Channel-Fidelity Partition Function

• Channel-Fidelity-Based Partition Functions are mathematical models that use partition sums to quantify the reliability of communication channels in classical and quantum systems.
• They integrate graphical models, Monte Carlo algorithms, and game-theoretic methods to evaluate and optimize performance under fidelity-driven constraints.
• These techniques support advanced protocol design in multi-user access, resource allocation, and quantum channel benchmarking through efficient partitioning and sampling methods.

A Channel-Fidelity-Based Partition Function is a rigorous framework and computational tool for characterizing, analyzing, and optimizing channel fidelity in both classical and quantum communication systems by leveraging the mathematical structure of partition functions. Channel fidelity, broadly defined as the ability of a channel to preserve or accurately convey information, can be captured and operationalized via partition functions that count or sum over channel configurations, user groupings, or system states under fidelity-driven constraints. This approach interfaces with graphical models, Monte Carlo algorithms, game-theoretic surplus allocation, and quantum distance metrics to enable both performance evaluation and practical protocol design in contemporary communication networks.

1. Mathematical Foundations and Definitions

In the context of classical constrained channels, the partition function is central for capturing fidelity-related performance. For a function $f$ defined over all configurations $x$ (e.g., for a 2D grid of binary variables), the partition function is

$Z_f = \sum_{x} f(x)$

with $f(x)$ typically acting as an indicator: $f(x) = 1$ if the configuration $x$ is permitted (e.g., satisfies a no-adjacent-1s constraint) or, in the presence of noise, contains likelihood terms reflecting channel law $p(y|x)$. For noiseless channels, the per-symbol capacity is directly

$C_m = \frac{1}{N} \log_2 Z_f$

with $N$ the grid size. In noisy channels, the mutual information rate depends on partition functions that encode averaged output likelihoods: $I(X;Y)/N = \frac{1}{N} \big[ H(Y) - H(Y|X) \big]$ with $H(Y)$ computable via partition functions over joint $p(x,y)$ factorizations (Molkaraie et al., 2011).

In quantum channels, fidelity is extended from quantum states to the channels themselves. The channel fidelity function $F_{QC}(K_1, K_2)$ is defined via minimization over unitary extensions: $$F_{QC}(K_1, K_2) = \cos \left[ \Theta_{QC}(K_1, K_2) \right]$$ where

$$\Theta_{QC}(K_1, K_2) = \arccos \left\{ \max_{\|W\| \leq 1} \frac{1}{2} \lambda_{\min}(K_W + K_W^\dagger) \right\}$$

and $K_W = \sum_{ij} w_{ij} F_{1i}^\dagger F_{2j}$, with $F_{1i}$ and $F_{2j}$ Kraus operators of the two channels (Yuan et al., 2015). Partition functions over channel operations (e.g., traces of exponentials of Hamiltonians encoding noise and decoherence) are central to quantifying channel fidelity in complex systems.

2. Stochastic and Graphical Algorithms for Channel-Fidelity-Based Partition Functions

Efficient estimation of partition functions that embody channel fidelity is challenging due to the exponentially large configuration spaces in realistic systems. Two main methodological frameworks emerge:

• Tree-Based Gibbs Sampling: For 2D channels with complex cyclic dependencies, partitioning variables into disjoint sets enables cycle-free (tree) subgraphs. Alternating Gibbs sampling on the two sets accelerates mixing. Marginals over these subsets allow for effective estimation of $Z_f$ through summing marginalized functions, offering an alternative and robust estimator (Molkaraie et al., 2011).
• Multilayer (Multitemperature) Importance Sampling: To address poor convergence when $f$ is sharply peaked (e.g., high-fidelity or strongly constrained regimes), auxiliary functions of the form $g(x) = f(x)^\alpha$ $(0 < \alpha < 1)$ are introduced at each "layer" to interpolate between an easy-to-sample distribution and the target. The partition function is obtained by telescoping ratios across layers: $$Z_f \approx Z_{g_J} \prod_{j=1}^J \hat{R}_j, \quad \hat{R}_j = \frac{1}{K} \sum_{k} f(x^{(k)})^{\alpha_{j-1}-\alpha_j}$$ (Molkaraie et al., 2011). These methods deliver asymptotically unbiased, scalable estimates even for high-dimensional, multimodal partition functions tied to channel fidelity.

