Channel-Wise Entropy Dependencies

• Channel-wise entropy dependencies are measures quantifying uncertainty propagation across quantum channels with fixed input marginals.
• Recent advances establish precise chain rules and additive structures for both von Neumann and sandwiched Rényi entropies under marginal constraints.
• The framework underpins adaptive cryptographic protocols by ensuring robust entropy accumulation and enhancing security in complex quantum networks.

Channel-wise entropy dependencies characterize how uncertainty, quantified via entropic functionals, accumulates or interacts across compositions and sequences of quantum channels—especially when subject to constraints such as fixed marginal states at each stage. Recent advances enable precise chain rules and additive structure for both von Neumann and sandwiched Rényi conditional entropies of quantum channels under marginal constraints, generalizing earlier entropy accumulation theorems and allowing adaptivity in cryptographic protocols such as quantum key distribution. This framework, as developed in "Marginal-constrained entropy accumulation theorem" (Arqand et al., 4 Feb 2025), reveals how entropy bounds propagate through networks of channels and how security guarantees can be robustly established even under dynamically updated statistical assessments.

1. Channel Conditional Entropy under Marginal Constraints

Let $$\mathcal{M} \in \mathrm{CPTP}(A\widetilde{A} \to B C)$$ be a quantum channel and fix an input marginal $\psi_A \in S(A)$. The channel conditional entropy is defined for each Rényi index $\alpha \geq 1$ (sandwiched version, Def. 2.7) as: $$H_\alpha^\uparrow\!\bigl(\mathcal{M},B,[\psi_A]\bigr) = \inf_{\substack{\rho_{A\widetilde{A}\widetilde{R}} \in S(A\widetilde{A}\widetilde{R})\ \operatorname{Tr}_{\widetilde{A}\widetilde{R}} \rho = \psi_A }} \;H_\alpha^\uparrow(B|C\widetilde{R})_{\mathcal{M}(\rho)}$$ with $\widetilde{R} \cong A\widetilde{A}$, and the conditional entropy taken as the supremum over reference states for sandwiched Rényi divergences: $$H_\alpha^\uparrow(B|C\widetilde{R})_\omega = \sup_{\sigma_{C\widetilde{R}}} - D_\alpha\left(\omega_{BC\widetilde{R}} \Vert I_B \otimes \sigma_{C\widetilde{R}}\right)$$ The conventional von Neumann version arises for $\alpha = 1$: $$H(\mathcal{M},B,[\psi_A]) = \inf_{\rho:\operatorname{Tr}_{\widetilde{A}\widetilde{R}}\rho = \psi_A} H(B|C\widetilde{R})_{\mathcal{M}(\rho)}$$ This formulation specifies that entropy accumulation through a quantum channel can be controlled not just globally, but at the level of individual input marginals.

2. Additive Chain Rule for Channel Entropy

The key structural result is a chain rule for marginal-constrained channel entropies (Thm 3.9): Consider two channels,

$$\mathcal{E}_1 \in \mathrm{CPTP}(A_0 Y_0 \to X_1 Y_1),\; \mathcal{E}_2 \in \mathrm{CPTP}(A_1 Y_1 \to X_2 Y_2)$$

and input marginals $\psi_{A_0}$, $\phi_{A_1}$. For all $\alpha \in [1, \infty]$,

$$H_\alpha^\uparrow(\mathcal{E}_2 \circ \mathcal{E}_1, X_1 X_2, [\psi_{A_0}\otimes\phi_{A_1}]) \geq H_\alpha^\uparrow(\mathcal{E}_1, X_1, [\psi_{A_0}]) + H_\alpha^\uparrow(\mathcal{E}_2, X_2, [\phi_{A_1}])$$

For independent channels, this inequality becomes equality (Cor. 3.10): $$H_\alpha^\uparrow(\mathcal{E}_1\otimes\mathcal{E}_2, X_1 X_2, [\psi_{A_0}\otimes\phi_{A_1}]) = H_\alpha^\uparrow(\mathcal{E}_1, X_1, [\psi_{A_0}]) + H_\alpha^\uparrow(\mathcal{E}_2, X_2, [\phi_{A_1}])$$ Thus, entropy is additive across tensor products of channels under these constraints.

