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 $\sqrt{x}$ on $[0,1]$ , and two explicit families, binomial and $L^1$ -weighted, yield hierarchies of separability inequalities. Finally, we compare the criteria against one another and on PPT entangled states.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 $\sqrt{x}$ on $[0,1]$, and two explicit families, binomial and $L^1$-weighted, yield hierarchies of separability inequalities. Finally, we compare the criteria against one another and on PPT entangled states.

quant-pharXiv:2609.29471Sep 25, 20260 citations

Semidefinite descriptions of separable matrix cones

Roland Hildebrand

Let K, K' be convex cones residing in finite-dimensional real vector spaces E, E'. An element in the tensor product E \otimes E' is K \otimes K'-separable if it can be represented as finite sum \sum_l x_l \otimes x'_l with x_l \in K and x_l' \in K' for all l. Let S(n), H(n), Q(n) be the spaces of n x n real symmetric, complex hermitian and quaternionic hermitian matrices, respectively. Let further S_+(n), H_+(n), Q_+(n) be the cones of positive semidefinite matrices in these spaces. If a matrix in H(mn) = H(m) \otimes H(n) is H_+(m) \otimes H_+(n)-separable, then it fulfills also the so-called PPT condition, i.e. it is positive semidefinite and has a positive semidefinite partial transpose. The same implication holds for matrices in the spaces S(m) \otimes S(n), H(m) \otimes S(n), and for m < 3 in the space Q(m) \otimes S(n). We provide a complete enumeration of all pairs (n,m) when the inverse implication is also true for each of the above spaces, i.e. the PPT condition is sufficient for separability. We also show that a matrix in Q(n) \otimes S(2) is Q_+(n) \otimes S_+(2)-separable if and only if it is positive semidefinite.Let K, K' be convex cones residing in finite-dimensional real vector spaces E, E'. An element in the tensor product E \otimes E' is K \otimes K'-separable if it can be represented as finite sum \sum_l x_l \otimes x'_l with x_l \in K and x_l' \in K' for all l. Let S(n), H(n), Q(n) be the spaces of n x n real symmetric, complex hermitian and quaternionic hermitian matrices, respectively. Let further S_+(n), H_+(n), Q_+(n) be the cones of positive semidefinite matrices in these spaces. If a matrix in H(mn) = H(m) \otimes H(n) is H_+(m) \otimes H_+(n)-separable, then it fulfills also the so-called PPT condition, i.e. it is positive semidefinite and has a positive semidefinite partial transpose. The same implication holds for matrices in the spaces S(m) \otimes S(n), H(m) \otimes S(n), and for m < 3 in the space Q(m) \otimes S(n). We provide a complete enumeration of all pairs (n,m) when the inverse implication is also true for each of the above spaces, i.e. the PPT condition is sufficient for separability. We also show that a matrix in Q(n) \otimes S(2) is Q_+(n) \otimes S_+(2)-separable if and only if it is positive semidefinite.

arXiv:math/0701571Jan 20, 20071 citation

Convex cones in mapping spaces between matrix algebras

Mark Girard, Seung-Hyeok Kye, Erling Størmer

arXiv:2002.09614Linear Algebra and its ApplicationsSep 9, 202010 citations

An exact semidefinite characterization of the stoquasticity cone of permutationally invariant Bell operators

Jan Li, Owidiusz Makuta, Evert van Nieuwenburg +1 more

Bell correlations have been detected in atomic ensembles ranging from hundreds to hundreds of thousands of particles, witnessed by the permutationally invariant (PI) Bell operators. For all the operators realized so far, the maximally violating state can be brought, in a suitable basis, into a nonnegative-amplitude form. In that basis, the Bell operator is stoquastic. The coefficients parametrising these operators that at the same time lead to stoquasticity form a convex cone, the so-called stoquasticity cone. Its characterization had remained open beyond two-body operators, as for many-body operators the number of inequalities grows with the system size $n$ . In this work we show that, in the thermodynamic limit, the stoquasticity cone admits an exact semidefinite characterization, valid at any operator order. The key observation is that, along each off-diagonal, the matrix elements are the values of a single univariate polynomial. Being univariate, this condition is equivalent to a linear matrix inequality, with no relaxation hierarchy. Therefore, deciding stoquasticity becomes a single semidefinite feasibility problem. As its lowest-degree instances, this description recovers the known three-hyperplane characterization for two-body operators and yields, for three-body operators, a curved boundary.Bell correlations have been detected in atomic ensembles ranging from hundreds to hundreds of thousands of particles, witnessed by the permutationally invariant (PI) Bell operators. For all the operators realized so far, the maximally violating state can be brought, in a suitable basis, into a nonnegative-amplitude form. In that basis, the Bell operator is stoquastic. The coefficients parametrising these operators that at the same time lead to stoquasticity form a convex cone, the so-called stoquasticity cone. Its characterization had remained open beyond two-body operators, as for many-body operators the number of inequalities grows with the system size $n$. In this work we show that, in the thermodynamic limit, the stoquasticity cone admits an exact semidefinite characterization, valid at any operator order. The key observation is that, along each off-diagonal, the matrix elements are the values of a single univariate polynomial. Being univariate, this condition is equivalent to a linear matrix inequality, with no relaxation hierarchy. Therefore, deciding stoquasticity becomes a single semidefinite feasibility problem. As its lowest-degree instances, this description recovers the known three-hyperplane characterization for two-body operators and yields, for three-body operators, a curved boundary.

quant-pharXiv:2610.03438Oct 5, 20260 citations

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

\sqrt{x}

[0,1]

, and two explicit families, binomial and

L^1

-weighted, yield hierarchies of separability inequalities. Finally, we compare the criteria against one another and on PPT entangled states.

n

. In this work we show that, in the thermodynamic limit, the stoquasticity cone admits an exact semidefinite characterization, valid at any operator order. The key observation is that, along each off-diagonal, the matrix elements are the values of a single univariate polynomial. Being univariate, this condition is equivalent to a linear matrix inequality, with no relaxation hierarchy. Therefore, deciding stoquasticity becomes a single semidefinite feasibility problem. As its lowest-degree instances, this description recovers the known three-hyperplane characterization for two-body operators and yields, for three-body operators, a curved boundary.