Shannon’s channel coding theorem establishes the upper bound of reliable communication rate as the channel capacity C, while the rate R of practical codingschemes is usually strictly lower than C, and the difference between them constitutes the rate gap. Within the framework of constraint network informationgeometry, this paper encodes a discrete memoryless channel and a linear blockcode as a channel constraint network and a coding constraint network respectively,and proposes to strictly define the information dimension by the volume growthrate of the rigid kernel under the Fisher information metric. The core conclusionshows that the Shannon gap C−R is asymptotically exactly equal to the logarithmnormalized limit of the ratio of Fisher volumes of the rigid kernels of the channelconstraint network and the coding constraint network, i.e., the difference of thevolume growth rates of the two networks. Under the local asymptotic normalitycondition, this gap satisfies L2 = 2(C −R)+O(n−1/2logn), where L is the squareof the Fisher geodesic distance. For the repetition code (3,1) over the binary symmetric channel, the analytical probability distributions of all eight output patternsare given, the finite-length deviation of the KL divergence from the asymptoticgap is computed, and its sources are analyzed. For the Hamming (7,4) code overthe binary erasure channel, by combinatorially enumerating all one hundred andtwenty-eight erasure patterns over finite fields, the average residual non-recoverabledimension and its deviation from the asymptotic prediction n(C −R) are preciselygiven, verifying the error bound O(1/√n). Finally, a roadmap is provided to extendthe volume compression framework to LDPC codes via the peeling algorithm andto Polar codes via the polarization transform, and the applicability boundaries ofthe current framework in the cases of asymmetric channels, continuous alphabets,and non-linear codes are honestly declared.
Shannon’s channel coding theorem establishes the upper bound of reliable communication rate as the channel capacity C, while the rate R of practical codingschemes is usually strictly lower than C, and the difference between them constitutes the rate gap. Within the framework of constraint network informationgeometry, this paper encodes a discrete memoryless channel and a linear blockcode as a channel constraint network and a coding constraint network respectively,and proposes to strictly define the information dimension by the volume growthrate of the rigid kernel under the Fisher information metric. The core conclusionshows that the Shannon gap C−R is asymptotically exactly equal to the logarithmnormalized limit of the ratio of Fisher volumes of the rigid kernels of the channelconstraint network and the coding constraint network, i.e., the difference of thevolume growth rates of the two networks. Under the local asymptotic normalitycondition, this gap satisfies L2 = 2(C −R)+O(n−1/2logn), where L is the squareof the Fisher geodesic distance. For the repetition code (3,1) over the binary symmetric channel, the analytical probability distributions of all eight output patternsare given, the finite-length deviation of the KL divergence from the asymptoticgap is computed, and its sources are analyzed. For the Hamming (7,4) code overthe binary erasure channel, by combinatorially enumerating all one hundred andtwenty-eight erasure patterns over finite fields, the average residual non-recoverabledimension and its deviation from the asymptotic prediction n(C −R) are preciselygiven, verifying the error bound O(1/√n). Finally, a roadmap is provided to extendthe volume compression framework to LDPC codes via the peeling algorithm andto Polar codes via the polarization transform, and the applicability boundaries ofthe current framework in the cases of asymmetric channels, continuous alphabets,and non-linear codes are honestly declared.