3. Proof Structure and Additivity Mechanisms

Additivity is established via several supporting lemmas:

• Weak additivity (Lemma 3.4) for iid channels: $$H_\alpha^\uparrow(\mathcal{E}^{\otimes m}, X^m, [\psi^{\otimes m}]) = m\,H_\alpha^\uparrow(\mathcal{E}, X, [\psi])$$.
• Dual minimization (Lemma 3.6): Each channel conditional entropy can be represented as a regularized (infimum) sandwiched divergence minimization problem over reference channels and isometries.
• Measured-divergence chain rule (Lemma 3.7, Cor. 3.8): Minimizing over POVMs yields superadditivity of the underlying divergences, and these are transferred to entropy bounds through the duality framework. The combination of these mechanisms guarantees precise additivity and supports generalization to long channel sequences.

4. Marginal Constraints and Generalized Entropy Accumulation

The framework generalizes the classical entropy accumulation theorem (EAT) to cases where the input marginal to each channel is fixed or constrained and introduces "f-weighted entropies'' (Def. 4.4): $$H_\alpha^{\uparrow,f}(QC|CQ')_\rho = \frac{\alpha}{1-\alpha}\log\!\sum_{c\in\mathcal{C}} \rho(c)\! \Big[ \sum_{c'} \rho(c'|c)^\alpha\,2^{(1-\alpha)(-f(c,c')+H_\alpha^\uparrow(Q|Q')_{\rho|c,c'})} \Big]^{1/\alpha}$$ For a sequence of $n$ channels $\mathcal{M}_j$ each with a fixed marginal on $A_{j-1}$ and associated public/private outputs, the overall entropy bound accumulates as: $$H_\alpha^{\uparrow,f_\mathrm{full}}(S_1^n,C_1^n|C_1^n,E_n) \geq \sum_{j=1}^n \min_{c_1^{j-1}} \inf_{\rho_{\mathrm{in}}:\mathrm{Tr}_{>A_{j-1}}\rho_{\mathrm{in}}=\sigma^{(j-1)}} H_\alpha^{\uparrow,f_{|c_1^{j-1}}}(S_j,C_j|C_j,E_j,\widetilde{E})$$ Normalization by suitable offsets enforces non-negativity. Conditioning on an accept event $\Omega$ with probability $p_\Omega$ produces an explicit asymptotic key-rate lower bound: $$H_\alpha^\uparrow(S_1^n|C_1^n,E_n)_{\rho|\Omega} \geq n\,h_\alpha^\uparrow - \frac{\alpha}{\alpha-1}\log(\frac{1}{p_\Omega})$$ where $h_\alpha^\uparrow$ is a function of the minimal single-round entropy under the given marginal constraints.

5. Channel-wise Entropy Accumulation in Adaptive Cryptographic Protocols

In the context of prepare-and-measure QKD, the input source can be mapped to an entangled state having fixed marginal, and the adversarial channel and measurement processes are treated as sequential quantum channels with intermediate announcements and outputs. The entropy accumulation theorem with marginal constraints delivers:

• Security bounds with no limitation on repetition rate (extending beyond Markov or IID assumptions).
• Capability for "fully adaptive" key rate estimation, where tradeoff functions can be selected or tuned during protocol execution based on statistical tests, observed error rates, or other protocol history.
• Strict generalization over previous frameworks, including the Quantum Probability Estimation and Generalized EAT, which were limited in adaptivity or constraint structure.

6. Illustrative Calculations and Special Cases

Several scenarios provide concrete demonstrations of the framework:

• For binary-symmetric error tests per round, the asymptotic key-rate bound is

$$H_\alpha^\uparrow(S_1^n|C_1^n,E_n) \gtrsim n[1-h(e)] - O(\sqrt{n})$$

where $h(e)$ is the binary entropy of the error rate.

• For dephasing channels, e.g., $\rho \mapsto p Z\rho Z + (1-p)\rho$, the sandwiched Rényi-2 entropy is additive: $H_2^\uparrow(\mathcal{M}) = \log2 - H_2(p)$ respecting exact tensor-product additivity.
• When empirical noise is low in early rounds, later rounds can adjust the tradeoff $f_{|c_1^{j-1}}$ upward, and the entropy accumulation theorem guarantees the bound remains valid.

7. Impact and Implications for Quantum Information Theory

The marginal-constrained entropy accumulation theorem (Arqand et al., 4 Feb 2025) establishes a rigorous additive structure for quantum channel conditional entropies under sophisticated marginal constraints, broadening the analytic and operational toolkit for quantum cryptography, especially in dynamically varying or adversarial scenarios. Crucially, this enables the analysis of entropy flow and uncertainty accumulation across complex channel networks, supports adaptive protocol design, and sets the groundwork for future generalizations involving higher-order Rényi quantities, multipartite channels, and finely constrained input distributions. The chain rule and underlying additivity are foundational for entropy-based analysis in information-theoretic security and quantum system engineering.