We develop a self-contained finite-dimensional theory of relational-information functionals on hypergraphs whose hyperedges carry quantum registers. The central object is a weighted hierarchy of total correlations evaluated after local product compression. We prove an exact data-processing inequality under compression-compatible block-local product channels and obtain the corresponding monotonicity along local product GKSL semigroups. For the variational layer we separate the continuous base functional on the compact configuration space from its faithful-state dynamics: an extended-valued log-determinant regularization yields interior minimizers, and every smooth objective with compact sublevel sets generates a unique global Riemannian gradient flow whose omega-limit set is critical; isolated critical points give single-point convergence and nondegenerate minima are exponentially stable. We then study the maximum-entropy decomposition of total correlation. Every maximum-entropy level is locally continuous on faithful states with an explicit modulus, invariant under local unitaries, and nondecreasing under local unital channels, while the sum of the connected-information layers is nonincreasing under all local product channels. Individual layers, however, are not data-processing monotones: a known three-qubit counterexample creates irreducible three-party correlation by a local random-unitary channel. The paper therefore gives a rigorous finite-dimensional foundation and an explicit claim boundary; it does not assume global consistency of overlapping hyperedge states, a continuum limit, or a physical realization of the variational flow.
We develop a self-contained finite-dimensional theory of relational-information functionals on hypergraphs whose hyperedges carry quantum registers. The central object is a weighted hierarchy of total correlations evaluated after local product compression. We prove an exact data-processing inequality under compression-compatible block-local product channels and obtain the corresponding monotonicity along local product GKSL semigroups. For the variational layer we separate the continuous base functional on the compact configuration space from its faithful-state dynamics: an extended-valued log-determinant regularization yields interior minimizers, and every smooth objective with compact sublevel sets generates a unique global Riemannian gradient flow whose omega-limit set is critical; isolated critical points give single-point convergence and nondegenerate minima are exponentially stable. We then study the maximum-entropy decomposition of total correlation. Every maximum-entropy level is locally continuous on faithful states with an explicit modulus, invariant under local unitaries, and nondecreasing under local unital channels, while the sum of the connected-information layers is nonincreasing under all local product channels. Individual layers, however, are not data-processing monotones: a known three-qubit counterexample creates irreducible three-party correlation by a local random-unitary channel. The paper therefore gives a rigorous finite-dimensional foundation and an explicit claim boundary; it does not assume global consistency of overlapping hyperedge states, a continuum limit, or a physical realization of the variational flow.
We establish a deterministic discrete-to-continuum theorem for free-information fields whose values are finite collections of quantum-register density operators. The local state space is the compact convex configuration space introduced in the companion finite theory, embedded in a finite-dimensional Hilbert space with the Hilbert–Schmidt metric. A mesh field carries two contributions: a finite-range quadratic register-mismatch energy and a local free-information density consisting of an energetic term, the relational-entropy hierarchy, and an optional log-determinant faithfulness barrier. Under explicit assumptions on cell quadrature, finite interaction range, directional second moments, discrete Poincaré compactness, and divergence consistency, the discrete Dirichlet forms Mosco-converge to an anisotropic continuum form. We then prove equicoercivity, the full Gamma-liminf inequality, a recovery theorem that respects the density-matrix constraints and the barrier, convergence of minima and almost minimizers, and mesh universality: any two admissible mesh sequences with the same limiting second-moment tensor have the same continuum functional. Natural Neumann and imposed Dirichlet boundary conditions are separated at the form level. In the convex regime the convergence strengthens to Mosco convergence of the complete functionals and therefore to convergence of resolvents and variational semigroups. We also give sharp failure statements: without a convergent directional tensor there need not be a unique full-sequence limit; the local continuum theorem does not solve the overlapping quantum marginal problem; and static Gamma-convergence alone does not derive a physical open-system dynamics. No random-substrate, Lorentzian, gravitational, gauge, particle-physics, cosmological, or empirical claim is made.