This paper encodes the subsystem entanglement structure of boundary conformal field theory into a weighted graph and defines the spectral entanglemententropy via the spectral theory of the Dirichlet-to-Neumann map. Within the constraint network framework, a continuum limit and a mechanism for dimensionalemergence are introduced to establish a rigorous transition from discrete spectrato continuous holographic geometry. It is proved that, in the rigid limit, the spectral entanglement entropy strictly decomposes into the sum of a geometric areaterm and a topological correction term: the area term arises from a strictly positive lower bound on the spectral gap of the Dirichlet-to-Neumann map and ispinned by the curvature of the extremal surface; the topological correction termoriginates from the nullity of the coarse-grained rigid kernel, whose value is proportional to the absolute value of the Euler characteristic. It is further proved thatthe stability of a quantum extremal surface is equivalent to the algebraic connectivity of the quotient graph being strictly positive, a condition characterized bythe Fiedler value. This yields three testable predictions, encompassing a universal difference in the topological correction, multi-surface transitions for non-convexregions, and the finite-temperature spectral-gap scaling, and reveals a deep isomorphism with holographic quantum error correction. This framework unifies thegeometric origin, quantum corrections, and the recoverability of error correction ofthe Ryu-Takayanagi formula within a single spectral flow structure.
The energy gaps appearing in different physical systems– the anyonic excitation gap of Kitaev’s toric code, the first eigenvalue of the Laplacian on a compactRiemannian manifold, and the mass gap of Yang–Mills theory– traditionally belong to three isolated domains: topological order, spectral geometry, and quantumfield theory. In this paper we construct a covariant functor, called the spectralfunctor, between the category of weighted constraint networks and the category ofself-adjoint operators, and prove that the three types of gaps are not independentphenomena but manifestations of a common spectral structure on different physicalsource objects. The codomain of the spectral functor is the SA-Hilb category whichcarries self-adjoint operator information, thereby avoiding the categorical defect ofattaching spectral information externally to Hilbert spaces. Rigorous verification ofthe functor axioms ensures covariance between coarse-graining and spectral preservation. For the toric code, the gap ∆ = 4J is strictly derived from the stabilizeralgebra; under coarse-graining the effective coupling constant is strictly given bythe compression of matrix elements of the Galerkin projection, and the dimensionless gap remains invariant. For the Laplacian, the finite element discretisationresolves the conflict between negative stiffness matrix entries and the positivityaxiom of weights by distinguishing geometric weights from coupling operators; theclosed-form solution of the first eigenvalue (4/h2)sin2(h/2) converges exactly tothe continuum spectral gap, and the deviation from asymptotic commutativity ofcoarse-graining is controlled by O(H2). For Yang–Mills theory, we draw an honestboundary between classical constraint networks and quantum operators, formulatethe mass gap problem as a uniform positivity condition for the spectral gap of lattice constraint networks, and establish a one-way logical closure, clearly indicatingthe mathematical obstacles for the reverse reconstruction. The Bures informationgeometric metric based on thermal states further unifies the spectral gap and theinformation-geometric distance through two-way inequalities, making the physicalenergy gap, the operator spectral gap, and the information-geometric rigidity radius equivalent. The commutativity verification of the three examples– strictlycommutative, asymptotically commutative, and conditionally commutative– fullyconfirms the spectral functor as a categorical foundation for the transfer of stabilityacross scales.
The asymptotic rates of information-theoretic protocols - including error exponents, data-compression rates, and channel capacities - are traditionally derived under the idealised assumption that the underlying resources are independent and identically distributed (i.i.d.). Somewhat surprisingly, even slight departures from the exact i.i.d. structure can drastically alter the asymptotic behaviour predicted by the i.i.d. theory. If the precise nature of the perturbation is known, for instance in the case of a pointwise defect, one can design a bespoke protocol that compensates for it, e.g. by discarding the corrupted subsystem. In realistic physical settings, however, exact i.i.d. behaviour cannot be guaranteed, and deviations from the ideal regime cannot generally be identified precisely. This raises a fundamental question: which notions of almost i.i.d. structure are sufficiently robust to preserve the asymptotic predictions of quantum Shannon theory? We investigate this question for three central information-theoretic tasks: asymmetric hypothesis testing, classical and quantum data compression, and classical communication through quantum channels. Rather than designing protocols tailored to specific defects, we seek robust protocols that remain asymptotically optimal and that are universal within a broad class of almost i.i.d. resources whose precise deviations from the ideal regime are unknown. To this end, we study three inequivalent notions of almost i.i.d. structure, and determine which of them preserve the asymptotic rates and error exponents predicted by the i.i.d. theory. Along the way, we introduce the notion of an almost i.i.d. process and a new distance measure between quantum channels - the club distance - designed to capture stability under local perturbations. These notions may be of independent interest.
