Let
S be the von Neumann entropy of a finite ensemble
E of pure quantum states. We show that
S may be naturally viewed as a function of a set of geometrical volumes in Hilbert space defined by the states and that
S is monotonically increasing in each of these variables. Since
S is the Schumacher compression limit of
E, this monotonicity property suggests a geometrical interpretation of the quantum redundancy involved in the compression process. It provides clarification of previous work in which it was shown that
S may be increased while increasing the overlap of each pair of states in the ensemble. As a by-product, our mathematical techniques also provide an interpretation of the subentropy of
E.
The core of the black hole information paradox lies in whether Hawking radiation leads to information loss. Based on connectivity engineering, we construct a minimal, gravity-free spin network. By calculating the mutual information and fidelity across different bipartition sizes, we reproduce the complete Page curve. Numerical results show that the peak of mutual information in the geometrically decoupled network (k=3) exhibits a non-trivial scaling behavior with the connectivity k, and reaches saturation at k=3 (with a peak approximately 1.49 times that of the normal network). This not only reveals the reshaping of entanglement capacity by geometric topology but also establishes a topological upper bound for the Page curve peak. Fidelity analysis further reveals that in the late stages of evaporation, the residual subsystem tends towards purification, with the target state fidelity significantly rising to approximately 0.52, which is weakly sensitive to k. This indicates that information is not completely thermalized but exists in a highly compressed global correlation form within the entanglement structure of the black hole residue and external radiation. This model provides a deterministic and strictly reproducible discrete physical benchmark for studying black hole information dynamics.
Central phenomena in quantum information processing– from the coding gainof quantum error-correcting codes, to the information mapping between boundaryand bulk in the AdS/CFT holographic duality, to the Page curve of informationrecovery during black hole evaporation– can be unified under the concept of “information compression cost”. This paper proposes that any quantum informationoperation must pay an irreversible entropy production jointly determined by Landauer’s principle and the Bekenstein bound, and that systems under these costconstraints seek to minimize a cost functional representing the difference betweendescription complexity and predictive (or protective) capability. The encodingprocess of stabilizer codes is modeled as the information compression required toprotect logical qubits against noise. The Ryu-Takayanagi formula in holographicduality indicates that the boundary theory must pay an information maintenancecost proportional to the area of the extremal surface in order to maintain its effective description of the bulk geometry. The Page curve of the black hole informationparadox is interpreted as a first-order kinetic phase transition on the cost landscapebetween two minima– one corresponding to an information-lost state and the otherto an information-recovered state. This framework integrates quantum error correction, the holographic principle, and the black hole information problem into aunified picture of information economics, and presents quantitative relations thatare amenable to further testing.