliably transmitted using a single use of $\Phi$ such that the entanglement transmission fidelity $F_{\text{ent}}(\Phi; m) \geq 1-\varepsilon$ (0902.0158). In this setting, reliability is enforced for finite resources, with no recourse to infinite blocklength or i.i.d. assumptions.

Superadditivity refers to the property that for two channels (or two uses of the same channel) $\Phi_1$ and $\Phi_2$,

$Q(\Phi_1 \otimes \Phi_2) > Q(\Phi_1) + Q(\Phi_2),$

where $Q$ may be the one-shot quantum capacity (or its regularized asymptotic limit). This violation of additivity does not have a classical analog and reveals that joint coding over multiple channel uses, often via entangled inputs, can unlock communication rates unattainable by any product-state strategy (Elkouss et al., 2015, Leditzky et al., 2022, Leditzky et al., 2018). Formally, the capacity is controlled by optimizing over entropic functionals (e.g., smoothed 0-th order Rényi relative entropy, coherent information) that are themselves nonadditive under tensor products. As a consequence, the regularization (i.e., computing the limit $$\lim_{n \to \infty} \frac{1}{n} Q_{\mathrm{ent}}(\Phi^{\otimes n})$$) is needed to characterize the true asymptotic capacity.

2. Mathematical Framework: Entropic Bounds and Superadditivity Mechanisms

Central to quantitative analysis is the use of smoothed entropic quantities, in particular smoothed 0-coherent information. For a channel $\Phi$, an input state $\rho_A$, and a purification to $|\Psi\rangle_{RA}$, the output state is $$\omega_{RB} = (\mathrm{id}_R \otimes \Phi)(|\Psi\rangle\langle\Psi|_{RA})$$. The smoothed 0-coherent information is defined as

$$I_{0,\delta}(\omega_{RB}) = \max_{P \in p(\omega_{RB};\delta)} \min_{\sigma \in \mathcal{S}(H_B)} \left\{ -S_0(\omega_{RB} \Vert \mathbb{I}_A \otimes \sigma) \right\},$$

where $$S_0(\rho \| \sigma) = -\log \operatorname{Tr}[P_{\mathrm{supp}\ \rho} \sigma]$$ and the maximization is over projector sets forming "smoothing" neighborhoods (0902.0158). The main capacity bounds are:

$$Q_{\mathrm{ent}}(\Phi;\varepsilon) \geq \max_{S \subset H_A}\left\{ I_{0,\varepsilon/8}(\omega_{RB}) + \log(1+\varepsilon) \right\} - \Delta,$$

(with similar upper bounds involving $I_{0,2\varepsilon}(\omega_{RB})$). These expressions not only capture nonasymptotic (finite $n$) operation, but, crucially, allow for superadditivity: when maximizing over code subspaces $S$, entangled choices can increase the total spectral weight available for communication beyond the sum of independent contributions.

Operationally, superadditivity emerges because the smoothed entropic quantities (and hypothesis testing metrics) are not additive in general; optimizing jointly across multiple channel uses can strictly outperform the sum of single-use optima (1007.5456). Quantum hypothesis testing and the nonadditivity of the corresponding relative entropy play a central technical role in formalizing this mechanism.

3. Manifestations of Superadditivity: Channel Models and Examples

Several explicit families of channels exhibit superadditivity of one-shot quantum capacity. Notable instances include:

• Platypus and Generalized Platypus Channels: Channels constructed by hybridizing simple (degradable or useless) channels through an isometry display superadditivity when used jointly with qubit erasure channels, multilevel amplitude damping channels, or in higher dimensions. Strikingly, these effects persist even when both constituent channels individually have large quantum capacities (Wu et al., 30 May 2025, Leditzky et al., 2022). For example, for a platypus channel $O_{\vec{\mu}}$ parameterized by probability vector $\vec{\mu}$ and qudit erasure channel $\mathcal{A}_{\lambda,d}$:

$$Q(O_{\vec{\mu}} \otimes \mathcal{A}_{\lambda,d}) > Q(O_{\vec{\mu}}) + Q(\mathcal{A}_{\lambda,d}).$$

• Dephrasure Channels: The concatenation of dephasing and erasure channels exhibits superadditivity of coherent information at the two-letter level and a pronounced separation between coherent and private information. Experimentally, even when the single-letter coherent information vanishes, coding over additional channel uses can yield strictly positive rates (Leditzky et al., 2018, Yu et al., 2020).
• Generalized Erasure Channels with Polarization-Dependent Losses: For channels with biased loss parameters (e.g., due to differing transmission probabilities for polarization modes), the two-letter coherent information can surpass the single-letter value by a quantifiable margin—up to $7.197 \times 10^{-3}$ bits per qubit in reported scenarios (Filippov, 2021, Filippov, 2021).
• Disjoint Zero-Error Capacity Channels: There exist low-dimensional channels for which $Q_0(\Phi)=0$ but $Q_0(\Phi \otimes \Phi)>0$, the so-called "superactivation" of quantum zero-error capacity (Shirokov et al., 2013).