3. Partition Functions and Channel Fidelity in Quantum Information

Quantum channel fidelity incorporates both state preservation and the broader operational distance between quantum channels, enabling unified treatment of quantum metrology and discrimination problems:

• Bures Angle and Channel Distance: Using $F_{QC}(K_1, K_2)$, the Bures angle and Bures distance between quantum channels are defined, generalizing standard state measures: $B_{QC}(K_1, K_2) = \sqrt{2 - 2 F_{QC}(K_1, K_2)}$ (Yuan et al., 2015).
• Fisher Information and Precision Limits: The "distance" between adjacent parameterized channels $K_x, K_{x+dx}$ yields a quantum channel Fisher information

$$J_{QC}(K_x) = \lim_{dx \to 0} \frac{4 \Theta_{QC}^2(K_x, K_{x+dx})}{dx^2}$$

determining the ultimate metrological sensitivity (Yuan et al., 2015).

• Partition Functions in Channel Simulation and Verification: In physical and simulated open-system channels, partition function estimation based on the trace over exponentials of the relevant Hamiltonians parallels the estimation of channel fidelity in the presence of noise and decoherence. Quantum-algorithmic techniques, such as DQC1 protocols, are used to estimate such partition functions—including at complex temperatures—providing a route to quantifying the effective fidelity of implemented quantum channels (Jackson et al., 2022).

4. Game-Theoretic and Cooperative Networking Perspectives

Partition function-based approaches offer a powerful method for allocating channel resources—interpreted as "surpluses"—in cooperative multi-user communication settings:

• Partition Function as Surplus Distribution: Given a partition function $f$ defined on all possible coalition groupings (partitions) of user nodes, with atoms being unordered pairs (edges), Möbius inversion enables surplus allocation at an atomic (link) level (Rossi, 2018): $f(P) = \sum_{[ij] \leq P} \mu^f([ij])$ where $[ij]$ is an edge and $\mu^f$ its Möbius inversion.
• Novel TU-Sharing Rules: Two solution concepts, chain-uniform (CU) and size-uniform (SU), dictate how the surplus gets divided among edges, influencing overall channel fidelity. SU is distinguished by the fixed-point property, i.e., if $f$ is edge-atomic, SU returns $f$ exactly. These methods can be interpreted as distributing overall network fidelity among specific links, offering a structured, incentive-aligned allocation useful for multi-access and mesh network design (Rossi, 2018).
• Limitations of Supermodularity: In contrast to classical games, supermodularity is not sufficient for core nonemptiness in partition-function games; the surplus must also match the overall size (i.e., sum of atomic contributions), which has direct operational implications for robust, high-fidelity network design.

5. Channel-Fidelity-Based Partition in Multi-User Access Protocols

Practical partition functions based on channel fidelity also manifest concretely in multiple-access channel (MAC) protocols, especially in the context of collision resolution and resource allocation:

• Partitioning vs. Group Testing: In multi-access control, partitioning (assigning active users to distinct groups) requires less information than full identification (group testing). The partition information is formally

$W_N^I = \log \binom{N}{K} - K \log (N/K)$

compared with the group testing requirement of $\log \binom{N}{K}$ (Wu et al., 2014). This reduction results in higher achievable rates and fewer channel uses for partition-based access control, directly benefiting throughput and reliability.