Quantum information theory is built on entropic quantities; among them, the sandwiched Rényi relative entropy is a fundamental divergence with various applications, and its data processing inequality (DPI) under quantum channels is a cornerstone result. In this work, we present a Lean 4 library for quantum information, designed as a reusable formal infrastructure for theoretical analysis. As a central demonstration of the library, we formalize the DPI for the sandwiched Rényi relative entropy for positive semidefinite operators on finite-dimensional quantum systems. The library provides a basis-independent operator-theoretic framework for finite-dimensional quantum mechanics compatible with the standard mathematical library Mathlib, including reusable interfaces for finite-dimensional systems, states, channels, tensor products, partial traces, Choi operators, Kraus representations, and Stinespring representations. It also builds infrastructure for noncommutative trace inequalities, including operator monotonicity and convexity via the real continuous functional calculus, block-operator positivity, Hilbert-Schmidt operator spaces, Jensen's operator inequality, generalized perspectives, operator power means, and Lieb-Ando trace inequalities. On top of this framework, we formalize entropy-specific ingredients for the DPI: variational formulas for the sandwiched quasi-entropy via Young and reverse-Young inequalities, tensor-product compatibility of real powers, and Haar measures on unitary groups. Together, these components yield a Lean formalization of the DPI, give strong subadditivity as a corollary, and provide the last missing component needed to complete the Lean formalization of the generalized quantum Stein's lemma. More broadly, the development provides machine-checkable foundations for future formalized and AI-assisted research in quantum information theory.
Marginal information persistence is introduced as the condition that accumulated operational information under recursive composition approaches a finite, strictly positive asymptotic value only when the growth of available records exactly balances the loss of local distinguishability. In the canonical binary concatenated Information-Causality protocol, this criterion uniquely selects the Tsirelson bias η = 1/√2; for anisotropic coefficients, the corresponding critical set is ∑ᵢvᵢ² = 1. The exact Shannon recursion admits a positive series representation, while finite-depth corrections on the critical set are organized by the Rényi spectrum of the normalized squared coefficients, with collision entropy determining the leading correction rate. The central result concerns the physical pre-query register of the canonical singlet-based binary entanglement-assisted random access code (EARAC). For primitive bias v, the complete register satisfies the exact identity I(Aₖ:Rₖ) = G[(2v²)^(k/2)], whereas each target marginal satisfies I(Aⱼ:Rₖ) = G(vᵏ). At the critical value v = 1/√2, every individual target correlation vanishes exponentially with depth, while the mutual information between the complete database and the complete pre-query register remains exactly one bit at every depth. The difference between complete-register information and the sum of target marginals is resolved exactly as a sum of conditional mutual informations. The result therefore provides an exactly solvable example of marginal attenuation coexisting with depth-independent global information persistence in a concatenated quantum register. Two direct depth-two experiments on IBM ibm_marrakesh test this separation. A 128-circuit witness experiment yields the nonasymptotic certificate I(A₂:R₂) − ΣⱼI(Aⱼ:Yⱼ) ≥ 0.013245941 bit at 99.9% confidence, under explicitly stated repeated-shot, static-branch, and intended-observable assumptions. A second 864-circuit experiment performs full three-qubit tomography of the pre-query register. Physical maximum-likelihood reconstruction gives I(A₂:R₂) = 0.691632483 bit, ΣⱼI(Aⱼ:R₂) = 0.535829616 bit, and a reconstructed strong gap of 0.155802867 bit; an independent linear-inversion/PSD reconstruction agrees to within 4.06×10⁻⁴ bit. Bootstrap and matched-shot null simulations are reported only as estimator diagnostics; no strictly positive state-space confidence bound is claimed for the tomographic strong gap. The complete-register theorem motivates a quantum-information-processing perspective in which target-by-target readout can underestimate information retained by a composite quantum register. Connections to collective readout, information scrambling, quantum data hiding, and quantum error correction are discussed as related but distinct research directions. No computational speedup, fault-tolerance theorem, universal collective decoder, or characterization of the full quantum correlation set is claimed. This Zenodo record includes the manuscript together with the reproducibility package for the reported hardware analyses, including experimental data, reconstruction outputs, analysis scripts, and integrity checks.
