We study quadratic stability of entropy minimization under convex constraints with block-separable structure. For constraint sets of the formC = { ⊕ᵢ pᵢ ρᵢ : p ∈ Π, ρᵢ ∈ Cᵢ },we establish explicit stability estimates showing that, under a confining (fixed-support) hypothesis, the entropy gap controls the squared trace-norm distance to the set of entropy minimizers, with constants determined by the geometry of the constraint. The stability constant admits a natural decomposition into marginal and conditional components. The marginal contribution is governed by the curvature of Shannon entropy on the marginal polytope Π at extreme points, while the conditional contribution is governed by the curvature of von Neumann entropy at entropy-minimizing states within each block. We prove that the quadratic exponent is optimal by constructing explicit examples for which no linear stability bound holds uniformly. Exploiting the block-separable structure, the analysis reduces the global stability problem to independent marginal and conditional subproblems, yielding a geometric characterization of the stability constants in terms of constraint curvature. This stability phenomenon cannot be derived from Pinsker-type inequalities or standard entropy continuity bounds, since no reference state is fixed and the minimizer emerges intrinsically from the constraint geometry.
For block-diagonal quantum states with prescribed block weights and dimensions, we determine the minimum von Neumann entropy excess at each trace-norm distance from the blockwise-pure states. The one-block extremal is the classical reverse-Fano profile; the new problem is its constrained assembly when the block weights and dimensions cannot be redistributed. The resulting value function
Φq,n(D) is attained on the full admissible interval and is the lower envelope of finitely many rank-lattice edges. Every minimizing distance allocation has at most one block between consecutive integer ranks, and the active edge can switch nonmonotonically with
D. We also determine the lower convex envelope of
Φq,n explicitly: it is the polygonal curve through synchronized rank vertices and admits a one-parameter dual representation. The exact nonconvex envelope differs from this convex surrogate by at most
q⋄log(5/4), with a sharper stage-dependent bound, and the constant is optimal. For binomial charge-sector weights the uniform gap is
O(L−1/2). The framework yields a symmetry-sector purity certificate; in a three-qubit phase-damping example the fixed-weight certificate remains nontrivial where the universal classical entropy-error envelope is already vacuous.
We analyze the entropy behaviour of a discrete quantum channel defined by the composition of a unitary automorphism and a projective conditional expectation onto a finite-dimensional subalgebra. Let [Formula: see text], where [Formula: see text] is unitary and [Formula: see text] is the conditional expectation associated with an orthogonal projector decomposition of the identity. We note that the von Neumann entropy is monotone nondecreasing under repeated application of [Formula: see text]. Equality is characterized explicitly in terms of the invariant algebra of the channel. We further identify the fixed-point space as the intersection of the commutant of [Formula: see text] with the block-diagonal von Neumann algebra defined by the coarse-graining projectors. Finally, we describe the asymptotic structure under repeated application of the channel via spectral properties of completely positive trace-preserving maps. The results demonstrate that entropy growth in this setting arises solely from algebraic noncommutativity between unitary action and conditional expectation, without invoking thermodynamic limits, statistical ensembles, or environmental assumptions.
Strassen's asymptotic spectrum offers a framework for analyzing the complexity of tensors. It has found applications in diverse areas, from computer science to additive combinatorics and quantum information. A long-standing open problem, dating back to 1991, asks whether Strassen's support functionals are universal spectral points, that is, points in the asymptotic spectrum of tensors. In this paper, we answer this question in the affirmative for tensors over complex numbers by proving that the support functionals coincide with the quantum functionals - universal spectral points that are defined via entropy optimization on entanglement polytopes. We obtain this result as a special case of a general minimax formula for convex optimization on entanglement polytopes (and more general moment polytopes). Our formula can be interpreted as a very general "classical-quantum correspondence" analogous to the well-known relation between the Shannon entropy and the von Neumann entropy. Its proof is based on a recent Fenchel-type duality theorem on Hadamard manifolds due to Hirai. In addition to settling Strassen's question, our results yield a unified and simpler approach to several other tensor invariants, including the asymptotic slice rank, the symmetric quantum functional, the
G-stable rank, and the non-commutative rank.
We investigate the geometric structure of the set of spectra of bipartite absolute separable states (
ASEPm,n) and absolute positive partial transpose states (
APPTm,n), \emph{i.e.}, bipartite quantum states that remain separable or PPT, respectively, under all global unitary transformations. First, we establish general geometric properties of absolute convex sets of matrices, their spectra and extreme points. Regarding absolute separability, we present a permutation-symmetric reformulation of the absolute PPT criterion and use it to demonstrate that
APPTm,n is a spectrahedron for all
m≤n: in particular, all its faces are exposed. While
ASEP2,n is a spectrahedron, too, for
ASEPm,n we can in general only show that it is a semialgebraic set for all
m,n≥3. Furthermore, we provide a complete characterization of the faces and extreme points of
APPTm,n and demonstrate that the dimension of a face is determined by the rank of a certain matrix, with maximal proper faces having dimension
mn−m−1.
In the quantitative setting, we provide a rigorous lower bound on the maximal attainable purity of
APPTm,n via an inscribed polytope
Pm,n and conjecture that the maximal purity of
APPTm,n (along with its spectra) coincides with the polytope for arbitrary dimensions except when
m=n=2. Additionally, we also provide a rigorous upper bound on the minimal von Neumann entropy of
APPTm,n and demonstrate numerically that the minimum entropy eventually coincides with the polytope
Pm,n as the local system dimension
n increases. Finally, we show that the relative spectral volume of
APPTm,n decays exponentially in
n by a constant multiplicative factor of the relative volume of the inscribed polytope
Pm,n.
The von Neumann entropy plays a vital role in quantum information theory. The von Neumann entropy determines, e.g., the capacities of quantum channels. Also, entropies of composite quantum systems are important for future quantum networks, and their characterization is related to the quantum marginal problem. Furthermore, they play a role in quantum thermodynamics. In this thesis the set of quantum entropies of multipartite quantum systems is studied. The problem of its characterization is not new -- however, progress has been sparse, indicating that the problem might be hard and that new methods might be needed. Here, a variety of different and complementary approaches are taken. First, I look at global properties. It is known that the von Neumann entropy region -- just like its classical counterpart -- forms a convex cone. I describe the symmetries of this cone and highlight geometric similarities and differences to the classical entropy cone. In a different approach, I utilize the local geometric properties of extremal rays of a cone. I show that quantum states whose entropy lies on such an extremal ray of the quantum entropy cone have a very simple structure. As the set of all quantum states is very complicated, I look at a simple subset called stabilizer states. I improve on previously known results by showing that under a technical condition on the local dimension, entropies of stabilizer states respect an additional class of information inequalities that is valid for random variables from linear codes. In a last approach I find a representation-theoretic formulation of the classical marginal problem simplifying the comparison with its quantum mechanical counterpart. This novel correspondence yields a simplified formulation of the group characterization of classical entropies (IEEE Trans. Inf. Theory, 48(7):1992-1995, 2002) in purely combinatorial terms.