• Random Coding and Hypergraph Coloring: Random coding matrices and MAP decoding can efficiently realize optimal partitioning over Boolean MACs. In these protocols, the partitioning problem can be reformulated as hypergraph strong coloring, where channel actions correspond to hyperedge deletions and valid K–colorings equate to successful user separations (Wu et al., 2014). Extended Fibonacci structures appear in randomized algorithms for small $K$, providing tight analytical handles on error probabilities.
• Asymptotic Analysis in Noisy MAC: For noisy Boolean multi-access channels, robust partitioning protocols exhibit error probabilities $P_e^{(N)} \to 0$ when the number of protocol steps scales as $T \ge (C_1 + \xi_1) \log N$, with $C_1$ lower than the group testing counterpart. These protocols leverage joint typicality and graph coloring, explicitly tying channel fidelity to network protocol performance (Wu et al., 2015).

6. Optimization and Learning of Channel-Fidelity-Based Partitions

Optimal quantizer or classifier design for maximizing channel fidelity (or minimizing end-to-end loss) is achieved by formulating and solving partitioning problems that incorporate both fidelity objectives and operational constraints:

• Constrained Optimization and Cost Functions: For source $X$ corrupted to $Y$ and further processed via quantizer/classifier $Q: Y \rightarrow Z$ and deterministic channel $A: Z \rightarrow T$, the objective is to minimize a fidelity-related cost

$$F(X;T) = \sum_h p_{T_h} f(p_{X_1|T_h}, \dotsc, p_{X_N|T_h})$$

subject to a concave constraint $G(p_Z) \leq C$ on the output probabilities (Nguyen et al., 2020).

• Hard Partition and Hyperplane Cuts: The optimal $Q^*$ is always a "hard partition" (deterministic assignment), and the induced cluster boundaries correspond to hyper-planes in probability space $p(X|Y)$, analogously to generalized $k$-means clustering. The optimal assignment for $Y_m$ is determined by cost-gradient-based distance minimization, with polynomial-time algorithms available by reducing the assignment problem to hyperplane search (Nguyen et al., 2020).
• Generalization and Applicability: This framework subsumes previous results—spanning mutual information maximization, Gini-index minimization, and rate- or power-constrained coding—thereby supporting flexible design of channel-fidelity-optimal partitions under rich structural and resource constraints.

7. Channel-Fidelity Partition Functions in Quantum Algorithms

In quantum computation, the estimation of channel-fidelity-based partition functions—often tied to traces of exponentials of Hamiltonians—can be operationalized using both quantum and classical methods:

• Quantum (DQC1) Algorithms: A one-clean-qubit circuit can estimate partition functions at complex inverse temperatures (i.e., $$Z = \mathrm{Tr}[e^{-(\beta_R + i\beta_I)\mathcal{H}}]$$), employing Trotter decompositions and probabilistic synthesis of non-unitary evolution (Jackson et al., 2022).
• Quantum-Inspired Classical Algorithms: For real inverse temperatures, Clifford circuit structure enables efficient classical simulation (e.g., via stabilizer techniques) to approximate partition functions with guarantees on multiplicative error. These tools can be adapted or interpreted as estimators of quantum channel fidelity, since overlaps and traces central to physical channel discrimination or benchmarking have partition function structure.
• Complexity of Hamiltonian Decomposition: Efficient partition function and thus channel fidelity estimation requires that the system Hamiltonian be expressible as a sum of Pauli operators—however, producing this decomposition from general descriptions is itself DQC1-hard. This highlights a fundamental computational barrier for fidelity analysis in general quantum channels (Jackson et al., 2022).

Conclusion

Channel-Fidelity-Based Partition Functions integrate concepts from information theory, graphical models, quantum information, cooperative networking, and algorithmic optimization to facilitate rigorous quantification and practical optimization of fidelity in communication systems. These partition functions serve as both analytical tools for capacity and rate evaluation and algorithmic primitives for efficient sampling, resource allocation, and fidelity benchmarking—from classical multi-access channels to advanced quantum networks. They unify partitioning, coloring, and clustering methods with fidelity constraints, supporting scalable protocols and informing the limits and opportunities in both classical and quantum communication theory.