We study maximum purity and minimum von Neumann entropy of absolutely positive partial transpose (APPT) states, together with the relative volume of their spectral sets. A consequence of Hildebrand's criterion gives an explicit outer spectral polytope, whose vertices we classify. Optimizing purity over this polytope, together with the Song--Chen result for the 2⊗3 system, yields, for every mn≥6, an explicit upper bound on the purity of APPT states that is asymptotic to 4/(3mn). This improves the previous 2/(mn) upper bound for absolutely separable states. The new bound applies to every APPT state, a class containing all absolutely separable states, and is sharp for every 2⊗n system with n≥3. For every 3⊗n system with n≥3, however, the unique outer-polytope maximizer is not APPT, so the bound is strict and disproves the Dũng--Khôi qutrit--qudit conjecture. The polytope also gives an explicit entropy lower bound in arbitrary bipartite dimensions and, together with the Song--Chen extreme-point classification, the exact minimum entropy for every 2⊗n system. Finally, exact formulas for the relative volumes of an inner polytope and the outer spectral polytope give explicit two-sided bounds on the qubit--qudit relative spectral volume an whose ratio is less than 4 and tends to 3. Consequently, an=Θ(n(4/27)n), and the relative volume of the qubit--qudit APPT spectral set (equivalently, the absolutely separable spectral set) has exact exponential decay rate ln(27/4).
ABSTRACT This work introduces a discrete, non-continuous mathematical paradigm formalizing the universe as a countably discrete digital matrix spanning the absolute interval [0, ∞). We reject the classical assumption that the Planck spacetime scale forms the fundamental floor of reality. Instead, we propose that the Planck fabric is an emergent property woven by a nested, self-similar hierarchy of infinitely small, subatomic, string-like informational units vibrating from absolute zero upward. By mathematically modeling multi-scale hierarchical fractional entanglement, we demonstrate how these sub-Planckian string-like particles undergo localized sub-sampling, state collapsing, and integration transformations to stitch themselves into macroscopic classical realities. Furthermore, we redefine standard Von Neumann and Shannon entropy metrics into a singular, scale-dependent information delta. This redefinition proves that thermodynamic entropy, vacuum fluctuations, and background noise are not fundamental physical constants, but rather mathematical artifacts of an incomplete observer constrained by specific spatial, temporal, and spectral sensing windows.
Entropy of a quantum oscillator in the presence of a quantum environment (or heat bath) is studied here. The spectral density, which comprises the environmental influences, here corresponds to a quasi-monochromatic thermal harmonic noise. This further implies that the environment is a structured one compared to the usually employed unstructured strictly ohmic environment. The entropy is found to be vanishing as the temperature approaches its absolute zero value, which thereby guarantees the third law of thermodynamics. At very high temperatures the classical behavior is observed. Similar to the usually employed ohmic-Drude bath model, the coefficient of the leading temperature dependent term in the entropy (at low temperatures) is proportional to the damping strength but not on other parameters of the spectral density characterising the bath.