Quantum channel relative entropy can be nonadditive, making its regularization essential for asymptotic analysis. We study when the regularized value is attained at finite blocklength and how efficiently it can be approximated. For the first question, we establish a sharp dichotomy: the regularized value is either attained already at the single-copy level, implying additivity at every blocklength, or is never attained at any finite blocklength. Notably, a simple two-copy additivity test distinguishes between these alternatives for qubit channels and for channel pairs in arbitrary dimensions admitting an interior optimizer. We further show that both the interiority condition and finite nonattainment are generic, holding for almost every channel pair. For the second question, we establish explicit convergence rates for approximating the regularized channel relative entropy. For fixed channel dimensions, O(ε−3/2) channel copies suffice to approximate the regularized value to additive accuracy ε. Alternatively, a semidefinite hierarchy achieves the same accuracy at level O(ε−2), yielding a certified polynomial-time approximation. Together, these results reveal the intriguing structure and generic behavior of regularized channel relative entropy, sharpening our understanding of both its analytical and computational aspects.
We construct a composite variational functional K on the space of quantum density operators, built from three information-theoretic components: quantum relative entropy (measuring deviation from global consistency), record-consistency variance penalties (enforcing definiteness of macroscopic records), and boundary-compression divergence (penalizing deviations in boundary encoding). We prove that this single functional recovers three distinct pillars of physics in three limiting regimes: Helmholtz free-energy minimization in thermodynamic equilibrium (exactly, via the Jaynes identity), environment-induced superselection in the quantum measurement limit (for commuting record operators, reproducing Zurek's einselection), and the Einstein field equations in the semiclassical gravitational limit (via the Jacobson thermodynamic route, with the Clausius relation derived as a fixed-point condition rather than assumed). The gradient flow of K with respect to the Bures-Riemannian metric on density-operator space is proven to be well-posed for finite-dimensional systems, with exponential convergence following from the lambda-convexity of quantum relative entropy. We establish a uniqueness characterization: K is the unique functional (up to monotone reparameterization) satisfying scale universality, boundary sensitivity, record consistency, CPTP monotonicity, and reduction to known physics. The Lagrange multipliers in the record-consistency term are not free parameters but are determined by physically measurable decoherence timescales, yielding a framework with no adjustable constants beyond those already present in quantum mechanics and general relativity. This work provides a unified variational language for thermodynamics, decoherence, and gravitation, grounded in established mathematics: quantum information geometry, gradient flows in metric spaces, and thermodynamic derivations of gravitational dynamics.
Quantum systems may contain underlying correlations which are inaccessible to computationally bounded observers. We capture this distinction through a framework that analyses bipartite states only using efficiently implementable quantum channels. This leads to a complexity-constrained max-divergence and a corresponding computational min-entropy. The latter quantity recovers the standard operational meaning of the conditional min-entropy: in the fully quantum case, it quantifies the largest overlap with a maximally entangled state attainable via efficient operations on the conditional subsystem. For classical-quantum states, it further reduces to the optimal guessing probability of a computationally bounded observer with access to side information. Lastly, in the absence of side information, the computational min-entropy simplifies to a computational notion of the operator norm. We then establish strong separations between the information-theoretic and complexity-constrained notions of min-entropy. For pure states, there exist highly entangled families of states with extremal min-entropy whose efficiently accessible entanglement in terms of computational min-entropy is exponentially suppressed. For mixed states, the separation is even sharper: the information-theoretic conditional min-entropy can be highly negative while the complexity-constrained quantity remains nearly maximal. Under the cryptographic Learning with Errors hardness assumption, we construct efficiently preparable states exhibiting both separations, even when their preparation circuits are fully known. Overall, our results demonstrate that computational constraints can fundamentally limit the quantum correlations that are observable in practice.