We investigate entropy minimization problems for quantum states subject to convex block-separable constraints. Our principal result is a quantitative stability theorem: under a natural confining (fixed-support) hypothesis, if a state has entropy within ε of the minimum permitted by the constraint, then it must lie within O(ε^{1/2}) in trace norm of the set of entropy minimizers. We show that this rate is optimal and cannot be improved uniformly.
The analysis is entirely finite-dimensional and exploits the block-separable structure of the constraint set, which induces a natural decomposition of entropy into a marginal (classical) component and conditional (internal) components. Quadratic stability emerges from the curvature of Shannon entropy on the marginal polytope and of von Neumann entropy on the constrained block states, yielding explicit stability constants determined by the geometry of the constraint.
We further demonstrate that this stability phenomenon cannot be derived from Pinsker-type inequalities or standard entropy continuity bounds, since no reference state is fixed a priori and the entropy minimizer arises intrinsically from the constraint geometry. The framework is abstract and independent of any arithmetic input, and provides a general quadratic stability principle for entropy minimization under structured convex constraints.
We investigate entropy minimization problems for quantum states subject to convex block-separable constraints. Our principal result is a quantitative stability theorem: under a natural confining (fixed-support) hypothesis, if a state has entropy within ε of the minimum permitted by the constraint, then it must lie within O(ε^{1/2}) in trace norm of the set of entropy minimizers. We show that this rate is optimal and cannot be improved uniformly.
The analysis is entirely finite-dimensional and exploits the block-separable structure of the constraint set, which induces a natural decomposition of entropy into a marginal (classical) component and conditional (internal) components. Quadratic stability emerges from the curvature of Shannon entropy on the marginal polytope and of von Neumann entropy on the constrained block states, yielding explicit stability constants determined by the geometry of the constraint.
We further demonstrate that this stability phenomenon cannot be derived from Pinsker-type inequalities or standard entropy continuity bounds, since no reference state is fixed a priori and the entropy minimizer arises intrinsically from the constraint geometry. The framework is abstract and independent of any arithmetic input, and provides a general quadratic stability principle for entropy minimization under structured convex constraints.
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.
Let H=PH⊕QH and let Πρ=PρP⊕QρQ be the two-block pinching of a density matrix. Write A=PρP, B=PρQ, and C=QρQ. We study the block-coherence entropy D(ρ∥Πρ), the relative-entropy cost of removing cross-block coherence. First, we prove a midpoint Bogoliubov--Kubo--Mori coercivity estimate, giving the operator-kernel lower bound D(ρ∥Πρ)≥Tr[B∗ΩA,C−1(B)] for strictly positive diagonal blocks. The proof uses the block-sign involution U=P−Q to obtain a sign-symmetric midpoint lower bound for the BKM Hessian. Second, we determine the exact minimum in the associated constrained entropy-minimization problem: at fixed protected-sector floor, leakage mass, and quadratic coherence, the minimum can be achieved by a single active two-level block, independently of the ambient dimension. Third, under a uniform floor A≥a0P and 0<εQ≤a0/2, we obtain the logarithmic boundary law D(ρ∥Πρ)≥∥B∥F2log(a0/εQ), showing that the lower-bound coefficient multiplying the quadratic coherence diverges logarithmically as leakage mass tends to zero. As a finite-dimensional application, the estimates give an entropy certificate for coherent leakage from a protected subspace: PρQ measures protected--leakage coherence, and the constrained entropy minimum is attained by a state supported on a single effective two-level transition. The BKM estimate also yields a dissipation lower bound along an idealized uniform block-dephasing orbit. Finally, the midpoint mechanism extends to finite-dimensional trace-preserving Z2-graded automorphisms and bounded finite-trace semifinite corners.
We establish a sharp quantitative stability theorem for the von Neumann entropy under a natural class of convex constraints induced by orthogonal block decompositions of finite-dimensional Hilbert spaces. Given a block–convex set of block-diagonal quantum states, we prove that any state whose entropy is within ε of the minimal value must lie within O(ε)O(\sqrt{\varepsilon})O(ε) in trace norm of the manifold of entropy minimizers. The stability exponent is shown to be optimal, and purely quadratic stability is characterized precisely in the confining (fixed-support) regime. The proof relies on a canonical decomposition of entropy into classical (Shannon) and conditional components, combined with sharp relative entropy estimates and geometric properties of the marginal polytope. As an application, we obtain quantitative non-concentration bounds for spectral measures associated with finite spectral decompositions of self-adjoint operators. The framework is entirely finite-dimensional, independent of arithmetic or dynamical assumptions, and provides a general geometric stability principle for entropy minimization under structured convex constraints.
We study estimation of the von Neumann entropy of a d-dimensional quantum state from independent outcomes of a fixed rank-one measurement. Let κν∈[1,d+1] denote the normalized trace-zero frame spread; κν=1 for exact complex projective 2-designs. For any fixed rank-one measurement and n≤c2d2/κν, the minimax mean-squared error is at least c1log2{d2/(κνn)}. For bounded frame spread and every fixed 0<a<2, the minimax risk is of order log2d throughout n≲d2−a; hence fixed uniform accuracy requires Ω(d2) observations. Existing tomography bounds give bounded risk from O(d3log3d) observations for finite equal-weight exact designs and the Haar covariant measurement, while the intermediate minimax regime remains open. The results concern a fixed measurement repeated independently across copies; adaptive and collective protocols, and protocols that reuse common randomness across copies, are not covered.
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.
Randomized measurement protocols give experimental access to low-degree polynomial invariants of a quantum state rather than to the state itself, which raises the question of what the best entanglement criterion is that can be built from finitely many such invariants. In this work we answer this question through convex duality.
For each degree we introduce the cone of local unitary trace inequalities that are valid for separable bipartite states in all local dimensions, together with its one-matrix counterparts for Hermitian and positive semidefinite matrices. Each cone is the polar dual of a moment cone; we also determine the minimal combinatorial parametrization of the separable cone. The framework organizes the known moment criteria: the PPT criterion using the first three moments of the partial transposition is the image under an embedding of a Hankel determinant from the positive semidefinite cone to the bipartite cone, which we show to be an extremal ray, and we determine the first nontrivial separable cone completely, finding exactly four extremal rays.
The framework allows us to analyze the realignment criterion and its centered (or enhanced) variant. We prove that the even singular value moments of the (centered) realignment matrix are trace invariants, associated with explicit fixed-point-free involutions. Extracting the optimal trace-norm bound from m such moments thus becomes a dimension-free truncated moment problem, which we solve in closed form for m=2 and relax to a semidefinite program of size O(m) for arbitrary m. The dual solutions are polynomial minorants of x on [0,1], and two explicit families, binomial and L1-weighted, yield hierarchies of separability inequalities. Finally, we compare the criteria against one another and on PPT entangled states.