We study the extreme points of the convex set $\mathcal{C}(ρ_1,ρ_2)$ of bipartite quantum states with fixed marginals $ρ_1$ and $ρ_2$ . We construct extreme points in $(d,\,d+m)$ dimension, of rank $d+m$ , matching the highest possible value, for all $d\geq 3$ , $m > \frac{d^2-2d-2}{2}$ (when $d=2$ , $m\geq 1$ ). This proves the existence of extremal states with relatively large rank and also covers all the known examples. We further show that, in order to analyze the extreme points of $\mathcal{C}(ρ_1,ρ_2)$ , it is sufficient to study the special case $\mathcal{C}(\mathcal{D}_1,\mathcal{D}_2)$ , where the marginals are diagonal. Additionally, we observe that it is sufficient to consider $d_1\leq d_2$ . Thus, our results show that apart from possibly a few finite cases, for each $d_1$ , the maximal rank is achieved almost all times.We study the extreme points of the convex set $\mathcal{C}(ρ_1,ρ_2)$ of bipartite quantum states with fixed marginals $ρ_1$ and $ρ_2$. We construct extreme points in $(d,\,d+m)$ dimension, of rank $d+m$, matching the highest possible value, for all $d\geq 3$, $m > \frac{d^2-2d-2}{2}$ (when $d=2$, $m\geq 1$). This proves the existence of extremal states with relatively large rank and also covers all the known examples. We further show that, in order to analyze the extreme points of $\mathcal{C}(ρ_1,ρ_2)$, it is sufficient to study the special case $\mathcal{C}(\mathcal{D}_1,\mathcal{D}_2)$, where the marginals are diagonal. Additionally, we observe that it is sufficient to consider $d_1\leq d_2$. Thus, our results show that apart from possibly a few finite cases, for each $d_1$, the maximal rank is achieved almost all times.

quant-pharXiv:2605.26920May 27, 20260 citations

Quantum marginals, faces, and coatoms

Stephan Weis, João Gouveia

Many problems of quantum information theory rely on the set of quantum marginals. A precise knowledge of the faces of this convex set is necessary, for example, in the reconstruction of states from their marginals or in the evaluation of complexity measures of many-body systems. Yet, even the two-body marginals of just three qubits were only described in part. Here, we propose an experimental method to search for the coatoms in the lattice of exposed faces of the convex set of quantum marginals. The method is based on sampling from the extreme points of the dual spectrahedron. We provide an algebraic certificate of correctness, employing ground projectors of local Hamiltonians. Using this method, we present an explicit family of coatoms of rank five in the lattice of ground projectors of two-local three-qubit Hamiltonians (the rank is always six for bits). This family describes a family of coatoms in the lattice of exposed faces of the convex set of two-body marginals of three qubits. Besides introducing the experimental method, we show that the support sets of probability distributions that factor are the ground projectors of frustration-free Hamiltonians in the commutative setting. We also discuss nonexposed points of the set of marginals.Many problems of quantum information theory rely on the set of quantum marginals. A precise knowledge of the faces of this convex set is necessary, for example, in the reconstruction of states from their marginals or in the evaluation of complexity measures of many-body systems. Yet, even the two-body marginals of just three qubits were only described in part. Here, we propose an experimental method to search for the coatoms in the lattice of exposed faces of the convex set of quantum marginals. The method is based on sampling from the extreme points of the dual spectrahedron. We provide an algebraic certificate of correctness, employing ground projectors of local Hamiltonians. Using this method, we present an explicit family of coatoms of rank five in the lattice of ground projectors of two-local three-qubit Hamiltonians (the rank is always six for bits). This family describes a family of coatoms in the lattice of exposed faces of the convex set of two-body marginals of three qubits. Besides introducing the experimental method, we show that the support sets of probability distributions that factor are the ground projectors of frustration-free Hamiltonians in the commutative setting. We also discuss nonexposed points of the set of marginals.

arXiv:2103.08360Mar 15, 20210 citations

Entropic Inequalities and the Marginal Problem

T. A. Fritz, Rafael Chaves

Dec 20, 20113 citations

When do pieces determine the whole? Extreme marginals of a completely positive map

E. Haapasalo, Teiko Heinosaari, Juha-Pekka Pellonpää

We will consider completely positive maps defined on tensor products of von Neumann algebras and taking values in the algebra of bounded operators on a Hilbert space and particularly certain convex subsets of the set of such maps. We show that when one of the marginal maps of such a map is an extreme point, then the marginals uniquely determine the map. We will further prove that when both of the marginals are extreme, then the whole map is extreme. We show that this general result is the common source of several well-known results dealing with, e.g., jointly measurable observables. We also obtain new insight especially in the realm of quantum instruments and their marginal observables and channels.We will consider completely positive maps defined on tensor products of von Neumann algebras and taking values in the algebra of bounded operators on a Hilbert space and particularly certain convex subsets of the set of such maps. We show that when one of the marginal maps of such a map is an extreme point, then the marginals uniquely determine the map. We will further prove that when both of the marginals are extreme, then the whole map is extreme. We show that this general result is the common source of several well-known results dealing with, e.g., jointly measurable observables. We also obtain new insight especially in the realm of quantum instruments and their marginal observables and channels.

arXiv:1209.5933Reviews in Mathematical PhysicsFeb 7, 201422 citations

On the Shannon entropy and related functionals on convex sets

Armin Uhlmann

Reports on Mathematical PhysicsOct 1, 197041 citations

Extremal Marginal States of Maximal Rank in

(d, d+m)

We study the extreme points of the convex set

\mathcal{C}(ρ_1,ρ_2)

of bipartite quantum states with fixed marginals

ρ_1

and

ρ_2

. We construct extreme points in

(d,\,d+m)

dimension, of rank

d+m

, matching the highest possible value, for all

d\geq 3

m > \frac{d^2-2d-2}{2}

(when

d=2

m\geq 1

). This proves the existence of extremal states with relatively large rank and also covers all the known examples. We further show that, in order to analyze the extreme points of

\mathcal{C}(ρ_1,ρ_2)

, it is sufficient to study the special case

\mathcal{C}(\mathcal{D}_1,\mathcal{D}_2)

, where the marginals are diagonal. Additionally, we observe that it is sufficient to consider

d_1\leq d_2

. Thus, our results show that apart from possibly a few finite cases, for each

d_1

, the maximal rank is achieved almost all times.