These findings show that superadditivity can be robust to dimension, can occur between two "good" channels rather than only between useless and useful channels, and may demand multipartite entangled codes as optimal encodings (1110.2594).

4. Operational Implications and Communication Protocols

Superadditivity has profound impact on both the theoretical characterization and practical exploitation of quantum channels:

• Optimal Code Design: Realizing superadditive rates necessitates encoding across multiple uses with entangled input states—simple product or classical codebooks are generally suboptimal. In some network models (e.g., entanglement-assisted MACs), multipartite, rather than merely bipartite, entanglement may be required to saturate achievable rates (1110.2594, Zhu et al., 2017).
• Protocols for Estimation and Verification: Practical protocols for capacity verification in devices with arbitrarily correlated (including non-i.i.d.) noise can detect superadditivity by preparing and measuring in complementary bases and analyzing the resulting error statistics; observed rates above the i.i.d. bound signal the presence of superadditivity (Pfister et al., 2016).
• Trade-off Capacities and Resource Regions: Superadditivity extends to dynamic capacity regions in which classical, quantum, and entanglement resources are simultaneously present. Even when individual resource capacities are additive, the collective (e.g., CQE) region can exhibit superadditivity in combined capacities due to joint resource exploitation (Zhu et al., 2017).
• Activation and Superactivation: Zero-capacity channels can be "activated" when combined, with the composite channel supporting nonzero transmission; this property connects directly to the superadditive structure of one-shot capacity and quantum error-correcting codes (Shirokov et al., 2013, Wang et al., 2016).
• Perturbative and Analytical Criteria: Recent work establishes general criteria (based on perturbations of coherent information under small variations of the input state) to certify the presence of superadditivity or detect the gap between one-shot quantum and private capacities (Wu et al., 22 Jul 2025). These methods provide diagnostic tools for both analytical and numerical assessments in complex models.

5. Constraints, Decay, and Limitations of Superadditivity

While superadditivity is generic for many quantum channels, it is not universal:

• Degradable and Flagged Channels: Structural properties such as degradability, flagged (direct-sum) construction, or informational degradability impose additivity of coherent information and hence linear scaling of one-shot capacity over multiple uses. In these cases, $Q(\mathcal{N})=Q^{(1)}(\mathcal{N})$ for all $n$ (Smith et al., 5 Sep 2024).
• Dimension Dependence: For qudit depolarizing channels, the interval in depolarizing probability where superadditivity is nonzero shrinks as the dimension increases. In the limit $d \to \infty$, the capacity becomes additive and coincides with the one-shot coherent information:

$$\lim_{d \to \infty} Q_\mathrm{coh}^{(d)}(\Lambda_p^d) = 1 - 2p,$$

so $\xi(\Lambda_p^{d\to\infty}) = 0$ (Martinez et al., 2023).

• Limits of Superadditivity: Weak and strong additivity can be broken or maintained depending on structural channel features. Notably, relaxing output-environment conditions or combining nondegradable channels can induce superadditivity, whereas sufficiently strong constraints (e.g., flagged mixtures of degradable and anti-degradable channels under certain parameters) retain additivity (Smith et al., 5 Sep 2024).

6. Broader Impact and Future Directions

The superadditivity of one-shot quantum capacity is not a pathological rarity but pervasively shapes the landscape of quantum communication and error correction. It requires rigorous regularization procedures in channel capacity formulas and underlies exotic operational phenomena in both theoretical and experimental settings (Wu et al., 30 May 2025). The prevalence of superadditivity, even in simple channels, compels revisiting assumptions about channel additivity, code design, and protocol optimality.

Current and future avenues include:

• Extension to Other Capacities: Whether analogous superadditivity phenomena exist for private or classical capacities in broader classes of channels remains an active field of inquiry.
• Efficient Characterization: Developing computable upper and lower bounds (e.g., via semidefinite programming or hypothesis testing analogues) that accommodate superadditivity is vital for practical capacity estimation (Wang et al., 2016, 1007.5456).
• Multipartite and Network Scenarios: Design and analysis of distributed coding protocols exploiting multipartite entanglement and correlated noise structures are required to operationally realize the capacity enhancements predicted by superadditivity (1110.2594, 1107.0546).
• Experimental Verification: Physical realization and confirmation of superadditive capacity effects, as in the dephrasure channel model, remain a benchmark for the applicability of quantum Shannon theory and the resource requirements for advanced quantum communications platforms (Yu et al., 2020).

Through rigorous mathematical, operational, and experimental development, the phenomenon of superadditivity continues to deepen understanding of quantum information transmission and challenges canonical approaches inherited from the classical regime.