This paper establishes a strict upper bound theorem for spectral gap compression of random local transfer operators on finite-dimensional non-associativenormed algebras. Let the associator norm of the underlying algebra be δ and thelattice size be N. For the left-right averaged transfer operator defined under threebody multiplication gates, when δ > 0 (for sedenions and higher non-associativealgebras), there exists a universal constant Cd > 0 depending only on the algebra dimension such that the intensity-normalized spectral gap ∆int strictly satisfies∆int ≤ Cd · δ2 in the thermodynamic limit N → ∞. This upper bound is independent of system size, indicating that non-associativity imposes a ceiling on thespectral gap determined by the intrinsic algebraic structure, but does not forceit to vanish. The core of the proof uses the variational principle in the correctdirection: construct a trial state localized on only three lattice sites, quantify theinterference deficit under the action of the transfer operator by the associator normto obtain a lower bound on the expectation value, thereby obtaining a lower boundon the second largest eigenvalue and an upper bound on the spectral gap. On thephysical level, the theorem is mapped to the three-body perturbative expansion ofthe Kitaev honeycomb model, yielding three quantitatively observable predictions:compression of the magnon gap, a crossover behavior in low-temperature thermaltransport, and a lengthened time scale for entanglement growth. An independenttest scheme based on superconducting qubit arrays is designed. All predictionscontain no free fitting parameters; constants are predetermined by the algebra dimension; falsification conditions are declared in advance in Chapter 6, ensuring thetheory’s falsifiability.
The Quantum Reverse Shannon Theorem has been a milestone in quantum information theory. It states that asymptotically reliable simulation of a quantum channel, assisted by unlimited shared entanglement, requires a rate of classical communication equal to the channel’s entanglement-assisted classical capacity. In this paper, we study the error exponent of quantum channel simulation, which characterizes the optimal speed of exponential convergence of the performance towards the perfect, as the blocklength increases. Based on channel purified distance, we derive lower and upper bounds for the error exponent. Then we show that the two bounds coincide when the classical communication rate is below a critical value, and hence, we have determined the exact formula of the error exponent in the low-rate case. This enables us to obtain an operational interpretation to the channel’s sandwiched Rényi information of order from 1 to 2, since our formula is expressed as a transform of this quantity. In the derivation, we have also obtained an achievability bound for quantum channel simulation in the finite-blocklength setting, which is of realistic significance.
We study channel simulation under common randomness assistance in the finite-blocklength regime and identify the smooth channel max-information as a linear program one-shot converse on the minimal simulation cost for fixed error tolerance. We show that this one-shot converse can be achieved exactly using no-signaling-assisted codes, and approximately achieved using common randomness-assisted codes. Our one-shot converse thus takes on an analogous role to the celebrated meta-converse in the complementary problem of channel coding, and we find tight relations between these two bounds. We asymptotically expand our bounds on the simulation cost for discrete memoryless channels, leading to the second-order as well as the moderate-deviation rate expansion, which can be expressed in terms of the channel capacity and channel dispersion known from noisy channel coding. Our bounds imply the well-known fact that the optimal asymptotic rate of one channel to simulate another under common randomness assistance is given by the ratio of their respective capacities. Additionally, our higher-order asymptotic expansion shows that this reversibility falls apart in the second order. Our techniques extend to discrete memoryless broadcast channels. In stark contrast to the elusive broadcast channel capacity problem, we show that the reverse problem of broadcast channel simulation under common randomness assistance allows for an efficiently computable single-letter characterization of the asymptotic rate region in terms of the broadcast channel’s multipartite mutual information.